V A Pammi

M G Clerc

S Coulibaly

Sylvain Barbay

come

Extreme events prediction from nonlocal partial information in a spatiotemporally chaotic microcavity laser

Extreme events prediction from nonlocal partial information in a spatiotemporally chaotic microcavity laser V. A. Pammi, 1 M. G. Clerc, 2 S. Coulibaly, 3 and S. Barbay 1

1 Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120 Palaiseau, France 2 Departamento de Física and Millenium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile 3 Université de Lille, CNRS, UMR8523 -PhLAM -Physique des Lasers Atomes et Molécules, F-59000 Lille, France

The forecasting of high-dimensional, spatiotemporal nonlinear systems has made tremendous progress with the advent of model-free machine learning techniques. However, in real systems it is not always possible to have all the information needed; only partial information is available for learning and forecasting. This can be due to insufficient temporal or spatial samplings, to inaccessible variables or to noisy training data. Here, we show that it is nevertheless possible to forecast extreme events occurrence in incomplete experimental recordings from a spatiotemporally chaotic microcavity laser using reservoir computing. Selecting regions of maximum transfer entropy, we show that it is possible to get higher forecasting accuracy using nonlocal data vs local data thus allowing greater warning times, at least twice the time horizon predicted from the nonlinear local Lyapunov exponent.

The prediction of extreme events (EEs) occurrence, while having potentially a large impact in many fields of science and everyday life, remains a challenge especially in large and complex spatiotemporal systems [1][2][3][4][5][6][7][8][9]. EEs, which are rare and intense amplitude phenomena -as compared to the long-time average of an observable in a given system [START_REF] Nicolis | Foundations of complex systems: emergence, information and predicition[END_REF] -have been found in many types of systems [START_REF] Onorato | [END_REF], either natural or in laboratory experiments. In the latter case, optical systems have played a great role because of the analogy between oceanic rogue waves and optical pulses propagation in nonlinear optical fibers [12,13], allowing to generate and study these EEs with a large statistics and in a controlled environment. EEs have also been found in nonlinear optical dissipative systems displaying chaos [14][15][16][17][18] or spatiotemporal chaos [19][20][21]. Likewise, model-free prediction of low [22][23][24][25] and high [26][27][28] dimensional chaotic timeseries have been made possible thanks to the advent of machine learning techniques. However they usually require the precise knowledge of the whole spatiotemporal history of a dynamical field, which is often impossible in real situations where only a part of the dynamics is observable while some dynamical variables remain hidden and cannot be recorded. When a dynamical variable is observed and used to predict the outcome of another variable, the concept of cross-prediction has been introduced and tested [29,30]. The application of model-free techniques is more challenging when dealing with experimental and natural data [31], where the resolution of the measurements in time and space is limited. Recent results have been obtained in this area for the prediction of rogue solitons in supercontinuum generation in an optical fiber [32,33] and the space-time localization of of extreme wind speeds in the north Atlantic ocean [8]. In Ref. [33], the system is purely temporal and a spatiotemporal map is obtained by a pseudo-space reconstruction. In Ref. [8], the full spatiotemporal field is recorded and used for the forecast, thanks to the relatively slow time scale of the system's evolution.

In this work, we utilize a model-free reservoir computing approach for the prediction of EEs occurrence with experimental data from a spatiotemporal chaotic broad area laser [20], where only partial information of the past spatiotemporal field is known. The only accessible observable is the laser intensity (not the laser material dynamics), and the dynamics can only be known accurately and simultaneously at two given locations in space. This simulates the common situation in practice where the spatiotemporal field is only scarcely sampled in space. We identify the spatial locations of potential precursors using an information theoretic measure, namely transfer entropy [34]. At contrast with Ref. [33], the precursors cannot be identified reliably but are mostly hidden in the system's dynamical fluctuations and in the detection noise. A classification task is performed using reservoir computing to identify EEs in advance, using local and nonlocal information. We compare the prediction results and identify regimes where the nonlocal, cross prediction task yields better prediction accuracy than the local task.

We investigate a quasi-1D broad-area microcavity laser with integrated saturable absorber which has been shown, both experimentally and numerically. to display spatiotemporal chaos and and EEs [20,35]. The observed spatiotemporal chaos results from a chaotic alternation of amplitude and phase turbulence phenomena [36]. The microcavity laser pumped area is delimited by a clear aperture of 10×80 µm 2 and emits at λ c 980 nm. Transverse spatial coupling in the microresonator is obtained through light diffraction with a diffraction length w d 7.4µm [20]. The detailed optical setup is described in [20] and recalled in the Suppl. Mat. (SM) for completeness. This system has the advantage of having fast timescales, on the order of hundreds of picoseconds, thus facilitating the sampling of a large number of low probability events in a single experimental run. EEs are qualified using the standard definitions used in hydrodynamics where these phenomena are coined "rogue waves" [START_REF] Onorato | [END_REF]. The dynamics recorded at the center of the laser (see Fig. -1a) displays large amplitude fluctuations (Figs.-1b, 1d and 1e). These fluctuations of height H (defined as the maximum between the amplitudes measured at the left and right sides of the pulse) can be classified into two classes: extreme events (EE) or non-extreme events (NE). The classification criteria for EEs is H ≥ 2H s , where H s , the significant height, is simply the average of the height of the events in the highest tercile. For technical reasons, it is not possible to access the evolution of the whole section of the laser with the required detection bandwidth. Only a partial information is available, namely we detect the simultaneous evolution in two different points, one fixed located at the center of the laser I C (t) = I(x C , t), and one mobile across the transverse section I M (t) = I(x M , t). In Fig. -1b) the intensity of EEs simultaneously measured by the two photodetectors at the same location displays correlated time traces. The average time trace of EEs shows some oscillations around the peak value at time = 0, that quickly dampens away from it, evidencing a typical temporal pattern for EEs. In Fig. -1c) by comparison, NE are completely uncorrelated which results in a very flat average time trace. Away from the correlation width of an EE, at M 1 , an EE recorded in C is accompanied by no clear sign in the time trace at M 2 which displays a dynamics very similar to the one recorded for a NE in Fig. -1c). By contrast, the average signal recorded at M = M 2 (Fig. -1d) shows a small fluctuation for -1.5 ≤ t ≤ -0.5 ns which may point to the presence of a precursor. However, the precursor identification is rendered difficult since the signal fluctuations are large and on the same order of magnitude as the signal itself, as can be seen on the non-averaged timetrace. The identification of potential precursors can be made easier using the tool of transfer entropy described below.

The dynamical complexity of the dataset can be estimated from the Lyapunov spectra computed for the individual local recordings I M (t). From these, the largest Lyapunov exponent λ M can be extracted, giving access to a global, mean maximum prediction time τ p 1 λ M ln( ∆ δ0 ) [START_REF] Kantz | Nonlinear Time Series Analysis[END_REF][START_REF] Ding | [END_REF], with δ 0 the initial perturbation and ∆ the resolution of the measurement. The mean Kaplan-Yorke dimension D KY and fractal dimension D f of the attractor are respectively D KY 11 and D f 7.1 (Fig. -2a), which are consistent with a high-dimensional chaos. A more precise estimate of the prediction time horizon is given using the rate of growth of initial error rate Φ computed in Fig. -2, which can be extracted from the nonlinear local Lyapunov exponent [START_REF] Ding | [END_REF] (see SM). The prediction horizon time can be defined as the time at which log(Φ) reaches 90% of its saturation value and is of the order of 0.47 ns here. Using side results (see SM) we can also estimate δ 0 3 × 10 -4 and assuming ∆ 5 × 10 -3 , which corresponds to a SNR of 1, and extracting the mean maximum Lyapunov exponent from Fig.

-2 λ M = 7.6 ns -1 , we get a mean maximum prediction time τ p = 0.37 ns. This time is slightly smaller than the time obtained from log(Φ), as expected, but is still larger than the Lyapunov time usually considered as a time horizon indicator τ L = λ M -1 = 0.13 ns. Recent results showed that machine learning aided model-free predictions of high-dimensional chaotic systems was possible up to about 6 Lyapunov times [26,29,[39][40][41].

Early warning signs of EE, also called precursors, have been considered in many previous works (e.g. in [1,6,42]) including in low dimensional optical systems [43][44][45][46]. To identify potential regions of precurors, we consider transfer entropy [34], similarly as in [33], which measures the information transfer between two signals. It is more robust that e.g. a simple cross-correlation since it uses conditional probabilities instead of correlations.

We introduce the two-dimensional effective transfer entropy T eff M →C (x M , τ ) (see SM) which measures the information gained at point C (in bits) from the knowledge of a history of duration τ h in the past at M (see Fig. 1e), with τ parametrizing the time delay in the past. It is obtained by subtracting to the transfer entropy T M →C the transfer entropy for surrogate data in M, allowing comparison between transfer entropies computed using differ- ent τ h . T eff M →C (x M , τ ) is calculated and plotted in Fig.

-3a for a history of size τ h = 0.050 ns. It displays three regions of interest. large central lobe centered around x M x C which corresponds to causal information in the immediate spatiotemporal surrounding of the EE, and two disconnected regions almost symmetric about the temporal axis which we identify as the location of potential precursors (around P 1 and P 2 ). It is clearly seen that EEs extend over a finite length of 10µm width, as already noted in Ref. [20]. At lags around τ = -5∆t = -0.25 ns, there is a net transfer of information to the center of the laser at τ = 0. This corresponds to the immediate warning signal of the EE formation. More importantly, there are disconnected regions around P 1 and P 2 at delay times τ 0.9 ns where there is a net positive transfer of information, outside of the initial correlation length of the system. In the following, we are going to use this knowledge for a model-free prediction of the occurrence of EE given the past dynamical information.

