Jessica Bosseaux

The official quality signs influence on prices and volumes: the case of organic fresh eggs

Keywords: performance, price, market, store label Bio, performance, prix, marché, type de distributeurs

The use of organic official quality sign is growing for animal products production in France. Past researches have assessed the market performance of organic using attitudinal data. This paper investigates the impact of organic label on market performance using real market data. We use the hedonic price formation to understand the value of organic and the influence of store types in price formation. To complete the understanding of the organic official quality sign in the market, we calculate the price elasticity of organic among the other egg marks. Our results reveal a positive trend for higher quality products, and that core organic consumers are not price-sensitive, which increase market performance. This effect is mitigated by the store type which commercializes the organic product, highlighting the importance of the interaction between the two quality signals.

Key words: organic label,

Introduction

The intensification of animal products production required new policies regarding animal production diseases, sustainable intensification and animal welfare which incorporate consumer new priorities. Thereby, the table eggs are all marked according to an official marking system managed by a European directive concerning registration of establishments keeping laying hens Appendix 1 Code for the farming method (The commission of the european communities, 2002) (appendix 1). Each mark and official quality sign (OQS) (Organic and Label Rouge) dovetails with a production method whose ethics is assessed by consumers, according to their interests, lifestyle, opinions and attitudes [START_REF] Eisenhardt | Segmentation of the market for table eggs in Ontario[END_REF]. Over the last decades, the organic production shows an exponential growth rate, especially for animal products. In France, the laying hens are the second sector with the highest part of organic production (10,1% in 2017) (« Chiffres de la bio en France -Agence Française pour le Développement et la Promotion de l'Agriculture Biologique -Agence BIO », s. d.).

Literature review

The new consumers concerns grant a higher willingness-to-pay (WTP) [START_REF] Auger | Do social product features have value to consumers[END_REF][START_REF] Kehlbacher | Measuring the consumer benefits of improving farm animal welfare to inform welfare labelling[END_REF]. As consumers are not able to distinguish the production mode through tasting [START_REF] Guibert | 60 millions de consommateurs[END_REF], they essentially focus on in-store information including price ranges, marks, store types and brand types, but also beliefs towards animal welfare, healthcare, environmental and social concerns [START_REF] Shepherd | Determinants of consumer behavior related to organic foods[END_REF]. Eggs are credence goods which quality is signaled by labelling [START_REF] Roe | Credence good labeling: The efficiency and distributional implications of several policy approaches[END_REF][START_REF] Nelson | Information and consumer behavior[END_REF]. This study contributes to the body of research on the impact of organic label on market results [START_REF] Katsikeas | Assessing Performance Outcomes in Marketing[END_REF].

The OQS are signals that provides a better information to the consumers about the product, its production process or the company which produces it, and represent marketing levers [START_REF] Larceneux | Segmentation des signes de qualité: labels expérientiels et labels techniques[END_REF]. As a brand, the OQS has intangible resource characteristics. According to the resource-based view and more specifically the concept of dynamic capabilities, the competitive advantage of dynamic dimensions of resources "lies in their ability to alter the resource base: create, integrate, recombine, and release resources" [START_REF] Eisenhardt | Segmentation of the market for table eggs in Ontario[END_REF], p. 1116). The marketing actions influences the customer attitude and the market impacts and positions (Hanssens & Pauwels, 2016). The customer behavior toward the price and the volume of purchase is an indicator of the value addressed to the product [START_REF] Rust | Measuring Marketing Productivity: Current Knowledge and Future Directions[END_REF].

The organic OQS influences positively the purchase intention and the WTP [START_REF] Dufeu | Multi-labellisation socioenvironnementale et consentement a payer du consommateur[END_REF][START_REF] Larceneux | Why might organic labels fail to influence consumer choices? Marginal labelling and brand equity effects[END_REF][START_REF] Bernard | Comparing parts with the whole: Willingness to pay for pesticide-free, non-GM, and organic potatoes and sweet corn[END_REF].

Hypothesis 1: The organic OQS commands a price premium.

Moreover, the store type and its environment influence the organic purchase [START_REF] Gottschalk | Consumer reactions to the availability of organic food in discount supermarkets[END_REF][START_REF] Akaichi | Assessing consumers' willingness to pay for different units of organic milk: Evidence from multiunit auctions[END_REF][START_REF] Paul | Consumer behavior and purchase intention for organic food[END_REF] The store type and its environment influence the organic purchase [START_REF] Paul | Consumer behavior and purchase intention for organic food[END_REF]. For example, specialized food store are preferred for organic purchasers [START_REF] Akaichi | Assessing consumers' willingness to pay for different units of organic milk: Evidence from multiunit auctions[END_REF] and discount store stimulates the organic market [START_REF] Gottschalk | Consumer reactions to the availability of organic food in discount supermarkets[END_REF]. We investigate the weight of store type in price formation.

Hypothesis 2: The store type influences the organic eggs price premium.

To give a better understanding of the organic OQS effects in terms of price but also sales volumes, we test the price-elasticity. Previous studies have looked at the price-elasticity of organic products, and the results are heterogeneous [START_REF] Bezawada | What is special about marketing organic products? How organic assortment, price, and promotions drive retailer performance[END_REF][START_REF] Kiesel | Got Organic Milk? Consumer Valuations of Milk Labels after the Implementation of the USDA Organic Seal[END_REF]. Lower price has a negative impact on consumers purchase, according to a study on French grocery stores [START_REF] Ngobo | What Drives Household Choice of Organic Products in Grocery Stores[END_REF].

Hypothesis 3: The own-elasticity is greater for organic eggs than non-organic eggs.

Data

In order to test our hypotheses, we used two databases using the stated preference method. The first one is a consumer panel conducted by Kantar over 5 years, from January 2012 to September 2017, extracted from France Agrimer (appendix 2). The observation of the data highlights the profitable trend for high quality products including OQS in spite of a largely higher price. The consumers shift away from cage and barn categories.There are monthly mean prices and mean sales volumes for each egg mark.