A dataset is built after identifying events times t E of intensity maxima at C and recording the signal at M for a duration τ h corresponding to m samples, i.e. from [t E -τ w -τ h , t E -τ w ], τ w being the warning time (see Fig. 1e). Events at C are labelled as EE or NE. Since EEs are rare by definition, a balanced dataset is built by retaining all the N EEs and choosing an equal number of NEs at random. This allows us to use a standard metric for the loss function [7]. The dataset therefore consists of 2 × N time-traces associated to labels which identify their categories, 70% of which is used as training data and 30% as testing data. The prediction task is carried out using reservoir computing (RC). RC has been used for prediction on various low-and high-dimensional dynamical systems [24,26,[47][48][49]. It is particularly interesting as reservoirs are themselves dynamical systems making them ideal candidates to map other dynamical systems. While we have tested other machine learning algorithms (K-nearest neighbours, long-short term memory, logistic regression), none of them did show a significant superiority and RC happened to be the one with the most overall best performance [START_REF] Pammi | Photonic computing with coupled spiking micropillars and extreme event prediction in microcavity lasers[END_REF]. The reservoir generation and update follow standard procedures detailed in the SM. It comprises N = 50 nodes each with a hyperbolic tangent activation function and is initialized by a null state. Its parameters were optimized thanks to a hyperoptimization routine. At the end of the input sequence, the state of the reservoir nodes is stored forming an output vector of length N . Thus, an input time-series of m samples is converted into a vector of at most N values, which is a representation of the input data. A logistic classifier assigning a class EE or NE is then trained on all the training times sequences. The forecasting accuracy of an EE at C for a history τ h = 1.75 ns versus the warning time τ w is shown on Fig. -4a. It is displayed for different training data: using the local information at C, or the nonlocal data at M 1 or M 2 . Note that the accuracy does not depend significantly on the history length after a certain length is reached (see SM). In the first case, an accuracy close to 1 is obtained for small warning times, since this forecasting task is linear and simple. As τ w increases, the forecasting accuracy also decreases almost monotonically towards 0.5, i.e. to the absence of forecasting power. The same behavior occurs using nonlocal training data, though with important differences between M 1 and M 2 . At M 1 , the forecasting accuracy is always low since there is almost no information present at this location, as can be checked in Fig. -3. The forecasting accuracy using nonlocal data at M 2 , on the contrary, is close to 0.8 for small warning times and decreases steadily until about 1.2 ns where the accuracy drops considerably and is on par with the one computed using data at M 1 . However, most interestingly, there is a window of forecasting where it is possible to obtain slightly higher accuracy with the nonlocal data at M 2 rather than using the local data at C. This illustrates the importance of analyzing the transfer entropy pattern in Fig. -3a, which can allow to improve the prediction accuracy by evidencing the spatiotemporal location of potential precursors. The forecasting accuracy at M 2 drops at 1τ w 1 ns, which is more than twice the time horizon inferred previously and also about 7.5 times larger than the Lyapunov time. This corresponds also to the time at which the logistic regression alone gives comparable results with the RC approach. It also relates to the drop observed for the effective transfer entropy computed in Fig. -3b for different warning times. This means that no useful further information can be extracted from the input time series passed this timescale. For smaller warning times, the reservoir is able to improve slightly the forecasting accuracy with respect to a simpler logistic regression approach. By contrast when 1τ w 1 ns, very little information can be extracted for the prediction as testified by the low transfer entropy computed.

In Fig. -4b,c we analyze how our model-free approach classifies EEs depending on their actual heights. As shown on the testing dataset histogram of heights (Fig. -4b), despite the fact that the training sets have been balanced, large EEs are still far less frequent than smaller ones and will therefore participate less to the training. The probability P (Event = EE|H) of forecasting as an EE an event of actual height H at C, given the knowledge of a history of non-local data at M 2 characterized with τ w = 1 ns and τ h = 1.75 ns is shown on Fig. -4c). In the perfect case, the probability would evaluate to one above 2H s and zero below. It increases with H and generally reaches one for the largest EE heights values, while the complementary probability P (Event = NE|H) goes to zero (green and yellow histograms, respectively). This trend is true for both the results shown in Fig. -4b,c, obtained for two different training sets, and shows that while large EEs are less frequent in the training dataset, their prediction accuracy increases with their height, resulting in a usually very good prediction for the largest EEs. We note, however, that some statistical fluctuations can remain in the forecasting results as can be seen on the far right of Fig. -4c) where an isolated event has been misclassified in one realization of the train/test datasets partitions.

In conclusion, we have shown that a model-free approach based on reservoir computing can successfully classify with a reasonable accuracy the occurrence of EE in a dataset of an experimental system displaying high dimensional spatiotemporal chaos, from the partial knowledge of the history of the spatiotemporal field. Using the transfer entropy concept, we identify specific spatiotemporal regions with high information flow pointing to potential precursors, which in our specific case are hidden in the dynamical fluctuations or detection noise. We find that while the prediction using the local information gives generally the best accuracy, forecasting from the nonlocal precursor region can yield comparable to slightly higher accuracy in a window of large warning times. The forecasting ability extends to at least twice the time horizon computed from the nonlinear local Lyapunov exponent of the system and about 7.5 times the Lyapunov time, before dropping to a random prediction. We believe these results pave the way to extreme forecasting in other areas of science, with applications to many natural systems, such as in geoscience for the detection of earthquakes where the precurors are unknwon and the spatial detection is incomplete.

Figure 1

 1 Figure 1. a) Microlaser near-field image above lasing threshold. b-e) simultaneous recording of scaled intensities at C (fixed photodetector, upper) and M (mobile photodetector, lower) for specific events at C (placed at t = 0): an EE is present (b,d,e) or not present (c) at C. The photodetector is located at xM = 0 µm in b), xM 2 = +12.3 µm in d) and xM 1 = -13.8 µm in e). Red and black lines: average timetraces over the ensemble of EE or NE and associated standard deviations (orange and light blue shaded areas). Dark red: largest EE's timetrace, in b,d,e, (resp. single random NE in c) recorded at C with the simultaneous timetrace recorded at M (light blue). Red dashed line: EE threshold 2 × Hs (to be compared to the pulse height H from the trough to peak, not to the peak value). Yellow shaded area in e): information used for the prediction of an event at t = 0, with history time τ h and warning time τw.

Figure 2

 2 Figure 2. a) Largest Lyapunov exponent λM = max(λ) and Kaplan-Yorke dimension DKY of the time traces versus recording position. b) Initial error growth rate for the laser intensity versus time computed at C for both photodetector recordings. The extracted prediction time horizon (see text) is 0.47 ns.

Figure 3 .

 3 Figure 3. Effective transfer entropy T eff M →C as a function of xM and delay τ for τ h = 1∆t = 0.050 ns and b) at M2 for τ h = 0.050, 0.250, 0.500 ns. P1 and P2 locate the main precursor regions in the spatiotemporal diagram. T eff M →C is smoothed by a small Gaussian kernel (see SM for the original data).

Figure 4 .

 4 Figure 4. a) Mean EE forecasting accuracy at C with τ h = 1.75 ns versus the warning time, using either local (at C) or nonlocal (at M1 or M2) data as input. The mean and standard deviation are computed with 10 different realizations of the reservoir. Thin line: nonlocal forecasting at M2 using logistic regression alone. b) Histogram of the testing dataset heights distribution (blue, NE and red, EE events). c) classification probability as an EE (green) or a NE (complementary, orange) of an event of actual height H occurring τw = 1 ns in the future at C from the knowledge of a history of τ h = 1.75 ns duration of the non-local data at M2. Two different realizations of the testing dataset are illustrated (plain and empty histograms).

ACKNOWLEDGMENTS

MGC thanks for the financial support of ANID-Millenium Science Initiative Program-ICN17_012 (MIRO) and FONDECYT Project No. 1210353. This work was partially supported by the French Renatech network.


Robin Delbart

Aris Papasavvas

Colin Robert

Thuy Quynh

Truong Hoang

Francisca Martinez-Hergueta

An experimental and numerical study of the mechanical response of 3D printed PLA/CB polymers

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Nicolas Lanthier
email: nicolas.lanthier@saintluc.uclouvain.be

Valérie Lebrun

Olivier Molendi-Coste

Nico Van Rooijen

Isabelle A Leclercq

Carla Busquets-Cortés

Carlos López Roig

Liver Fetuin-A at Initiation of Insulin Resistance

Keywords: fetuin-A, hepatokine, immunofluorescence, nonalcoholic fatty liver disease, fatty liver, steatosis, macrophage, insulin resistance, prediabetes, Kupffer cell

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Metabolic dysfunction-associated fatty liver disease (MAFLD) occupies, today, a large place in the spectrum of chronic liver diseases, as it is now the first reason of visit at the hepatology clinic and one of the two main causes of liver transplantation [START_REF] Eslam | A Consensus-Driven Proposed Nomenclature for Metabolic Associated Fatty Liver Disease[END_REF][START_REF] Francque | The Belgian Association for Study of the Liver Guidance Document on the Management of Adult and Paediatric Non-Alcoholic Fatty Liver Disease[END_REF][START_REF] Méndez-Sánchez | Global Multi-Stakeholder Endorsement of the MAFLD Definition[END_REF][START_REF] Lanthier | Metabolic Dysfunction-Associated Fatty Liver Disease: A New Clearer Nomenclature with Positive Diagnostic Criteria[END_REF][START_REF] Gill | Metabolic Associated Fatty Liver Disease: Addressing a New Era in Liver Transplantation[END_REF]. In parallel, a similar increasing prevalence of insulin resistance (IR) which can lead to type 2 diabetes is also observed [START_REF] Petersen | Increased Prevalence of Insulin Resistance and Nonalcoholic Fatty Liver Disease in Asian-Indian Men[END_REF].