The second data base (appendix 3) completes the examination of organic profitability in terms of margin according to the store type (appendix 4). At stores, the cheapest organic eggs option is unsurprisingly available in the discount stores. There are weekly mean prices for three store types, namely conventional stores, discount stores and specialized stores (organic stores). We use the standard deviation to indicate the price volatility. We note that the price volatility is small at specialized stores compared to others.

The entire statistic treatment has been realized with the R Studio Software [START_REF] Developmentcoreteam | R: A language andenvironmentforstatisticalcomputing, R FoundationforStatistical Computing[END_REF].

Study 1. Hedonic price method: the price premium of organic and the store type influence

Our first study is based on the hedonic price function that determines the retail price in the fresh eggs markets. [START_REF] Rosen | Hedonic prices and implicit markets: Product differentiation in pure competition[END_REF] defined hedonic prices as "the implicit prices of attributes and are revealed to economic agents from observed prices of differentiated products and the specific amounts of characteristics associated with them". It allows us to estimate the implicit prices of each characteristic that define the nomenclature of table eggs. Differences in price levels between eggs should reflect differences in eggs characteristics. We selected four characteristics that are claimed by the marking system (appendix 5) to elaborate the hedonic price equation: the cage free breading, the free-range breading, the presence of a label, either organic or Rouge, and the organic characteristic. Our extended semi-log model is expressed as:

𝑙𝑛𝑃 𝑖 =∝ +𝛽 1 𝐶𝐹 𝑖 + 𝛽 2 𝐹𝑅 𝑖 + 𝛽 3 𝑂𝑄𝑆 𝑖 + 𝛽 4 𝑂𝑅 𝑖 + 𝜀 𝑖
With CF for the cage free characteristic, FR for the free-range characteristic, OQS for the fact that the egg wear an official quality sign whether it is a Label Rouge or an Organic label, and OR for the organic characteristic. Test results suggest the presence of heterosckedasticity; consequently White's heteroskedasticity-consistent standard errors has been used. The model fit the data well with an adjusted R² of 0,977. Every single characteristic significantly adds value to the product. We tested the multicollinearity by calculating the variance inflation factors, and none were relevant. We notice that the free-range characteristic is the most important in term of market price determination, followed by the presence of an OQS. The organic characteristic does not play a key role according to our results. This confirm the study of Karapidis and al (2005) which expressed that free-range is more powerful in price formation than organic. The cage-free characteristic is not as much important as free-range and OQS in price formation, and it confirms the study of [START_REF] Chang | The price of happy hens: A hedonic analysis of retail egg prices[END_REF]. The price premium for organic is significant, supporting H1. Nevertheless, we temper this result as the organic products has a low coefficient.

Based on the second data base, we investigate the price function of organic eggs according to the implicit prices of store types (Appendix 6). In our equation, the independent variable is the logarithm of the table eggs prices, including organic and non-organic quality. Our extended semi-log model is expressed as:

𝑙𝑛𝑃 𝑖 =∝ +𝛽 1 𝐺𝑀𝑆𝐵𝐼𝑂 𝑖 + 𝛽 2 𝐷𝐼𝑆𝐵𝐼𝑂 𝑖 + 𝛽 3 𝑆𝑃𝐸 𝑖 + 𝜀 𝑖
The model fits the data well with an adjusted R² of 0,977. The results confirmed that in any store type, organic quality creates value. We tested the multicollinearity by calculating the variance inflation factors, and none were relevant. Nevertheless, this value is unequal. The weight of specialized stores in organic price formation is the highest, followed closely by the conventional store. The discount store stays positive but is way weaker than others. The price premium for organic eggs is modified according to the store type. Our results support H2, in line with the literature about price premium at conventional stores [START_REF] Rondán Cataluña | Are hypermarket prices different from discount store prices?[END_REF].

Study 2. The price-elasticity of organic eggs

We analyze the sales volumes thanks to a log-log functional form that provides direct estimates of the respective elasticity of the independent variables with respect to the dependent variable.

(2)

(3)

𝑙𝑛𝑌 𝑖 = 𝑏 0 + 𝑏 1 𝑙𝑛(𝑋 1 ) + 𝑏 2 𝑙𝑛(𝑋 2 ) + 𝑏 3 𝑙𝑛(𝑋 3 ) + 𝑏 4 𝑙𝑛(𝑋 4 ) + 𝑏 5 𝑙𝑛(𝑋 5 ) + 𝑏 6 𝑙𝑛(𝑋 ( 6) )

Where 𝑙𝑛𝑌 𝑖 is the predicted sales volume, b0 is the model intercept, and all the following factors are price variation for each production methods: b1 for cage production methods, b2 for barn production methods, b3 for Label Rouge production methods, b4 for other free-range production methods, b5 for organic production methods and b6 for no signaled production methods. Barn and no sign categories are not analyzed deeply because of the very small amount of sales volumes they represent. This table shows a blatant asymmetrical and heterogeneous elasticity, which confirms that price is not the main concern for consumers. That may be the effect of a lower price premium that the consumer consent to pay to access the higher quality product. In that sense, the organic attribute is valuable for consumers. This support H3. The consumers pay a price premium to accede to the quality of organic eggs in spite of the price. A certain weakness of the Label Rouge is pointed out by the data. Its sales can be negatively impacted by the free-range price.

As the Label Rouge quality sign belongs to the free-range category, the free-range without quality sign might be perceived as a substitute of Label Rouge.

Discussion and implications

First, we find that consumers are globally in a social trend that shifts their purchase to highquality products, regardless of the price. Both hedonic analyses report the implicit prices of organic in eggs price formation and the store type has an influence on the organic price.

Although our results give evidence about the organic value creation, we must interpret them with cautious as other criteria may count such as GM-free or local production features.