Those two entities (MAFLD and IR) are, in fact, closely related [START_REF] Kotronen | Fatty Liver: A Novel Component of the Metabolic Syndrome[END_REF][START_REF] Stefan | A Global View of the Interplay between Non-Alcoholic Fatty Liver Disease and Diabetes[END_REF]. Indeed, numerous data show a causal relationship between hepatic steatosis and IR both in animal models [START_REF] Samuel | Mechanism of Hepatic Insulin Resistance in Non-Alcoholic Fatty Liver Disease[END_REF][START_REF] Lanthier | Kupffer Cell Activation Is a Causal Factor for Hepatic Insulin Resistance[END_REF] and in patients with MAFLD [START_REF] Petersen | Increased Prevalence of Insulin Resistance and Nonalcoholic Fatty Liver Disease in Asian-Indian Men[END_REF][START_REF] Petersen | Reversal of Nonalcoholic Hepatic Steatosis, Hepatic Insulin Resistance, and Hyperglycemia by Moderate Weight Reduction in Patients with Type 2 Diabetes[END_REF][START_REF] Seppälä-Lindroos | Fat Accumulation in the Liver Is Associated with Defects in Insulin Suppression of Glucose Production and Serum Free Fatty Acids Independent of Obesity in Normal Men[END_REF][START_REF] Lanthier | The Role of the Liver in Insulin Resistance[END_REF][START_REF] Binet | Non-Invasive Screening, Staging and Management of Metabolic Dysfunction-Associated Fatty Liver Disease (MAFLD) in Type 2 Diabetes Mellitus Patients: What Do We Know so Far?[END_REF][START_REF] Busquets-Cortés | Fatty Liver Index and Progression to Type 2 Diabetes: A 5-Year Longitudinal Study in Spanish Workers with Pre-Diabetes[END_REF]. We and others have evidenced that a short-term high-fat feeding (3 days) in animals induced hepatic steatosis as well as rapid hepatic IR [START_REF] Samuel | Mechanism of Hepatic Insulin Resistance in Non-Alcoholic Fatty Liver Disease[END_REF][START_REF] Lanthier | Kupffer Cell Activation Is a Causal Factor for Hepatic Insulin Resistance[END_REF]. This is concurrent with activation of the innate immune system (liver macrophages) [START_REF] Lanthier | Kupffer Cell Activation Is a Causal Factor for Hepatic Insulin Resistance[END_REF][START_REF] Knudsen | Hepatoprotective Effects of Indole, a Gut Microbial Metabolite, in Leptin-Deficient Obese Mice[END_REF]. Interestingly, depletion of liver macrophages significantly ameliorated hepatic insulin sensitivity [START_REF] Lanthier | Kupffer Cell Activation Is a Causal Factor for Hepatic Insulin Resistance[END_REF][START_REF] Huang | Depletion of Liver Kupffer Cells Prevents the Development of Diet-Induced Hepatic Steatosis and Insulin Resistance[END_REF]. A deleterious activation of the innate immune system specifically in the liver is, thus, playing a pivotal role in hepatic IR in response to high-fat diet (HFD) [START_REF] Lanthier | Kupffer Cell Activation Is a Causal Factor for Hepatic Insulin Resistance[END_REF][START_REF] Huang | Depletion of Liver Kupffer Cells Prevents the Development of Diet-Induced Hepatic Steatosis and Insulin Resistance[END_REF]. Liver macrophage expansion is also the first difference seen in liver biopsies of patients with steatosis compared to control patients [START_REF] Gadd | The Portal Inflammatory Infiltrate and Ductular Reaction in Human Nonalcoholic Fatty Liver Disease[END_REF]. In healthy 2 of 10 individuals, a short-term high-fat overfeeding (5 days) also induces liver-specific insulin resistance (proven in clamp experiments) with a 26% increase of fasting hepatic glucose production, associated with increased transaminase levels and without any change in body composition or peripheral insulin resistance [START_REF] Brøns | Impact of Short-Term High-Fat Feeding on Glucose and Insulin Metabolism in Young Healthy Men[END_REF].

Whether the liver (and liver macrophages) contribute to the onset of whole body IR in distant organs such as skeletal muscles and white adipose tissue remains a matter of debate [START_REF] Armandi | Insulin Resistance across the Spectrum of Nonalcoholic Fatty Liver Disease[END_REF] even though there is strong evidence to support it [START_REF] Lanthier | The Role of the Liver in Insulin Resistance[END_REF][START_REF] Lanthier | Liver and Systemic Insulin Resistance[END_REF][START_REF] Cai | Local and Systemic Insulin Resistance Resulting from Hepatic Activation of IKK-β and NF-KB[END_REF]. The liver plays a major role in amino acid and protein metabolism and provides most of the circulating proteins present in the blood. Steatosis or liver injury could alter the liver proteic secretome with potential (pathological) consequences on distant organs [START_REF] Clarembeau | Cirrhosis and Insulin Resistance: Current Knowledge, Pathophysiological Mechanisms, Complications and Potential Treatments[END_REF][START_REF] Meex | Linking Nonalcoholic Fatty Liver Disease and Insulin Resistance[END_REF][START_REF] Watt | The Liver as an Endocrine Organ-Linking NAFLD and Insulin Resistance[END_REF]. Fetuin-A, a liver-secreted protein, is indeed described as implicated in adipose tissue IR [START_REF] Pal | Fetuin-A Acts as an Endogenous Ligand of TLR4 to Promote Lipid-Induced Insulin Resistance[END_REF]. This glycoprotein can be secreted by the steatotic liver following the influx of free fatty acids (FFA), circulated in the blood in high concentration, and act as an endogenous ligand of the toll-like receptor 4 (TLR4) present on the adipocytes [START_REF] Pal | Fetuin-A Acts as an Endogenous Ligand of TLR4 to Promote Lipid-Induced Insulin Resistance[END_REF]. The blocking or administration of fetuin-A, therefore, makes it possible to reduce or worsen, respectively, the IR of the animals [START_REF] Pal | Fetuin-A Acts as an Endogenous Ligand of TLR4 to Promote Lipid-Induced Insulin Resistance[END_REF][START_REF] Mathews | Fetuin-Null Mice Are Protected against Obesity and Insulin Resistance Associated with Aging[END_REF][START_REF] Mathews | Improved Insulin Sensitivity and Resistance to Weight Gain in Mice Null for the Ahsg Gene[END_REF]. By analogy with adipose tissue proteins called "adipokines" and described as playing a key role in systemic IR in the context of obesity-associated adiposity (such as tumor necrosis factor-alpha or plasminogen activator inhibitor-1), liver proteins such as fetuin-A are named "hepatokines" [START_REF] Lanthier | Adipose Tissues as Endocrine Target Organs[END_REF][START_REF] Kim | Hepatokines and Non-Alcoholic Fatty Liver Disease: Linking Liver Pathophysiology to Metabolism[END_REF]. We have recently detected increased circulating fetuin-A in foz -/-mice fed an HFD and in patients with MAFLD, associated with IR [START_REF] Etienne | Fetuin-A in Activated Liver Macrophages Is a Key Feature of Non-Alcoholic Steatohepatitis[END_REF]. We have shown, in nonalcoholic steatohepatitis (NASH, the severe form of the disease), the presence of fetuin-A within activated liver macrophages forming crown-like structures or lipogranulomas [START_REF] Etienne | Fetuin-A in Activated Liver Macrophages Is a Key Feature of Non-Alcoholic Steatohepatitis[END_REF]. However, we know that IR may appear earlier on an HFD and involve liver-resident macrophages (Kupffer cells) before lipogranuloma formation and the development of NASH [START_REF] Lanthier | Kupffer Cell Activation Is a Causal Factor for Hepatic Insulin Resistance[END_REF].

We, therefore, decided to characterize a short-term HFD model with IR that we previously described [START_REF] Lanthier | Kupffer Cell Activation Is a Causal Factor for Hepatic Insulin Resistance[END_REF] by evaluating fetuin-A levels in the blood, the liver, the muscle, and the white adipose tissue, three key insulin-sensitive organs involved in IR pathogenesis. We paid specific attention to the connections with liver macrophages, known factors in the onset of hepatic IR [START_REF] Lanthier | Kupffer Cell Activation Is a Causal Factor for Hepatic Insulin Resistance[END_REF][START_REF] Huang | Depletion of Liver Kupffer Cells Prevents the Development of Diet-Induced Hepatic Steatosis and Insulin Resistance[END_REF][START_REF] Lanthier | Targeting Kupffer Cells in Non-Alcoholic Fatty Liver Disease/Non-Alcoholic Steatohepatitis: Why and How?[END_REF].

Materials and Methods

Animals, Diets, and Treatment

After 1 week acclimatizing, 5 weeks old male C57BL/6J mice were fed ad libitum for 3 days the HFD in which 60% of calories are derived from fat (D12492 from Research Diets, New Brunswick, NJ, USA) or the normal diet (ND) (10% of calories from fat, Carfil Quality, Oud-Turnhout, Belgium). The day of culling, the liver, epididymal white adipose tissue, and right quadriceps were rapidly dissected. Portions of tissue were immersed in formalin 4%; the remaining tissue was snap frozen in liquid nitrogen and kept at -80 • C until analyses [START_REF] Leclercq | Intrahepatic Insulin Resistance in a Murine Model of Steatohepatitis: Effect of PPARγ Agonist Pioglitazone[END_REF]. Intravenous liposome-encapsulated clodronate was used as previously described to selectively deplete liver macrophages [START_REF] Lanthier | Kupffer Cell Activation Is a Causal Factor for Hepatic Insulin Resistance[END_REF][START_REF] Lanthier | Clodronate Liposomes: All Sites of Injection Are Not Equal[END_REF]. Analyses were performed on 8 animals per group (24 animals in total).

Hyperinsulinemic-Euglycemic Clamp Study

In another set of 18 animals, an intravenous catheter was implanted, and insulin sensitivity was assessed by the euglycemic-hyperinsulinemic clamp as described [START_REF] Lanthier | Kupffer Cell Activation Is a Causal Factor for Hepatic Insulin Resistance[END_REF][START_REF] Lebrun | Impact of PPAR-α Induction on Glucose Homoeostasis in Alcohol-Fed Mice[END_REF][START_REF] Lanthier | Kupffer Cell Depletion Prevents but Has No Therapeutic Effect on Metabolic and Inflammatory Changes Induced by a High-fat Diet[END_REF]. Briefly, 5 h fasted mice were infused with insulin at a rate of 2.5 mU•kg -1 •min -1 for 2.5 h. Glucose was infused at a variable flow rate to maintain euglycemia. For glucose turnover measurements, [3-3 H] glucose (Perkin Elmer, Boston, MA, USA) was infused at a rate of 0.33 µCi•min -1 . Analyses were performed on 6 animals per group.

Biochemical Analyses

Total liver lipids were extracted with methanol and chloroform and quantified by the vanillin-phosphoric acid reaction [START_REF] Leclercq | CYP2E1 and CYP4A as Microsomal Catalysts of Lipid Peroxides in Murine Nonalcoholic Steatohepatitis[END_REF]. Serum levels of fetuin-A were measured by a mouse fetuin-A enzyme linked immunosorbent assay (ELISA) kit (R&D systems) [START_REF] Etienne | Fetuin-A in Activated Liver Macrophages Is a Key Feature of Non-Alcoholic Steatohepatitis[END_REF].

Immunofluorescence

Detection of F4/80 and fetuin-A was performed on formalin-fixed, paraffin-embedded sections treated with proteinase K using a primary rat antimouse F4/80 monoclonal Ab (1/200, Serotec, Oxford, UK) and a primary goat antimouse/human fetuin-A (1/100, Santa Cruz, CA, USA). Secondary antibodies were donkey antirat/AlexaFluor 594 and donkey antigoat/AlexaFluor 488 (dilution 1/2000, Invitrogen, Merelbeke, Belgium). Hoechst (dilution 1/10,000) was used to reveal the nuclei.