Second, the price variation investigation reveals a lack of elasticity for organic. Pricing does not affect sales, whereas other high-quality ranges react differently. This confirms the work of [START_REF] Ngobo | What Drives Household Choice of Organic Products in Grocery Stores[END_REF] which found that consumers tend to stay with organic products rather than shift to another quality. Also, the organic volumes benefit from free-range and Label Rouge price increase. This result confirms the work of Bernard and Bernard (2010, p. 473) which found that a price increase for non-organic products shifts consumers toward the organic version, more than the opposite.

Third, the marking system gives information about a non-commercial attribute (the production process) that enhance the value [START_REF] Kehlbacher | Measuring the consumer benefits of improving farm animal welfare to inform welfare labelling[END_REF][START_REF] Auger | Do social product features have value to consumers[END_REF]. Nevertheless, the strength of the organic OQS must come from both process methods and health aspect or other beliefs.

Fourth, lower prices is not a competitive advantage for organics, and increasing may have positive outcomes for organics. The results confirm the work of [START_REF] Ngobo | What Drives Household Choice of Organic Products in Grocery Stores[END_REF] and [START_REF] Bezawada | What is special about marketing organic products? How organic assortment, price, and promotions drive retailer performance[END_REF]: reducing organic prices is ineffective, and increasing non-organic eggs is in favor of organic products. The stores' managers should consider increasing or stabilizing their organic eggs prices. Otherwise, the non-specialized stores must use the low price to attract noncore organic consumers, as a loss-leader product, and benefits from the consumers' traffic in the organic section for creating margins with other products.

Fifth, the organic label acts as an intangible resource that contributes to a market performance [START_REF] Hooley | The performance impact of marketing resources[END_REF]. The firms employ OQS as a resource to create value, defined as an "additional revenue minus the costs of generating the additional revenue" [START_REF] Mcwilliams | Creating and capturing value: Strategic corporate social responsibility, resource-based theory, and sustainable competitive advantage[END_REF], p. 1492). We consider the organic OQS as a competitive advantage in the sense that it shifts the consumers in certain case and does not suffer from price increasing. The stores must promote organic products in the hope of margins and store revenues increase.

Finally, our study also has implication for policymakers. The official marking system creates value as it gives better information to consumers. This system may be tested on other agrifood products.

Conclusion

The French egg market witness an impressive growth between 2012 and 2017. Using market metrics instead of consumer metrics, we avoid the biases associated with hypothetical responses and unrealistic scenarios [START_REF] Breidert | A REVIEW OF METHODS FOR MEASURING WILLINGNESS-TO-PAY[END_REF] and complete the literature on market results of organic label in terms of price and volumes. We assess the market performance through hedonic price and the price elasticity of organic OQS, in order to strive to identify how the organic certification influences value creation. We took into consideration the other marks and the store types to understand better the OQS.

The framework we have proposed assesses the importance of the interaction between the store type and the organic label. Future research must analyzing the congruence with store name, brand equity and other in-store information. The acceptance of a higher price for a higher quality revealed a market advantage for the organic label. The future of organic eggs is bright for specialized stores and must conduct non-specialized stores to modify their strategies to take advantage of this market.

Table 1

 1 Semi-log hedonic model estimates for marks

			(1)
	Estimate	Std.Error	t-value

Table 3

 3 Price elasticity for eggs according to the categories[START_REF] Developmentcoreteam | R: A language andenvironmentforstatisticalcomputing, R FoundationforStatistical Computing[END_REF]

				Sales' Volumes
			Cage	Free-range	Label Rouge	Organic
	Prices	Cage Free-range Label Rouge	X X X	-0.811** X X	X -0,455* X	X 0.239* 1.080**
		Organic	X	X	X	X
	Note : ***sig < 0,01 ; ** < 0,01 ; *sig < 0,05

Appendix 1 Code for the farming method (The commission of the european communities, 2002) The farming methods as defined in Regulation (EEC) No 1274/91 and the Regulation (EEC) No 2092/91 shall be indicated by the following code:


D Rontani
email: damien.rontani@supelec.fr

M Sciamanna
email: marc.sciamanna@supelec.fr

D S Citrin
email: david.citrin@ece.gatech.edu

S Ortin
email: ortin@ifca.unican.es

Time-delay identification in the chaotic output of a semiconductor laser with optical feedback: a dynamical point of view D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin

Abstract-A critical issue in optical chaos-based communications is the possibility to identify the parameters of the chaotic emitter, and hence to break the security. In this paper, we study theoretically the identification of a chaotic emitter that consists of a semiconductor laser with an optical feedback. The identification of a critical security parameter, the external-cavity-round trip time (the time delay in the laser dynamics), is performed using both the auto-correlation function and delayed mutual information methods applied to the chaotic time-series. The influence on the time-delay identification of the experimentally tunable parameters, the feedback rate, the pumping current and the time-delay value, is carefully studied. We show that difficult time-delay-identification scenarios strongly depend on the timescales of the system dynamics as it undergoes a route to chaos, in particular on how close is the relaxation oscillation period from the external-cavity-roundtrip time.

Index Terms-Semiconductor laser, optical feedback, nonlinear dynamics, time-delay identification

I. INTRODUCTION

S EMICONDUCTOR lasers with an external cavity (ECSL) have been considered as rich sources of optical chaos, with well-known chaotic regimes such as the so-called coherence collapse [START_REF] Lenstra | Coherence collapse in single mode semiconductor lasers due to optical feedback[END_REF] and low-frequency fluctuations (LFF) regimes [START_REF] Risch | Self pulsation in the output intensity and spectrum of GaAs-AlGaAs cw diode lasers coupled to a frequency selective external optical feedback[END_REF]. Chaotic regimes from ECSL have received considerable attention since they constitute key elements of optical secure chaotic communications [START_REF] Mirasso | Synchronization of chaotic semiconductor lasers: Application to encoded communications[END_REF]. Indeed, the addition of an external cavity introduces an infinite numbers of degree of freedom through the time-delay in the dynamical representation of the semiconductor laser, leading to the generation of highdimensional chaos [START_REF] Vicente | Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop[END_REF]. The unpredictable and noisy appearance of the signals emitted by hyper-chaotic optical systems has been used to enhance security and privacy in communications: A chaotic carrier conceals and transmits an information bearing message, which is decrypted by synchronization at the receiver end (a physical copy of the chaotic emitter) [START_REF] Mirasso | Synchronization of chaotic semiconductor lasers: Application to encoded communications[END_REF], [START_REF] Ikeda | High-dimensional chaotic behavior of systems with time-delayed feedback[END_REF]- [START_REF] Argyris | Chaos-based communications at high bit rates using commercial fibre-optic links[END_REF]. The security of such a chaos-based communication scheme relies mainly on the difficulty to identify the emitter parameters, and on the sensitivity of synchronization to parameter mismatch [START_REF] Li | Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization[END_REF]. Therefore, it is of particular importance to study the capability of an eavesdropper to retrieve information about the parameters of a particular chaotic generator from its time series transmitted through the communication channel.