Protein Studies-Western Blotting

Proteins from liver, white epidydimal adipose tissue, and muscle homogenates were analyzed by Western blotting using a goat fetuin-A antimouse/human antibody (1/2000, Santa Cruz, CA, USA). The immunoreactivity was detected with a horseradish peroxidaseconjugated secondary antigoat Ab (1/40,000) and enhanced chemiluminescence reagents (Western Lightning Chemiluminescence Reagent Plus, Perkin-Elmer, Boston, MA, USA). Ponceau-S red staining was used as a loading control. One membrane was sequentially probed with the fetuin-A antibody and then with the HSP-90, beta-actin, glyceraldehyde 3-phosphate dehydrogenase (GAPDH) or Akt antibodies to control for protein loading.

RNA Extraction, Reverse Transcription, and RT-qPCR

Total RNA was extracted from frozen liver, epididymal fat, and muscle samples using TRIpure Isolation Reagent (Roche Diagnostics Belgium, Vilvoorde). cDNA was synthesized. Quantitative real-time PCR analysis was carried out as previously described [START_REF] Leclercq | Intrahepatic Insulin Resistance in a Murine Model of Steatohepatitis: Effect of PPARγ Agonist Pioglitazone[END_REF]. Primer pairs for transcripts of interest F4/80, CD68, α2-HS glycoprotein (AHSG/fetuin-A), TNF-α, CD68, and RPL-19 chosen as an invariant standard were designed using the Primer Express design software (Applied Biosystems, Lennik, Belgium). Primer sequence for AHSG was 5 -TGGCCTGCAAGTTATTCCAAA-3 (forward) and 5 -GCTGTGGGTACGGGACCTACT-3 (reverse) [START_REF] Etienne | Fetuin-A in Activated Liver Macrophages Is a Key Feature of Non-Alcoholic Steatohepatitis[END_REF]. The sequences of the other primers (F4/80, TNF-α, CD68, and RPL-19) have already been described [START_REF] Lanthier | Kupffer Cell Activation Is a Causal Factor for Hepatic Insulin Resistance[END_REF]. All experimental tissues and standard curve samples were run in duplicate in a 96-well reaction plate (MicroAmp Optical, Applied Biosystems). Results are expressed as fold expression relative to expression in the control group using the ∆∆Ct method.

Statistical Analysis

All the data are presented as means ± SD. Statistical analysis was performed by using GraphPad Prism ® for Windows (v.6.01, La Jolla, CA, USA). All data were checked for normality using the Shapiro-Wilk normality test. Data were analyzed using a one-way ANOVA with Tukey's post hoc tests for multiple comparisons. Data determined to be non-normal were analyzed using a Kruskal-Wallis test with Dunn's multiple comparisons post-tests. We considered p ≤ 0.05 to be statistically significant.

Results

Upregulation of AHSG mRNA Expression in the Liver under Short-Term High-Fat Feeding

As previously described [START_REF] Lanthier | Kupffer Cell Activation Is a Causal Factor for Hepatic Insulin Resistance[END_REF], short-term HFD mice gained weight (Figure 1A) and developed liver steatosis (Figure 1B) and hepatic IR proven at the clamp study (Figure 1C,D). We measured by RT-qPCR the expression of fetuin-A, a potential hepatokine suspected to contribute to peripheral IR pathogenesis. The liver mRNA level of AHSG was significantly increased in mice fed a 3-day HFD compared to controls fed the ND (Figure 1E). However, the protein content of fetuin-A within the liver did not change (Figure 1I). In accordance with previous findings, we confirmed the activation of resident liver macrophages, evi-denced by a significant upregulation of liver F4/80, a cell surface glycoprotein of Kupffer cells (Figure 1F), and an upregulation of liver CD68, a transmembrane glycoprotein of macrophages associated with lysosomal compartment (Figure 1G).

(Figure 1E). However, the protein content of fetuin-A within the liver did not change (Figure 1I). In accordance with previous findings, we confirmed the activation of resident liver macrophages, evidenced by a significant upregulation of liver F4/80, a cell surface glycoprotein of Kupffer cells (Figure 1F), and an upregulation of liver CD68, a transmembrane glycoprotein of macrophages associated with lysosomal compartment (Figure 1G).

Deletion of Hepatic Macrophages Modulates AHSG Expression

Knowing the roles of both liver macrophages and fetuin-A on insulin sensitivity, we then wanted to explore the effect of liver macrophage depletion on fetuin-A protein expression. This was achieved by an intravenous injection of liposome encapsulated clodronate one day prior to starting the mice on the HFD. As previously shown, intravenous clodronate liposome injection selectively depleted liver macrophage [START_REF] Lanthier | Clodronate Liposomes: All Sites of Injection Are Not Equal[END_REF].

Deletion of Hepatic Macrophages Modulates AHSG Expression

Knowing the roles of both liver macrophages and fetuin-A on insulin sensitivity, we then wanted to explore the effect of liver macrophage depletion on fetuin-A protein expression. This was achieved by an intravenous injection of liposome encapsulated clodronate one day prior to starting the mice on the HFD. As previously shown, intravenous clodronate liposome injection selectively depleted liver macrophage [START_REF] Lanthier | Clodronate Liposomes: All Sites of Injection Are Not Equal[END_REF]. Indeed, clodronate treatment eliminated liver F4/80 and CD68 transcripts (Figure 1F,G). Consistent with the removal of macrophages, hepatic tumor necrosis factor-alpha (TNF-α) was significantly downregulated in clodronate-treated animals (Figure 1H). We then measured the mRNA level of AHSG in liver-macrophage-depleted animals compared with other groups. Liver macrophage depletion was associated with a significant downregulation of the liver AHSG mRNA level, to a level comparable to that of animals on a control diet (Figure 1E).

Fetuin-A Circulating Form and Distribution within the Liver

High AHSG transcript levels in the liver of short-term HFD fed mice coincided with a significantly higher amount of the circulating fetuin-A protein (Figure 1J). However, while macrophage depletion significantly downregulated AHSG mRNA expression within the liver, it did not lower serum levels (Figure 1J).

On liver histological sections, fetuin-A was mainly located in the centrilobular hepatocytes, showing the granular appearance of secretory vesicles (Figure 2A). The resident macrophages of the liver positive for the F4/80 glycoprotein of the surface (Kupffer cells) were mainly located in the sinusoids adjacent to hepatocytes from the intermediate zone (Figure 2B). Costaining of fetuin-A with the F4/80 antibody did not reveal any colocalization between fetuin-A and liver macrophages (Figure 2C). The same situation was evidenced after the short-term HFD (Figure 2F-H).

Indeed, clodronate treatment eliminated liver F4/80 and CD68 transcripts (Figure 1F,G). Consistent with the removal of macrophages, hepatic tumor necrosis factor-alpha (TNFα) was significantly downregulated in clodronate-treated animals (Figure 1H). We then measured the mRNA level of AHSG in liver-macrophage-depleted animals compared with other groups. Liver macrophage depletion was associated with a significant downregulation of the liver AHSG mRNA level, to a level comparable to that of animals on a control diet (Figure 1E).

Fetuin-A Circulating Form and Distribution within the Liver

High AHSG transcript levels in the liver of short-term HFD fed mice coincided with a significantly higher amount of the circulating fetuin-A protein (Figure 1J). However, while macrophage depletion significantly downregulated AHSG mRNA expression within the liver, it did not lower serum levels (Figure 1J).

On liver histological sections, fetuin-A was mainly located in the centrilobular hepatocytes, showing the granular appearance of secretory vesicles (Figure 2A). The resident macrophages of the liver positive for the F4/80 glycoprotein of the surface (Kupffer cells) were mainly located in the sinusoids adjacent to hepatocytes from the intermediate zone (Figure 2B). Costaining of fetuin-A with the F4/80 antibody did not reveal any colocalization between fetuin-A and liver macrophages (Figure 2C). The same situation was evidenced after the short-term HFD (Figure 2F-H).

Fetuin-A in Other Insulin-Sensitive Tissues

We then compared the fetuin-A levels between the liver and the other two main insulin-sensitive tissues: the white adipose tissue and the skeletal muscle. We loaded the same amount of total protein on a gel for Western blot analysis (100 μg). Ponceau-S red staining confirmed equal protein loading amongst samples and tissues (not shown). The amount of beta-actin or glyceraldehyde 3-phosphate dehydrogenase (GAPDH), B,C,E,G,H) and nuclei (blue) (C,E,F) in liver sections of a normal diet (ND)-fed animal (A-E) and a high-fat diet (HFD)-fed animal (F-H). The magnification (D,E) allows the visualization of the granular aspect of fetuin-A within the hepatocytes.

Fetuin-A in Other Insulin-Sensitive Tissues

We then compared the fetuin-A levels between the liver and the other two main insulin-sensitive tissues: the white adipose tissue and the skeletal muscle. We loaded the same amount of total protein on a gel for Western blot analysis (100 µg). Ponceau-S red staining confirmed equal protein loading amongst samples and tissues (not shown). The amount of beta-actin or glyceraldehyde 3-phosphate dehydrogenase (GAPDH), commonly used for intersample normalization, greatly varied according to the nature of the tissue (Figure 3A). By contrast, Akt showed a more stable expression across tissues (Figure 3A). Although Akt is part of the insulin signaling pathway, the total form is not affected by the experimental conditions (HFD or clodronate administration) [START_REF] Lanthier | Kupffer Cell Activation Is a Causal Factor for Hepatic Insulin Resistance[END_REF] and can, therefore, serve as a reference. The highest amount of fetuin-A was observed in the adipose tissue compared with the liver or the muscle (Figure 3A). However, as in the liver (Figure 1I), HFD was not associated with an increased amount of fetuin-A in the adipose tissue or in the muscle (Figure 3B).

commonly used for intersample normalization, greatly varied according to the nature of the tissue (Figure 3A). By contrast, Akt showed a more stable expression across tissues (Figure 3A). Although Akt is part of the insulin signaling pathway, the total form is not affected by the experimental conditions (HFD or clodronate administration) [START_REF] Lanthier | Kupffer Cell Activation Is a Causal Factor for Hepatic Insulin Resistance[END_REF] and can, therefore, serve as a reference. The highest amount of fetuin-A was observed in the adipose tissue compared with the liver or the muscle (Figure 3A). However, as in the liver (Figure 1I), HFD was not associated with an increased amount of fetuin-A in the adipose tissue or in the muscle (Figure 3B). Finally, we compared the mRNA levels of AHSG within those three tissues. The ribosomal protein L19 (RPL19) showed minimal variation in expression across the samples and tissues and was, thus, chosen as the housekeeper gene (Figure 3C). In sharp Finally, we compared the mRNA levels of AHSG within those three tissues. The ribosomal protein L19 (RPL19) showed minimal variation in expression across the samples and tissues and was, thus, chosen as the housekeeper gene (Figure 3C). In sharp contrast with the Western blot data, the mRNA expression of AHSG was 800 times higher within the liver compared with the adipose tissue (Figure 3D). The muscle mRNA expression of AHSG was also very low compared with the liver (Figure 3D).