In delayed hyper-chaotic systems, the security assumption is based on the computational complexity to reconstruct a high-dimensional attractor from the time series. However, the knowledge of the time delay allows for the projection of the high-dimensional attractor onto a reduced-dimensional phase space, which makes the system vulnerable to lowcomputational-complexity identification techniques [START_REF] Hegger | Identifying and modeling delay feedback systems[END_REF]. It is thus crucial that the delay not be easily identifiable in a cryptosystem. Until recently, ECSLs with a single optical feedback were considered as weakly secure systems in terms of time-delay identification, such that the use of several external cavities has been suggested [START_REF] Lee | Dynamical characterisation of laser diode subject to double optical feedback for chaotic optical communications[END_REF]. However, we have shown recently that a simple ECSL with a single optical feedback could, with a careful choice of parameters, hide its time-delay signature when standard estimators are employed [START_REF] Rontani | Loss of timedelay signature in the chaotic output of a semiconductor laser[END_REF]. Interestingly the region of laser and feedback parameters where such a difficult time-delay identification occurs does not necessarily correspond to the situation where the chaos complexity (dimension and entropy) is higher [START_REF] Vicente | Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop[END_REF]. Indeed in laser diodes with optical feedback high-dimensional chaos is typically found where the optical feedback strength is large, but then the time-delay parameter is easily retrieved from the analysis of the chaotic output using straightforward techniques [START_REF] Rontani | Loss of timedelay signature in the chaotic output of a semiconductor laser[END_REF]. Time-delay identification should therefore be considered as an additional argument to appreciate the level of security of chaos-based identification, besides chaos complexity and robustness of chaos synchronization.

In this paper, we extend our earlier work on time-delay identification in laser diode with optical feedback. Identification is processed using both conventional technique based on autocorrelation function (ACF) and more sophisticated technique such as delayed mutual information (DMI). We analyze further the range of laser and feedback parameters that lead to difficult parameter identification and link these bad identification scenarios to the laser dynamics as it undergoes a bifurcation cascade to optical chaos. More specifically we show that the time-scales of the laser dynamics in its route to chaos influence the time-delay identification and therefore also the security of such a chaotic emitter. A careful tuning of the external cavity round-trip time with respect to the intrinsic laser relaxation oscillation frequency leads to situations where the time-delay signature is lost in the laser chaotic output. Finally we discuss the robustness of our results using other signal processing techniques applied to the laser chaotic timeseries.

The paper is organized as followed. Section II presents the model and specificities of the various identification techniques used to retrieve the time-delay value. Section III details the influence of the different operational parameters (the feedback rate, the time-delay and the pumping current) on the identification. Section IV gives a dynamical interpretation of the identification scenarios observed, based on the studies of the frequencies that appear in the ECSL dynamics as it undergoes a route to chaos. Section V discusses the influence of key internal parameters that modify the ECSL model and the possible application of our results to chaos-based communications. Finally, we summarize our main conclusions in section VI.

II. THEORETICAL FRAMEWORK

A. Rate equation model

In this work, we consider a single-mode semiconductor laser with optical feedback modelled by the Lang-Kobayashi rate equations [START_REF] Lang | External optical feedback effects on semiconductor injection lasers properties[END_REF]. This model has been successful in reproducing dynamical behaviors experimentally observed in ECSLs. The rate equations are

dE (t) dt = 1 2 (1 + iα) G N,|E| 2 - 1 τ p E (t) , +γe iω0τ E (t -τ ) + F (t) (1) dN dt =pJ th - N τ s -G N,|E| 2 |E (t)| 2 , ( 2
)
where t) is the slowly varying complex electric field, N is the average carrier density in the active region, α is the linewidth-enhancement factor that describes the amplitude phase coupling,

E (t) = |E (t)| e iφ(
G N,|E| 2 = G N (N -N0) 1+ε|E| 2
is the optical gain where ε is the saturation coefficient, N 0 is the carrier density at transparency, ω 0 is the angular frequency of the solitary laser, γ is the feedback rate, τ p is the photon lifetime, τ s is the carrier lifetime, J th is the threshold current, p is the pumping factor, and τ is the delay corresponding to the round-trip time of light in the external cavity. The force F (t) = F D (t)+F L (t) added in the field equation models the spontaneous emission occurring in the semiconductor laser. It is composed of a deterministic part

F D (t) = 2β N (t) |E(t)| e iφ(t) and a stochastic part F L (t) = 2βN (t)ζ (t) which cor- responds to a Langevin term where ζ = ζ E (t) + iζ φ (t)
is a complex Gaussian white-noise term with zero mean and correlation:

ζ E (t) ζ * φ (t ) = δ Eφ δ (t -t ).
β is the spontaneous-emission rate. The relaxation-oscillation period is an intrinsic damping time of the free-running laser and is defined by

τ RO = 1 τpτs (µ -1) -µ 2 4τ 2 s -1/2 with µ = G N τ s τ p pJ th -N0
τ s . We consider the following parameter values: α = 5 , ω 0 τ = 0 rad, τ p = 2 ps, τ s = 2 ns, G N = 7.5 × 10 -13 m 3 s -1 , N 0 = 3 × 10 24 m -3 , J th = 1.83 × 10 33 m -3 s -1 , ε = 2.5 × 10 -23 m 3 , and β = 10 3 s -1 .

B. Time-delay estimator and time-delay signature

The time delay is a key parameter of delay systems. Its estimation is of particular importance with regard to computationally efficient identification of all the parameters of the nonlinear system under consideration. Standard techniques, such as the ACF and the DMI, can be used to identify the time-delay.

1) The autocorrelation function (ACF): If a random process X(t) is ergodic and wide-sense-stationary, then the ACF is defined by

Γ (θ) = 1 σ2 X (x (t) -μX ) (x (t + θ) -μX ) , (3)
where x(t) and x(t + θ) are sampled from the random process X(t), μX = x (t) , and σX = (x (t) -μX )

2 1/2
with • denoting time average. The ACF measures for a given value θ, the tendency of the cloud of points (x (t) , x (t + θ)) to be aligned along a straight line, and thus measures a linear relationship between x (t) and x (t + θ).

2) The delayed mutual information (DMI): The mutual information, H, is a quantity originally used in information theory [START_REF] Cover | Elements of information theory[END_REF]. Given two continuous variables X and Y with joint probability density function, f XY (x, y) and marginal probability density functions f X (x) and f Y (y), the mutual information of X and Y is defined as

H (X, Y ) = E ln f XY (x, y) f X (x) f Y (y) , ( 4
)
where E(•) is the expectancy operator, the two variables X and Y are obtained by sampling the random process X (t) at two times t and t + θ , and such process is assumed to be stationary and ergodic. The probability density functions, f XY , f X , and f Y , will be estimated by their respective histogram fXY , fX , and fY computed from the time series and leading to the approximate mutual information estimator, also called delayed mutual information (DMI),

H (θ) = E ln fX(t)X(t+θ) (x, y) fX(t) (x) fX(t+θ) (y) . ( 5
)
The mutual information corresponds intuitively to the quantity of information that the two random variables X (t) and X (t + θ) are sharing. In time-delay systems, the presence of a delayed feedback term induces a nonlocal time dependence in the time evolution of its state variables. The integral definition of the estimators under consideration allows for the detection of nonlocal time dependencies, linear for the ACF and nonlinear for the DMI. Therefore, if a particular timescale, such as a time delay, is present in a given time series, it should manifest itself in the estimator through a local extremum. This signature of the time scale, depending on its sign, will be referred to as peak or valley later on in this article. The time location of a peak (respectively valley) will be considered as a possible estimation of the time delay.

III. IDENTIFICATION OF TIME DELAY IN LANG-KOBAYASHI EQUATIONS

In this section, the security in terms of time-delay identification is performed by analyzing the intensity time series of a chaotic ECSL. The intensity is

I(t) = |E(t)| 2 .
The timedelay signatures in the ACF and the DMI are analyzed with the evolution of the following operational parameters of the ECSL: the feedback rate γ, the external-cavity-roundtrip time τ , and the pumping factor p. Throughout this study, the spontaneousemission rate β is taken equal to zero.

A. Influence of the feedback rate

The feedback rate controls the optical power reinjected in the laser cavity. As a consequence, it drives the contribution of the delayed intensity I (t -τ ) in the time-evolution of I (t). We thus expect that the stronger the feedback rate, the more important the information shared between I (t -τ ) and I (t) will be.

Figure 1 analyzes a first case of time-delay identification, for increasing values of the feedback rate γ. Large values of the feedback rate produce chaotic time series with a clear signature of the time delay in the estimators: a significant peak is present in both the ACF [Fig. 1(b4)] and the DMI [Fig. 1(c4)], at a time corresponding to the time-delay. Similar results have been obtained experimentally, from the analysis of the autocorrelation function [START_REF] Bünner | Estimation of delay times from a delayed optical feedback laser experiment[END_REF]. The progressive decrease in feedback rate produces the following: first, a decrease of the peak's amplitude close to the time delay location and its multiples, in both the ACF and DMI [Fig. 1(b3) and c(3)] until it reaches a minimum value [Fig. 1(b2) and (c2)]; second, a qualitative change of behavior appears. For relatively weak feedback rates, numerous peaks and valleys, separated by τ RO /2 from each other, appear and are amplified in the vicinity of the origin (θ = 0) and the time delay (θ ≈ τ ) [Fig. 1(b1) and (c1)]. They obscure the time-delay identification. A quantitative description of the feedback-rate influence in terms of time location and amplitude of the time-delay signature is given in Fig. 2 for both the ACF and the DMI. Figure 2 is obtained by considering the most significant peak (or valley) in a vicinity W (τ ) of the time delay which is defined by max θ∈W (τ ) |Γ(θ)| for the ACF and max θ∈W (τ ) |H(θ)| for the DMI. In Fig. 2, the curves with triangle markers are drawn for the same parameters as the ones of Fig. 1. The curves confirm the tendency already observed in Fig. 1: as the feedback rate is decreased, the amplitude starts to diminish until a global minimum value is reached, and then it increases [Fig. 2(a1)-(a2)]. The decreasing region of the curve corresponds to weak chaotic regime; therefore, the ACF and DMI may still find a structural relationship within the intensity time-series I (t). Then for a sufficiently strong feedback rate, the chaotic regime becomes well established and most of the timescale signatures are weakened due to equivalent driving action of the intrinsic nonlinearity of the ECSL and the linearly introduced feedback term. However, if γ is further increased, then due to this linear introduction of the delayed term, γe iω 0 τ E (t -τ ) e iφ(t-τ ) , in the rate equations, the influence of the time delay is enhanced. The decrease in feedback rate also influences the time location of the largest peak at a time close to the time-delay; see Figs. 2(b1)-(b2). For small feedback rates, the largest autocorrelation or mutual information is obtained for a time close to a high order multiple of τ RO /2, when analyzing a small time window in the vicinity of the time delay W (τ ). Then the identification gives a value of time which is quite shifted from the expected value (τ ). However, as the feedback rate increases, the greatest autocorrelation and mutual information values are achieved for a time closer to the time-delay. This explains the monotonic decreasing behavior of the time shift identification curve in Fig. 2

(b1)-(b2).