Discussion

In this study, we characterized fetuin-A expression and distribution in the liver, the blood, the adipose tissue, and the muscle in a short-term HFD model of IR.

We showed that the circulating levels and the liver mRNA expression of AHSG (fetuin-A) are increased after three days of high-fat feeding in mice. This occurred in parallel with the development of liver steatosis, hepatic IR, and liver macrophage activation and underlines the central role of the steatotic liver in the onset of IR. This confirmed a concept previously inferred by the analysis of a large cohort [START_REF] Sun | Plasma Levels of Fetuin-A and Hepatic Enzymes and Risk of Type 2 Diabetes in Women in the U[END_REF]; namely, that fetuin-A is not only a marker of diabetes, IR [START_REF] Mori | Association of Serum Fetuin-A with Insulin Resistance in Type 2 Diabetic and Nondiabetic Subjects[END_REF][START_REF] Song | Serum Fetuin-A Associates with Type 2 Diabetes and Insulin Resistance in Chinese Adults[END_REF], or MAFLD [START_REF] Stefan | A2-Heremans-Schmid Glycoprotein/Fetuin-A Is Associated with Insulin Resistance and Fat Accumulation in the Liver in Humans[END_REF][START_REF] Filardi | Circulating Levels of Fetuin-A Are Associated with Moderate-Severe Hepatic Steatosis in Young Adults[END_REF] but also a trigger for diabetes development [START_REF] Sun | Plasma Levels of Fetuin-A and Hepatic Enzymes and Risk of Type 2 Diabetes in Women in the U[END_REF]. Indeed, in their large prospective study in which nondiabetic women were included, the authors found a positive association between baseline plasma fetuin-A levels and the occurrence of type 2 diabetes during follow-up [START_REF] Sun | Plasma Levels of Fetuin-A and Hepatic Enzymes and Risk of Type 2 Diabetes in Women in the U[END_REF]. A recent interventional study in humans pointed in the same direction [START_REF] Willis | Acute Hyperenergetic, High-Fat Feeding Increases Circulating FGF21, LECT2, and Fetuin-A in Healthy Men[END_REF], showing that a shortterm HFD, already known to induce liver steatosis [START_REF] Van Der Meer | Effects of Short-Term High-Fat, High-Energy Diet on Hepatic and Myocardial Triglyceride Content in Healthy Men[END_REF] and hepatic IR [START_REF] Brøns | Impact of Short-Term High-Fat Feeding on Glucose and Insulin Metabolism in Young Healthy Men[END_REF], also induced a low but significant increase in plasma fetuin-A level and is associated with reduced whole body insulin sensitivity [START_REF] Willis | Acute Hyperenergetic, High-Fat Feeding Increases Circulating FGF21, LECT2, and Fetuin-A in Healthy Men[END_REF]. In a large scale population-based study, fetuin-A levels correlated with the fatty liver index (FLI, a noninvasive test based on waist circumference, body mass index, level of triglycerides, and γ-glutamyl transpeptidase) indicating hepatic steatosis [START_REF] Huang | Serum Fetuin-A Associated With Fatty Liver Index, Early Indicator of Nonalcoholic Fatty Liver Disease[END_REF]. Importantly, FLI predicts the risk of type 2 diabetes development in people with prediabetes [START_REF] Busquets-Cortés | Fatty Liver Index and Progression to Type 2 Diabetes: A 5-Year Longitudinal Study in Spanish Workers with Pre-Diabetes[END_REF]. Our results on the rapid increase of both liver AHSG mRNA levels and fetuin-A serum levels reinforced our belief that the steatotic liver plays a central role in the pathogenesis of IR and supported the concept that this phenomenon could be mediated by fetuin-A.

Second, our results highlighted that fetuin-A, at least in part, is produced by the liver. Indeed, the high mRNA content of fetuin-A within the liver (compared with levels in other insulin-sensitive tissues) and the strong immunofluorescence signals in centrilobular hepatocytes are consistent with the liver being a major site for fetuin-A production. Such detection of fetuin-A by immunofluorescence in wild-type mice demonstrated fetuin-A production by hepatocytes and its accumulation in granular structures consistent with Golgi apparatus or secretory granules. The detection of fetuin-A in the liver has already been attempted in the past [START_REF] Kahraman | Fetuin-A MRNA Expression Is Elevated in NASH Compared with NAFL Patients[END_REF]. We recently demonstrated its histological pattern in foz -/- mice and in patients with MAFLD showing the same results in the case of isolated steatosis, before the development of NASH [START_REF] Etienne | Fetuin-A in Activated Liver Macrophages Is a Key Feature of Non-Alcoholic Steatohepatitis[END_REF]. Although we did not provide evidence that these are secretory vesicles, this picture of a granular formation resembles that of other secreted proteins such as albumin [START_REF] Yokota | Immunocytochemical Localization of Albumin in the Secretory Apparatus of Rat Liver Parenchymal Cells[END_REF].

Third, our results showed that the hepatic expression of AHSG is modulated by hepatic inflammation. A short exposure to HFD induced both liver inflammation and fetuin-A production. Moreover, blunting HFD-induced inflammation by depleting liver macrophages decreased AHSG mRNA levels to a level found in ND-fed animals. Together with the immunofluorescent data, it confirms that fetuin-A is produced by hepatocytes (not by macrophages) and that its production is mediated by the inflammatory milieu. Despite this, high fetuin-A serum concentration does not drop upon macrophage depletion. The reason for this remains unknown. This might be in link with a relatively long half-life of the protein in the circulation and our very short experimental design. One alternative explanation is that the clearance mechanisms of fetuin-A involve liver macrophages. Concerning the discrepancy between the high level of the protein in the blood and the stable level of the protein in the liver (in immunofluorescence and Western blot), we imagine that it is related to the fact that the protein is rapidly secreted by the liver. This explains why we observed an increase in hepatic AHSG mRNA on a high-fat diet, a stable level of liver fetuin-A protein, and an increase in the circulating form in the blood. This increase in circulating protein correlating with the increase in hepatic messenger RNA has been described for other secreted hepatic proteins such as selenoprotein-P [START_REF] Misu | A Liver-Derived Secretory Protein, Selenoprotein P, Causes Insulin Resistance[END_REF].

The high fetuin-A protein presence in the adipose tissue with a low AHSG mRNA level is consistent with a trapping of circulating fetuin-A in the adipose tissue, rather than with a local production. Cell culture experiments showing fetuin-A in the supernatant of cultured hepatocytes but not in cultured adipocytes support this hypothesis [START_REF] Etienne | Fetuin-A in Activated Liver Macrophages Is a Key Feature of Non-Alcoholic Steatohepatitis[END_REF][START_REF] Khadir | Fetuin-A Levels Are Increased in the Adipose Tissue of Diabetic Obese Humans but Not in Circulation[END_REF]. The study of humans with obesity offered similar data with a high fetuin-A protein level measured in the adipose tissue while its mRNA was undetectable [START_REF] Khadir | Fetuin-A Levels Are Increased in the Adipose Tissue of Diabetic Obese Humans but Not in Circulation[END_REF]. Collectively, our results supported the concept that fetuin-A potentially acts as a hepatokine stimulated by high-fat feeding, secreted into the bloodstream, and that it accumulates in peripheral sinks such as the adipose tissue [START_REF] Etienne | Fetuin-A in Activated Liver Macrophages Is a Key Feature of Non-Alcoholic Steatohepatitis[END_REF]. These data provided insight into the early mechanisms of development of IR following initiation of an HFD and confirm the central role of the steatotic liver in the pathogenesis of complications. Hepatic macrophages and the liver secretome should be considered both as markers of IR and as potential therapeutic targets in MAFLD [START_REF] Lanthier | Targeting Kupffer Cells in Non-Alcoholic Fatty Liver Disease/Non-Alcoholic Steatohepatitis: Why and How?[END_REF][START_REF] Stefan | A2-Heremans-Schmid Glycoprotein/Fetuin-A Is Associated with Insulin Resistance and Fat Accumulation in the Liver in Humans[END_REF][START_REF] Jenkins | Plasma Fetuin-A Concentrations in Young and Older High-and Low-Active Men[END_REF][START_REF] Lanthier | Targeting the Gut Microbiome to Treat Metabolic Dysfunction-Associated Fatty Liver Disease: Ready for Prime Time?[END_REF][START_REF] Lefere | Macrophages in Obesity and Non-Alcoholic Fatty Liver Disease: Crosstalk with Metabolism[END_REF][START_REF] Alharthi | Macrophages in Metabolic Associated Fatty Liver Disease[END_REF]. Tissue selective depletion of liver fetuin-A will support our findings and provide evidence for a causal role of liver fetuin-A production in insulin resistance and accumulation of this protein in adipose tissue.

In conclusion, the present study demonstrated that hepatic mRNA levels and circulating fetuin-A levels are elevated in response to a short-term high-fat feeding in association with steatosis and hepatic insulin resistance. Our findings also showed that this fetuin-A production is localized at the hepatocyte level and its expression is modulated by liver macrophages.

Figure 1 .

 1 Figure 1. Liver fetuin-A is increased upon short-term high-fat diet and is modulated by liver macrophages. Body weight (A), hepatic lipid content (B), glucose infusion rate (C), and hepatic glucose production (HGP) (D) changes in normal diet (ND) mice, 3-day high-fat diet (HFD) mice and 3-day HFD mice with liver macrophage depletion obtained by intravenous (i.v.) clodronate (CLO) injection. Liver mRNA expression of AHSG (E), F4/80 (F), CD68 (G), and TNF-α (H). Liver fetuin-A protein (I) and serum fetuin-A (J) protein expression. Data are expressed as means ± SD. n = 6-8 per group. * p < 0.05, ** p < 0.01, *** p < 0.001, **** p < 0.0001.

Figure 1 .

 1 Figure 1. Liver fetuin-A is increased upon short-term high-fat diet and is modulated by liver macrophages. Body weight (A), hepatic lipid content (B), glucose infusion rate (C), and hepatic glucose production (HGP) (D) changes in normal diet (ND) mice, 3-day high-fat diet (HFD) mice and 3-day HFD mice with liver macrophage depletion obtained by intravenous (i.v.) clodronate (CLO) injection. Liver mRNA expression of AHSG (E), F4/80 (F), CD68 (G), and TNF-α (H). Liver fetuin-A protein (I) and serum fetuin-A (J) protein expression. Data are expressed as means ± SD. n = 6-8 per group. * p < 0.05, ** p < 0.01, *** p < 0.001, **** p < 0.0001.

Figure 2 .

 2 Figure 2. Fetuin-A distribution in the liver. Representative immunofluorescence detection of fetuin-A (green) (A,C,D-F,H), F4/80 positive cells (red) (B,C,E,G,H) and nuclei (blue) (C,E,F) in liver sections of a normal diet (ND)-fed animal (A-E) and a high-fat diet (HFD)-fed animal (F-H). The magnification (D,E) allows the visualization of the granular aspect of fetuin-A within the hepatocytes.

Figure 2 .

 2 Figure 2. Fetuin-A distribution in the liver. Representative immunofluorescence detection of fetuin-A (green) (A,C,D-F,H), F4/80 positive cells (red) (B,C,E,G,H) and nuclei (blue) (C,E,F) in liver sections of a normal diet (ND)-fed animal (A-E) and a high-fat diet (HFD)-fed animal (F-H). The magnification (D,E) allows the visualization of the granular aspect of fetuin-A within the hepatocytes.

Figure 3 .

 3 Figure 3. Fetuin-A in insulin sensitive tissues. Comparison of liver, adipose tissue, and muscle protein content of fetuin-A, beta-actin, glyceraldehyde 3-phosphate dehydrogenase (GAPDH), and Akt evaluated by Western blot analysis in mice fed the ND (A,B) or the HFD for 3 days (B). Mean cycle threshold values of RPL19 in the liver, adipose tissue, and muscle (C) evaluated by qPCR analysis. Comparison of liver, adipose tissue, and muscle AHSG mRNA expression with RPL19 mRNA chosen as an invariant standard (D). Data are expressed as means ± SD. n = 3-6 per group. ns: nonsignificant, **** p < 0.0001.

Figure 3 .

 3 Figure 3. Fetuin-A in insulin sensitive tissues. Comparison of liver, adipose tissue, and muscle protein content of fetuin-A, beta-actin, glyceraldehyde 3-phosphate dehydrogenase (GAPDH), and Akt evaluated by Western blot analysis in mice fed the ND (A,B) or the HFD for 3 days (B). Mean cycle threshold values of RPL19 in the liver, adipose tissue, and muscle (C) evaluated by qPCR analysis. Comparison of liver, adipose tissue, and muscle AHSG mRNA expression with RPL19 mRNA chosen as an invariant standard (D). Data are expressed as means ± SD. n = 3-6 per group. ns: nonsignificant, **** p < 0.0001.

Acknowledgments:

The authors would like to thank Marie-Pierre Berghmans, Marcellin Mazziotta, and Simon Ravau for their technical support.

Data Availability Statement: All data analysed during this study are included in this published article. The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Informed Consent Statement: Not applicable.

Conflicts of Interest:

The authors declare no conflict of interest.


George I Stegeman
email: george@creol.ucf.du

Robert Stegeman

Clara Rivero

Kathleen Richardson

Thierry Cardinal

Michel Couzi

Glasses for Raman nonlinear optics

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The Raman effect has been used since its discovery in 1928, both to investigate the structure of molecules and to demonstrate phenomena in nonlinear optics [1]. When light interacts with a vibrational mode, optical phonons are excited and light is emitted down-shifted in frequency from the incident light by an amount equal to the vibrational frequency. In the area of nonlinear optics, various phenomena based on this effect have been demonstrated. For example, Bowden and colleagues pioneered the understanding of temporal soliton formation in stimulated Raman scattering [2]. Other effects include Raman gain and Raman lasers, the soliton self-frequency shift, and nonlinear Raman spectroscopies such as RIKES, coherent Raman scattering, etc. [START_REF] Agarwal | Nonlinear Fiber Optics[END_REF][START_REF] Levenson | Introduction to Nonlinear Laser Spectroscopy[END_REF].

Out of all of these different effects, Raman gain used for amplifying a communications signal appears to be the most promising application [START_REF] Ikeda | [END_REF]6]. A weak signal can be amplified by a strong pump beam whose frequency is up-shifted from the signal by the Raman response of a material. A key feature of this interaction is that the exponent of the gain depends on the product of the material's Raman gain coefficient and the intensity of the pump beam, so that there is no dependence on the relative phase between the signal and pump [6]. The ideal medium would have (1) a spectrally broad and uniform Raman response which is determined primarily by the Raman susceptibility coefficient ∂α/∂Q 0 | Q = 0 , where Q 0 is the vibrational amplitude and α is the molecular polarizability, and (2) a large value for this coefficient. Usefully broad spectral bandwidths are only obtained in disordered materials such as glasses.

The material currently used for Raman amplification in optical systems is based on silica glass, which suffers from both limited bandwidth and relatively small gain, i.e., small ∂α/∂Q 0 | Q = 0 [6,[START_REF] Stolen | [END_REF]. This has led to progressively more investigations into new glass materials [8][9][10][11][START_REF] Stolen | Issues in Raman Gain Measurements[END_REF][START_REF] Mori | [END_REF][14][15][16][17][18][19]. One of the requirements for local area networks is broad bandwidth, high-gain materials in which low loss is desirable but not a critical issue, the socalled Raman in a box [START_REF] Schafer | Forum-Raman Amplifiers[END_REF]. There is also the historic requirement of materials for "long-haul communications" where a combination of low loss, high gain, and large bandwidth is desirable [START_REF] Stolen | Issues in Raman Gain Measurements[END_REF]. To date, silica has been the material of choice, only because of its favorable tradeoff between gain and loss.

The search for better materials has been ongoing since the pioneering work of Lines and coworkers in the 1980s [8,9]. However, it is the insatiable appetite of the internet for more bandwidth that has recently accelerated the demand for better materials. A surprisingly large number of glass families have been proposed, and some have been probed experimentally [8][9][10][11][START_REF] Stolen | Issues in Raman Gain Measurements[END_REF][START_REF] Mori | [END_REF][14][15][16][17][18][19]. The two standard characterization methods have been spontaneous Raman scattering, in which thermally excited phonons are involved in the process, and direct measurement of Raman gain in fibers [21,22]. In principle, it is possible to obtain absolute values of ∂α/∂Q 0 | Q = 0 from Raman scattering experiments, but it is very difficult to fix and/or measure all of the factors involved. Instead, the Raman spectra are compared to the Raman spectra obtained from fused silica measured at the same wavelength. The values for fused silica are reasonably well known [START_REF] Stolen | [END_REF][START_REF] Stolen | Nonlinearity in Fiber Transmission[END_REF]. Hence, this process yields the Raman susceptibility of the new material being investigated at that wavelength. Well-known wavelength corrections are then applied to estimate the Raman gain in the communications bands. Obviously, the best approach has been to measure gain directly in fibers at communications wavelengths using various techniques [22]. However, this approach does not lend itself to rapid characterization of materials, since the issue of fabricating low-loss fibers must be solved first.

Recently, we developed an apparatus for making absolute Raman gain measurements for a pump at 1064 nm [START_REF] Stegeman | [END_REF]. Unlike the spontaneous Raman scattering technique, all of the beam parameters are easy to define, and in fact this apparatus reproduced the value for silica glass accurately as an independent check on its accuracy. Using this apparatus and a relatively standard spontaneous Raman scattering experiment equipped with different excitation (pump) wavelengths, we investigated a number of different glass families with multiple glass compositions in each family. Here, we describe our experiments and the results obtained.

BASIC CONCEPTS

The Raman effect can occur when the polarizability of a molecule is modulated by one of the vibrational phonon modes of a material. In the molecular frame of reference of a glass made up of "k" distinct units, (1) where (ω p -ω k, r )/ is the Raman molecular susceptibility, ω p is the frequency of the incident light, and "r" identifies the different electronic transitions within species "k" which are located at the frequencies ω k, r . Thus, the argument (ω p -ω k, r ) describes the frequency dependence of the different contributions to the polarizability, including the Raman susceptibility. The vibrational modes of the kth species are identified by "β" and have a frequency , so that

Due to the disordered nature of a glass, for each mode there is a distribution of vibrational frequencies ( -

) which peaks at with a total species number density N k , so that the number density in a frequency interval is given by

α ij k α ij k r , ω p ω k r , - (
)

r ∑ = + ∂α ij k r , ω p ω k r , - ( ) ∂Q β k --------------------------------------- Q β k 0 = ⎩ ⎭ ⎨ ⎬ ⎧ ⎫ q β k , β ∑ r ∑ ∂α ij k r , ∂Q β k Ω β k q β k 1 2 ---Q β k iΩ β k t [ ] exp c.c. + = f β k Ω β k Ω β0 k Ω β0 k dΩ β k (3)
with the normalization (4)

Note that here we assume that the spectral breadth of the distribution function is much larger than the natural linewidth due to the decay of the individual modes. The frequency dispersion in the refractive index reflects the behavior of the linear susceptibility averaged over all of the species present in the glass, their orientations (denoted by 〈 〉) and their electronic transitions. From Eq. ( 1), the refractive index of the material is given by [START_REF] Ikeda | [END_REF] (The absorption spectrum is given by the imaginary component.) Hence, the wavelength dispersion in the refractive index is a summation of the dispersion due to all of the electronic transitions in all the component species.

The frequency dispersion of the Raman susceptibility (ω p -ω k, r )/ associated with species (k) and a specific Raman line (β) in a multicomponent glass clearly does not mirror that of the refractive index except under very special circumstances. Such an equivalence could occur for single-component glasses or glasses with a dominant component, which exhibit a single vibrational mode coupled to the single dominant electronic transition responsible for the linear polarizability. In general, every vibrational mode in species k could couple to multiple electronic transitions in that species with the coupling strength depending on the direction of the transition electric dipole moment and the direction of the vibrational displacement. This is in contrast to the frequency dispersion in refractive index, which is an average over all of the electronic transitions in all of the species. However, in fused silica, Stolen has shown that the measured frequency dispersion in the dominant Raman line mirrors that of the refractive index for wavelengths longer that 500 nm [6,[START_REF] Stolen | Issues in Raman Gain Measurements[END_REF]. In that case, there is a single specie with a dominant electronic transition which couples to the vibration with a shift of 440 cm -1 (∆ν = 13.2 THz), and all the Raman spectra were measured far from the absorption edge, where dispersion and resonant enhancement in the Raman susceptibility is minimal.