In summary this first analysis shows that when the feedback rate is relatively weak, the chaotic output of the laser diode shows only small evidence of the time-delay signature, both in the ACF or the DMI. The largest contribution to the ACF or DMI, in a time window around the time-delay, is in fact closer to a high multiple of the relaxation oscillation period τ RO and this blurs the good identification of the expected value τ .

Figure 2 also shows the influence of the pumping factor p, which controls the injection current J = pJ th as well as the value of the relaxation oscillation period τ RO . When increasing p, chaotic dynamics are observed for larger values of the feedback rate γ but similar conclusions hold for the identification of the time-delay. The amplitude of the highest peak (or valley) in the ACF and DMI in the vicinity of τ exhibits a minimum value when the feedback rate increases, and the feedback rate corresponding to this minimum increases with the increase of p. Moreover the time value obtained from the ACF or DMI identification techniques is shifted from the expected value (τ ) when the feedback rate is small but gets closer to τ as the feedback rate increases. The maximum time shift is however smaller when p increases, hence progressively leading to a better identification of τ .

B. Influence of the time delay relatively to the relaxationoscillation period

In this subsection, we study how the value τ affects the time-delay identification based on the ACF and DMI. This influence is analyzed relatively to the value of the relaxationoscillation period τ RO , the other significant ECSL timescale in terms of identification. This line of reasoning is supported by the results of Section III.A, where the two timescales coexist and τ RO proves to have a blurring effect on the time-delay signature [Fig. 1(b1)-(c1)]. Here, we will show that τ RO may have an even stronger effect. If it is taken sufficiently close to τ and provided that the pumping current is small (p ≈ 1, which leads to relatively large values of τ RO ), then the time- delay signature will not be easily retrievable either in the ACF or the DMI.

Figure 3 illustrates this fact for τ = 1.2 ns and keeping τ RO = 0.75 ns.

At large feedback rate, the reduction of the time separation between τ and τ RO does not produce a qualitative change of the time-delay identification at strong feedback rates. Both estimators still exhibit an easy-retrievable time-delay signature: a sharp peak close to the time-delay location [Fig. 3(b3)-( c3) and (b4)-(c4)]. Then, similar to Fig. 1, a diminution of the feedback rate, promotes the effect of τ RO . This leads to a time-delay signature hardly retrievable for sufficiently close values of τ and τ RO [Fig. 3(b2)-(c2)], whereas it was still possible to retrieve it when the two timescales were sufficiently disparate [Fig. 1(b2)-(c2)]. We notice also that the ACF or DMI evolve from a pulse-like shape at high feedback rates to an oscillating shape at low feedback rate [Figs 3 (b1) and (c1)]. The transition between these two shapes is achieved for a feedback rate close to the one that leads to a minimum in the curves of figs. 2 (a1)-(a2), which corresponds to a particularly adapted situation to conceal the time-delay signature, since the two timescales equivalently drive the dynamics of the intensity time series. The exponentially-damped oscillations of the ACF mask the time-delay signature if τ and τ RO are sufficiently close. Indeed, the peaks and valleys are temporally located at multiples of τ RO and multiples of τ RO /2, respectively, and no signature of the time delay remains, by contrast to the situation where the difference between τ RO and τ was larger [Fig. 1

(b1)-(c1)].

IV. TIME-DELAY IDENTIFICATION: A DYNAMICAL POINT OF VIEW

The previous section has shown that the identification of the time-delay in the chaotic output of a laser with optical feedback strongly depends on the operational parameters of the ECSL: the feedback rate γ, the pumping current pJ th , and the value of τ relatively to τ RO . At high feedback rates, the estimators always possess a predictable behavior: a pulse-like shape with a clear signature of τ . However, when the feedback rate is weak, a large variety of behaviors has been reported and detailed in the previous section. These behaviors are largely dependent on the choice of p and τ relatively to τ RO .

These first results suggest a closer analysis of the timescales involved in the early stage of the laser chaotic dynamics, at small feedback rates. Different dynamical scenarios are expected depending on the ratio between τ and τ RO , which then also lead to the different time-delay identification scenarios we reported in the previous section.

A. Influence of the frequency generation on the time-delay signature 1) Disparate timescales scenario:

In this section we analyze the ECSL dynamics when the feedback rate γ is taken as the bifurcation parameter, all the other parameters remaining identical to the ones of Fig. 1. Figure 4(a) shows a bifurcation diagram of the laser intensity. A quasi-periodic (QP) route to chaos is observed. A projection of the system trajectory in the phase plane (N, E), the power spectrum, the ACF and the DMI, are plotted in Fig. 5 for specific values of the feedback rate. As we increase the feedback rate, the stationary solution of the ECSL is destabilized through a first Hopf bifurcation (H1) which induces a time-periodic dynamics [Fig. 5(a1)], with a frequency f H1 = 1.32 GHz almost equal to the relaxation-oscillation frequency f RO = 1/τ RO = 1.33 GHz [Fig. 5(b1)]. The intensity I(t) being time-periodic, the ACF and the DMI exhibit an oscillating shape at the frequency of the limit cycle oscillation. In the ACF, the maximum correlation Γ (θ) = 1 and anti-correlation Γ (θ) = -1 are reached at θ ≈ kτ RO and θ ≈ kτ RO /2, respectively. The DMI presents two series of peaks with different amplitudes: one located at θ ≈ kτ RO and the other at θ ≈ kτ RO /2.