The geometries used in probing the Raman response are shown schematically in Fig. 1. In both cases, the incident (pump) field of frequency ω p and wave vector k p is written as

N k f Ω β k Ω β0 k - ( )dΩ β k , f Ω β k Ω β0 k - ( ) Ω β k d ∞ - ∞ ∫ 1. = n 2 1 1 ε 0 ----N k ℜeal α ij k r , ω p ω k r , - ( ) r ∑ ⎩ ⎭ ⎨ ⎬ ⎧ ⎫ . k ∑ + = ∂α ij k r , ∂Q β k (6)
Similarly, the Raman signal (or scattered) field takes the form [START_REF] Stolen | [END_REF] where, in this case, ω s is the signal (scattered light) frequency and k s is the signal (or scattered) wave vector, which lies along the z axis for Raman gain and, for example, for 90° scattering lies along the x axis in spontaneous Raman. In the Raman gain case, the gain occurs in the overlap region between the copropagating beams, whereas, in the Raman scattering case, light is scattered into all directions, and the scattering volume is defined by the light-gathering optics. However, there are fundamental differences between the physics of the Raman gain and the spontaneous Raman scattering processes. In Raman gain, the optical phonons are coherently driven by the mixing of optical fields, whereas, in spontaneous Raman scattering, the phonons arise from noise and are hence uncorrelated. For Raman gain, ∝ E pi E si with ∝ [START_REF] Hopf | Nonlinear Optics, Wiley Series in Pure and Applied Optics[END_REF]. As a result, the phonons are driven at the frequency ω p -ω s = Ω ≅ Ω β (Stokes transition). On the other hand, in spontaneous scattering, the phonon modes are excited by thermal noise and are uncorrelated from molecule to molecule and between different vibrational modes in the same molecule [START_REF] Hopf | Nonlinear Optics, Wiley Series in Pure and Applied Optics[END_REF]. In this case, due to the statistical nature of the mode excitation, in the high-temperature limit (k

B T ӷ ), (8)
For the anti-Stokes case, a phonon is annihilated and the frequency is shifted by ω p -ω s = -Ω ≅ -. Here, is the effective mass for the vibration. Details can be found in [START_REF] Hopf | Nonlinear Optics, Wiley Series in Pure and Applied Optics[END_REF].

From this point on, the derivations follow the classical routes described in [START_REF] Hopf | Nonlinear Optics, Wiley Series in Pure and Applied Optics[END_REF]. In the limit of broad distri-

E pi 1 2 ---E pi iω p t k p z - ( ) exp c.c. + = E si 1 2 ---E si iω s t k s r ⋅ - ( ) exp c.c., + = q β k Q β k E pi E si * បΩ β k Q β k Q β k * 〈 〉 k B T m β k Ω β k [ ] 2 ---------------------. = Ω β k m β k butions in
, i.e., dominated by disorder rather than phonon lifetime, the spontaneous Raman spectrum is given by (9) where ∆Ω is the solid angle subtended by the detector and the optical parameters are values inside the material. The Raman gain coefficient for the intensity in the plane wave limit, i.e., I(ω s , z) = I(ω s , 0)exp[γ R (Ω)I p (ω p , z)z] is given by (10) where Ω = ω p -ω s . Note that the frequency spectra are essentially identical in the limits assumed, except for small differences in the tails. The distributions usually take the form of Voigt functions, which are characterized in the tails of the distribution by Gaussian-like behavior. Hence, the distributions do not "blow up" at low frequencies, despite the 1/Ω and 1/Ω 2 dependence evident in Eqs. ( 9) and ( 10) at small frequency shifts. Finally, we note that there is an explicit dependence of these spectra on the pump frequency via and ω s for spontaneous Raman scattering and Raman gain, and implicitly on (ω p -ω k, r )/ in both cases, and on n s and n p for Raman gain.

EXPERIMENTAL DETAILS

3.1. Samples The samples were prepared by conventional melting methods. In the case of the oxide compositions, the samples were melted in platinum crucibles at temperatures Raman active medium Raman active medium

Ω β k I ω s ( ) ∆ΩI ω p ( ) ---------------------- N k ω s 4 8c 4 4π ( ) 2 ε 0 2 --------------------------- β0 ∑ r ∑ k ∑ = × ∂α ij k r , ω p ω k r , - ( ) ∂Q β0 k --------------------------------------- Q β0 k 0 = 2 k B T m β k Ω 2 --------------f β0 k Ω Ω β0 k - ( ), γ R ω p Ω - ( ) N k ω s π 2m β0 k n s n p c 2 ε 0 2 --------------------------------- β0 ∑ r ∑ k ∑ = × ∂α ij k r , ω p ω k r , - ( ) ∂Q β0 k --------------------------------------- Q β0 k 0 = 2 1 2Ω -------f β0 k Ω Ω β0 k - ( ), ω s 4 ∂α ij k r ,
ω p ω p ω s ω s
of 900 and 1150°C for 30 min for the tellurite-and borophosphate-based glasses, respectively. In the case of the chalcogenide glass composition, the sample was melted in a rocking furnace in an evacuated, sealed silica tube to avoid any oxygen contamination. After the melt, the samples were annealed at 40°C below their glass transition temperature and optically polished. Figure 2 illustrates the absorption spectra of four different glasses. As one can clearly see from the figure, the absorption band gap of the four different compositions, representative of different glass systems, spans throughout the visible region of the spectrum. In the case of the chalcogenide glass composition, the absorption band edge is in the red, followed by tellurite-based glasses, whose absorption bands go from 400 to 450 nm (shown in the graph are the upper and lower limits of the absorption band-edge positions representative of all the different tellurite compositions tested to date), and finally the borophosphate-based glass composition, with its absorption band edge in the blue (UV) region.

Spontaneous Raman Scattering

The spontaneous Raman cross-section measurements were conducted using the micro-Raman setup shown in Fig. 3. Two lines from an Ar + laser (458 and 514 nm), the 752-nm line from a Kr + laser, and the 1064-nm line from a Nd : YAG laser were used as the excitation sources. The CCD detector used for visible excitation is from ANDOR, model FI UV. This CCD is cooled to -75°C by the Pelletier effect. It is made of 1024 pixels, each one 26 µm wide. The CCD used for near-infrared experiments is an InGaAs array from Jobin-Yvon. It is cooled to liquid-nitrogen temperature and is made of 512 pixels, each 50 µm wide. In all cases, the incoming polarized (V) laser beam was focused onto the front polished surface of the sample via a 100 X microscope objective, with a spatial resolution of about 2 µm. A polarizer was used to select the polarization direction (vertical V or horizontal H) of the scattered light. A backscattering geometry was used to collect the Raman signal, which is spectrally analyzed with a spectrometer and the appropriate CCD detector, with a typical resolution of about 6 cm -1 . The Rayleigh line was suppressed with a holographic notch filter. Furthermore, by measuring the Raman spectra of a test glass under the same experimental conditions as for fused silica at a laser wavelength for which the peak Raman gain for fused silica is known, the peak Raman gain of the test glass at that pump frequency for a Raman active mode can be deduced from (11) where R(ω p -) is the Fresnel reflection coefficient [26]. The prime parameters belong to fused silica, for which is the peak Raman frequency shift at 440 cm -1 (∆ν = 13.2 THz). ----------------------------------- -------------------------------------------- ------------------------------------------------------------------------- --------------------------------I inc ' ω 1 ( ) ----------------------------------,

γ β r k , ω p Ω β r - ( ) γ β' r' k' , ω p Ω β' r' - ( )
ω 1 Ω β' r' - ( ) 3 ω 1 Ω β r - ( ) 3 --------------------------- n ω 1 Ω β r - ( )n' ω 1 ( ) n' ω 1 Ω β' r' - ( )n ω 1 ( )
= × 1 R' ω 1 Ω β' r' - ( ) - [ ]1 R' ω 1 ( ) - [ ] 1 R ω 1 Ω β r - ( ) - [ ]1 R ω 1 ( ) - [ ]
- × I β k r , ω 1 Ω β r - ( )
I inc ω 1 ( )
I β' r' k' , ω 1 Ω β' r' - ( )
Ω β r Ω β'

Direct Raman Gain Measurement Apparatus

The setup used to measure absolute values of the Raman gain coefficient with a 1064-nm pump is shown in Fig. 4 and described in detail in [START_REF] Stegeman | [END_REF].

There are a number of features that allow such absolute measurements using a pulsed laser system consisting of a 10 Hz Nd : YAG pump and an OPG/OPA for the signal. Both the Nd : YAG and OPG/OPA pulse widths were measured using autocorrelation techniques, and their temporal overlap inside the sample was optimized. The beams were found to be approximately Gaussian in space and time and were gently focused at the center of the sample (optimized at each wavelength of the OPG/OPA). Thus, a plane-wave approximation was valid over the sample. The OPG/OPA pulse width varied weakly with wavelength (13-16 ps) over the wavelength of interest (1064-1230 nm) and the YAG pulse width was (33 ± 2 ps). The glass samples, typically 1-2 mm thick, had their input and output facets polished. Detailed analysis of the spatial and temporal overlap led to the following formula for the measured Raman gain coefficient in the small net signal gain region: [START_REF] Stolen | Issues in Raman Gain Measurements[END_REF] where τ p and τ s and w p and w s are the pump and signal pulse widths and minimum spot sizes, respectively [START_REF] Stegeman | [END_REF]. Knife-edge techniques were used to measure the spot sizes; the corresponding Rayleigh ranges were greater than 10× the sample size, and the input signal beam spot size was approximately 60% of the pump beam spot size. Furthermore, ∆E T (Ω, L/2), ∆E S (Ω, 0) and ∆E P (ω P , -L/2) are the total pulse energy (Raman + input signal) at the output of the sample, the input signal energy, and the input pump energy, respectively. Surface reflections are obtained from the experimen- ---------------------------------------------------------------

γ R Ω ( ) ∆E T Ω L/2 , ( ) ∆E S Ω 0 , ( ) - ∆E S Ω 0 , ( )∆E P L/2 - ( )
= × w p 2 w s 2 + ( ) τ s 2 τ p 2 + ( ) π 2 --- ⎝ ⎠ ⎛ ⎞ 3 2 --- ,
tally determined index of refraction data. This is a correction factor which is applied to the measurements along with the depolarization ratio of the material under investigation. Signal pulse energies were measured for input signals both parallel and orthogonal to the pump laser, and depolarization values from the spontaneous Raman experiments were used to estimate ∆E S (Ω, 0) from the measured orthogonal output signal. The dispersion with wavelength of all of the optical components, monochromator, etc., was measured and used to correct the Raman gain coefficient. The peak value and wavelength distribution of the Raman gain of silica was measured (0.9 × 10 -13 m/W) with this apparatus and found to agree with the accepted value to within the experimental error.