Increasing the feedback rate, the limit cycle destabilizes to a quasiperiodic dynamics [Fig. 5 Although the frequency content of the chaotic output is broad, the ACF and DMI functions still retain signatures of the first time-scales that appear in the bifurcation cascade: the undamped relaxationoscillation period τ H1 ≈ τ RO that appears after the first Hopf bifurcation and the time delay τ that appears when crossing a torus bifurcation to quasiperiodic dynamics. The undamped relaxation oscillations are responsible for the oscillatory shape of the ACF and the DMI functions, and then the time delay time-scale appears from a bifurcation to quasiperiodicity and slowly modulates the shapes of the ACF and DMI functions.

The fast oscillations at the first Hopf bifurcation, close to RO frequency, perturb the identification of the time-delay at weak feedback rate, close to the onset of chaos. Only high feedback rates sufficiently weaken the relaxation-oscillation signature, namely short-time correlation, and offers a clear time-delay signature.

2) Close timescales scenario: Similarly, the influence of the bifurcation cascade is investigated for close values of the two timescales τ and τ RO . In this case, the analysis is performed using parameters identical to the ones in Fig. 3. The bifurcation diagram in Fig. 4(b) shows a period-doubling (PD) route to chaos. We observe in this case that the frequencies generated by the ECSL nonlinear dynamics also significantly influence the shape of the estimators. However, unlike the previous case, the time-delay does not manifest itself in the early stages of the dynamics.

As previously, the progressive increase of feedback rate leads to a time-periodic dynamics with a frequency f H1 = 1.34 GHz [Fig. 6 The ACF function exhibits fast oscillations at the period τ H1 = 1/f H1 but the correlation is lost on longer timeextension. In this identification scenario, and by contrast to the previous case, the time delay time-scale is not present in the early stages of the ECSL nonlinear dynamics. This prevents its clear signature to appear in the estimators at Fig. 5. Projection of the attractor in (N, E) plane (1st column), Power spectrum (FFT of |I(t)| 2 ) (2nd column) , autocorrelation function (3rd column) and mutual information (4th column) for increasing value of the feedback rate γ = 0.35GHz (1st row), γ = 0.55GHz (2nd row), γ = 0.85GHz (3rd row), γ = 1GHz (4th row) with a time-delay value τ = 5ns and τ RO = 0.75ns. The vertical dotted-dashed and dashed lines give the time location of τ RO and τ , respectively. low feedback rates. Both the ACF and the DMI give only information concerning the value of the undamped relaxationoscillation period through an oscillating behavior.

In conclusion, we have shown that the cascade of bifurcations is responsible for the observed behavior of the estimator. The first frequency appearing in the ECSL leads to oscillations in both the ACF and DMI. Depending on the choice of the parameters, the time-delay signature may or may not appear early in the first stage of the ECSL nonlinear dynamics [Fig. 5(b2)]. This presence influences the security of the ECSL cryptosystem in terms of time-delay identification. However, the early appearance of τ is not the only aspect to consider: the ACF and DMI starting to oscillate at the period of the undamped relaxation oscillation τ H1 , it is of importance to also study the particular role of this frequency and its remaining presence as the ECSL undergoes its cascade of bifurcations.

B. Influence of the first Hopf frequency and diversity of timedelay-identification scenarios

The undamped relaxation-oscillation period τ H1 emerging from H1 is the first timescale that appears in the ECSL. This makes f H1 = 1/τ H1 ≈ f RO play the role of a fundamental frequency of the ECSL. The presence of this fundamental frequency seems to be systematically responsible for the oscillating behavior of the estimators and can blur or even mask the time-delay signature [Figs. 1 and3]. However, in this section, we will underline two important facts: first, the frequency f H1 can be significantly shifted from f RO ; second, the frequency emerging from H1 in the ECSL will not always play the role of the fundamental frequency responsible for the oscillating behavior of the ACF and DMI.

1) Frequency shift and influence on the time-delay identification:

In the previous cases of Figs. 1 and3, f H1 was found to be extremely close to the value of f RO . We have made in Fig. 7 a more systematic analysis of the first Hopf bifurcation frequency f H1 as a function of the time delay τ . The evolution of f H1 with τ is not monotonic; it periodically oscillates around a value corresponding to f RO (dashed line) and the period of oscillation is close to τ RO . A similar conclusion has been obtained for other sets of parameters in previous ECSL studies [START_REF] Murakami | Stability analysis of semiconductor laser with phase-conjgate feedback[END_REF]- [START_REF] Masoller | Effect of the external cavity length in the dynamics of a semiconductor laser with optical feedback[END_REF]. This property can be used to increase the security of the ECSL. Indeed, the oscillating behavior of the ACF and DMI is characteristic of the presence of the undamped relaxation-oscillation timescale. Our previous choice of the parameters τ = 1.2 ns or τ = 5 ns and τ RO = 0.75 ns has led to values of f H1 both close to f RO . Now if we choose a value of the time delay τ close to τ RO , the fundamental frequency at which the ECSL will oscillate will be significantly shifted from f RO ; when combined with a low feedback rate, we expect a damping oscillating behavior of both the ACF and DMI at a time-scale that corresponds neither to τ nor to τ RO , hence preventing a good time-delay identification.

2) Diversity of time-delay-identification scenarios:

In the previously investigated cases, the shape of the estimator (ACF or DMI) unveils the influence of the first timescale appearing in the ECSL, namely the frequency f H1 . However it appears that the frequency issued from the first Hopf bifurcation does not systematically play the role of the fundamental frequency that drives the shape of the ACF or DMI for larger feedback rates. The following example taking τ RO = 0.75 ns and τ = 0.85 ns illustrates this fact. Figure 4(c) shows the corre- and second make the ECSL locks on a period-doubled limit cycle with fundamental frequency f = 0.88 GHz [Fig. 8(b3)]. This induces new oscillating behaviors for ACF and DMI with fundamental frequencies largely shifted either from f H1 or f RO [Fig. 8(c2)-(c3)]. Finally, the system enters in a weakly developed chaotic regime that inherit the spectral contents of the last stable attractor [Fig. 8(b3)-(b4)]. This explains also the inherited shape of the ACF between Fig. 8(c3) and (c4). In Fig. 8(c4), the ACF exhibits a largely shifted peak issued from the fundamental frequency of the last stable limit cycle [Fig. 8(a3)] which can be considered as a possible timedelay signature. In conclusion, we have shown that the first Hopf frequency is not systematically responsible for the fast oscillating shape of the ACF or DMI. We have also illustrated, again, the key role of the other frequencies born from the cascade of bifurcation occurring in the ECSL: they shape in a non-trivial way the ACF and the DMI, and are responsible for blurring the time-delay signature. The resulting identification scenario is not completely similar to the previous ones in Figs. 5 and6. This different identification scenarios take their origin in the diversity of routes to chaos existing in the ECSL, which influence directly on the laser time-series and shape the estimators accordingly.