HIGH-GAIN TELLURITE GLASS SYSTEMS:

RESONANT ENHANCEMENT OF THE RAMAN SUSCEPTIBILITY Tellurite glasses have been of interest to the glass-NLO communities because they have been predicted, and shown, to have large optical nonlinearities [27]. In fact, their Raman gain coefficients have been measured by all three techniques, namely, spontaneous Raman scattering (515-nm pump), with the apparatus discussed above (1064-nm pump), and in fibers (14XX-nm pumps) [START_REF] Mori | [END_REF].

Shown in Fig. 5 are the Raman gain spectra obtained for two members of the tellurite family [16]. Note that, depending on the details of the composition, the gain coefficient can be as high as 40-50 times that of the peak of fused silica [14,16]. Furthermore, the bandwidth over which significant gain can be obtained is more than twice that of fused silica. Raman gain coefficients measured by spontaneous Raman scattering with a 515-nm excitation (pump) beam have been reported in tellurite glasses that are a factor of two larger than those measured by the direct NLO method described in Subsection 3.3. These were obtained by ratioing their Raman spectra to that of fused silica and using the known value of the silica gain coefficient at 515 nm to deduce that for the tellurites. The question of these and similar discrepancies has been raised by a number of authors [14,[17][18][19].

The band edge of two tellurite samples similar to those reported elsewhere is shown in Fig. 2. Given the fact that nonlinear susceptibilities in general are known to disperse with wavelength and that these measurements in the blue-green spectral range are near the band edge, this possibility was investigated by taking the spontaneous Raman spectrum at four different wavelengths, 458, 515, 752, and 1064 nm, and taking the ratio to the fused silica spectrum as described by Eq. (11). In the case of fused silica, its band edge is deep in the UV, and one would not expect significant dispersion in ∂α/∂Q 0 | Q = 0 in the visible for fused silica.

The results reproduced in Fig. 6 show a large dispersion in the peak tellurite Raman spectra with pump wavelength between the blue-green visible and 1064 nm. Note, however, that the relative response with increasing pump wavelength becomes wavelengthindependent already in the near-infrared. Because this ratio removes all of the usual wavelength dependence except that found in the refractive index, which is small, and the Raman susceptibility, this dispersion reflects the dispersion in the Raman susceptibility ∂α/∂Q 0 | Q = 0 . Additional proof in which the shape of the Raman spectrum changes with wavelength in the visible in multicomponent glasses was also reported [26]. Finally, as indicated in Table 1, the values of the Raman gain obtained by the direct NLO measurement technique and that from spontaneous Raman, both measured at 1064-nm pumping, are in excellent agreement. Therefore, this apparent discrepancy has been resolved. , m/W molar quantities, it can also serve as an intermediate or partner former in some glass compositions. PbO is a network participant in these ternary glasses.

Below in

5. CHALCOGENIDES Very large values for the Raman gain coefficient (many hundreds of times that of fused silica) have been reported for chalcogenide glass fibers at 1550 nm [28]. We measured the Raman gain spectrum of a chalco-genide glass sample in the series Ge-Sb-S-Se at a 1064-nm pump. The result for a Ge-Sb-S sample is shown in Fig. 7. A peak gain coefficient measured at seventy times that of the peak in fused silica was obtained. The introduction of selenium into the glass matrix red-shifts the absorption band edge; due to this fact, photoinduced structural damage was observed during the experiment. One can account for this issue by using lower pump intensities, since, as has been shown, there is a threshold intensity that is required to activate these light-induced structural changes, previously observed in other selenium-containing glasses [START_REF] Lopez | Evaluation of Photo-Induced Structural Mechanisms in Chalcogenide Glass Materials[END_REF]. In this case, operation with lower pump intensities is feasible for obtaining reasonable Raman gain signals, since previous spontaneous Raman scattering experiments have indicated that the peak Raman gain was on the order of a hundred times that of fused silica for these compositions. This glass composition had the largest Raman gain coefficient measured in this study.

BROADBAND GLASS: BOROPHOSPHATE-BASED GLASSES

Experiments were performed on a limited number of glasses in the 90%[(100 -x)NaPO 3 -xNa 2 B 4 O 7 ]-5%TiO 2 -5%Nb 2 O 5 family, which was designed for broad spectral response. As shown in Fig. 8, although the Raman gain coefficient is comparable to fused silica glass, the gain extends out to almost 40 THz and varies by only about 4 dB across the full band [START_REF] Rivero | [END_REF]. This is an increase in bandwidth of more than 5 times that of SiO 2 . Also, by changing the ratio of the glass constituents, we can tailor the Raman spectrum in order to obtain a more uniform, flat spectral bandwidth, desirable for this broadband Raman amplification application.

CONCLUSIONS

Both spontaneous Raman scattering and a new nonlinear optics apparatus capable of absolute Raman Gain measurements at 1064 nm have been used to measure the Raman gain coefficient in different glasses. One of the key results obtained has been to show that there is dispersion with wavelength in the Raman susceptibility when the measurement wavelength is near the band edge for the corresponding absorption spectrum. As a consequence, measurements of the Raman gain are resonantly enhanced in this case. When the Raman scattering measurements are far from the band edge, the agreement with the direct nonlinear optical measurements is excellent.

The Raman gain properties of a number of glass families have been investigated. It was found that tellurite glasses with different compositions of additional constituents have a Raman gain about 30-50 times that of fused silica, as well as superior bandwidth. The largest gains >70 were measured in chalcogenide glasses, although there are issues with photoinduced damage. Borophosphate glasses were found to have gains marginally larger than those of fused silica but with far superior spectral bandwidths, up to 5 times that of fused silica. A number of other glass families are currently being investigated.

Fig. 1 .

 1 Fig. 1. Schematic representation of the stimulated Raman (gain) process (left) and 90°-geometry spontaneous Raman scattering process (right).

5 α

 5 NaPO 3 -5% Na 2 B 4 O 7 ]-5% TiO 2 -5% Nb 2 O

Fig. 2 .

 2 Fig. 2. Example of the relative absorption band edge for a chalcogenide glass (square), TeO 2 -based glasses (circle and triangle curves represent the extreme band-gap positions for different TeO 2 compositions tested to date), and a borophosphate glass matrix (star).Fig. 3. Geometry of micro-Raman setup. Various lasers and detectors were used at different wavelengths.

Fig. 3 .

 3 Fig. 2. Example of the relative absorption band edge for a chalcogenide glass (square), TeO 2 -based glasses (circle and triangle curves represent the extreme band-gap positions for different TeO 2 compositions tested to date), and a borophosphate glass matrix (star).Fig. 3. Geometry of micro-Raman setup. Various lasers and detectors were used at different wavelengths.

Fig. 4 .

 4 Fig. 4. Apparatus used to measure absolute values of Raman gain.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Raman gain curves of 59.5% TeO 2 -25.5% TlO 0.5 -15% PbO (right-hand side) and 66.5% TeO 2 -28.5% TlO 0.5 -5% PbO (lefthand side). The dashed lines indicate where the notch filter cuts off the spectrum. The spontaneous Raman spectra were normalized to the peak at a 21.3-THz frequency shift.

γ R × 10 - 13

 1013

Fig. 7 .

 7 Fig. 7. Raman gain spectrum of the chalcogenide glass Ge-Sb-S at 1064 nm.

Fig. 8 .

 8 Fig. 8. Measured Raman gain (square) by the direct NLO technique and the normalized spontaneous Raman spectrum for 90%[95% NaPO 3 -5% Na 2 B 4 O 7 ]-5% TiO 2 -5%Nb 2 O 5 (circle). Also shown for comparison is the Raman gain spectrum of fused silica (solid light gray curve).

Table 2

 2 is a summary of the directly measured values for the Raman gain coefficient for a variety of glasses in a different tellurite family, which contains lead oxide and thallium oxide. This table illustrates how the peak at ∆ν

Table 1 .

 1 Raman gain coefficients with 1064-nm pumping at the peak Raman resonance at 665 cm -1 (δν = 20 THz) measured by taking the ratio of the spontaneous Raman spectrum to that of fused silica and by direct NLO measurements[26]

	Composition, molar percent	Sample code	Calculated peak Raman gain coefficient, from spontaneous Raman cross-section	Directly measured peak Raman gain coefficient
	85%TeO 2 -15%WO 3	W	40 × 10 -13 m/W ± 15%	38 × 10 -13 m/W ± 10%
	85%TeO 2 -10%Nb 2 O 5 -5%MgO	Nb	26 × 10 -13 m/W ± 15%	26 × 10 -13 m/W ± 10%

Table 2 .

 2 Peak Raman gain coefficients of TeO 4 (∆ν = 20 THz) units and TeO 3 and/or TeO 3 + 1 units (∆ν = 21.3 THz)[16]

	Glass composition, Molar percent	∆ν = 20 THz Gain coefficient (10 -13 m/W)	∆ν = 21.3 THz Gain coefficient (10 -13 m/W)
	66.5%TeO 2 -28.5%TlO 0.5 -5%PbO	30 ± 1.5	34 ± 1.7
	63%TeO 2 -27%TlO 0.5 -10%PbO	23 ± 2.6	38 ± 2.5
	59.5%TeO 2 -25.5%TlO 0.5 -15%PbO	25 ± 2	42 ± 1.5
	75%TeO 2 -25%TlO 0.5	25 ± 4	19 ± 3
	70%TeO 2 -30%TlO 0.5	21 ± 4	23 ± 5
	60%TeO 2 -40%TlO 0.5	21 ± 5	30 ± 7
	50%TeO 2 -50%TlO 0.5	14 ± 3	52 ± 3

ACKNOWLEDGMENTS

This work was carried out with the support of numerous research, equipment, and educational grants, including NSF grants ECS-0123484 and ECS-0225930 and NSF Integrative Graduate Education and Research Training (IGERT) grant DGE-0114418. The authors also acknowledge the assistance and financial support of the College of Optics and Photonics and the Student Government Association (SGA) at the University of Central Florida. The authors are also thankful for an equipment donation from JDS Uniphase. Finally, the authors would like to acknowledge Phillipe Thomas and Laeticia Petit for providing us with some glass samples for the Raman gain testing.