V. DISCUSSION

In this section, we discuss the robustness of the time-delay concealment with respect to other identification techniques and to the influence of laser spontaneous emission noise and gain compression coefficient, as well as potential applications of our results for chaos-based communications.

A. Robustness of time-delay concealment to other identification techniques

Numerous methods exist to identify the time-delay, besides the well known ACF and DMI. The most common ones are: the filling factor analysis [START_REF] Bünner | Recovery of the timeevolution equation of time-delay systems from time series[END_REF], the local linear models (LLMs) [START_REF] Hegger | Identifying and modeling delay feedback systems[END_REF], global nonlinear models such as those obtained with modular neural networks [START_REF] Ortin | Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction[END_REF], and statistics of time-series extrema [START_REF] Bezruchko | Reconstruction of time-delay systems from chaotic time series[END_REF]. It is however interesting to analyse whether these time-delay-identification techniques would allow to retrieve any information on the time-delay value.

We present here a time-delay extraction based on modular neural networks (MNN). In order to mimic the structure of the equation driving the evolution of the light intensity I(t), the MNN incorporates two modules, one for the non delayed part (ND) of I(t) and a second one for its delayed part (D). A feed-forward neural network [START_REF] Ortin | Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction[END_REF] is used for each of the modules. The first module is fed with input data: x N D = [I(t -τ e ), ..., I(t -m 1 τ e )] and the second module is fed with

x D = [I(t -θ -m 2 τ e ), ..., I(t -θ), ..., I(t -θ + m 2 τ e )].
The parameter τ e is the embedding time, θ is a time-delay candidate, m 1 is the number of inputs of the ND part, and 2m 2 + 1 is the number of required inputs for the delayed part [START_REF] Hegger | Identifying and modeling delay feedback systems[END_REF]. The output of the neural network is

x nn = f (x N D ) + h(x D ), (6)
where f and h are nonlinear functions associated to ND and D respectively. The forecast error of the MNN is defined by σ(θ) = I(t) -x nn , and is a function of θ. If there exists a value θ 0 such that σ(θ 0 ) is minimum, corresponding to a valley in the forecast-error curve, then θ 0 will be considered a time-delay estimation. We apply the MNN technique to the intensity time series corresponding to the operational parameters τ = 1 ns, τ RO = 0.75 ns, and γ = 2 GHz. In previous sections, we have shown that such a set of parameter leads to a chaotic laser output from which it is not possible to retrieve the information on the time-delay using ACF or DMI. Figure 9 presents the results of a time-delay extraction based on a MNN composed of 6 neurons in the first layer and 3 neurons in the second layer for the delayed module and only one neuron for the non delayed module. The plot of the forecast error reveals only a minimum for values θ close to 0, that corresponds to the linear correlation time [START_REF] Ortin | Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction[END_REF]. We do not observe, however, any other minimum in the vicinity of the time delay τ . As a consequence, the time delay cannot be inferred from the intensity time-series using MNN.

The other identification techniques described above have also been tested and do not give more insight about the timedelay value. However, it is interesting to note that the statistics of time-series extrema is among all methods the one that identifies the time delay of the system for the smallest value of the feedback rate γ.

In conclusion, we have shown that independently from the method of identification, if the operational parameters of the ECSL are carefully chosen, the time-delay signature remains hidden.

B. Influence of internal parameters 1) Influence of the spontaneous-emission noise:

In our study, we have considered that the ECSL is fully deterministic; as a consequence, a description based on a deterministic delaydifferential equation was sufficient. However, in real lasers, stochastic processes, such as spontaneous emission, occur in the gain medium. Such phenomena therefore require an adapted model. The influence of the spontaneous-emission noise is modelled by a Langevin force and a deterministic component added to the equation describing the time evolution of the complex electric field E (t) [see Eq. ( 1)]. This additional stochastic part of the LK equations blurs the cascade of bifurcations that has proven to be directly responsible, in a fully deterministic approach, for the blurred time-delay signatures observed in both the ACF and DMI. Indeed, when noise is present at very weak feedback rates, its contribution in the ECSL dynamics is comparable to that of the delayed optical feedback term. The Langevin force F L (t) = 2βN (t)ζ (t) acts as a driving force for the dynamics that weakly excite the intrinsic nonlinearity of the ECSL. This stochastic excitation, only visible at low feedback rate, does not influence the time-delay identification: the signature is still blurred. For large values of feedback rates, the contribution of the noise is negligible in comparison with the delayed-feedback term. This implies that the time-delay signature remains identical to the one in Fig. 1(b4)-(c4). Figure 10 shows this result by comparing the evolution for increasing feedback rate strength of a model of ECSL with and without spontaneous-emission noise. In conclusion, the presence of noise in the equation does not affect the time-delay identification, and the results of section III remain true.

2) Influence of gain saturation: Gain saturation is phenomenologically introduced in the rate equations by considering an explicit intensity dependence of the gain. Its value is typically smaller or equal than 2.5 × 10 -23 m 3 . It has been reported elsewhere that the saturation gain has a stabilizing effect on the ECSL dynamics: the progressive increase of gain saturation leads to an increasing feedback-strength threshold above which the transition to chaotic regime occurs [START_REF] Masoller | Comparison of the Effects of Nonlinear Gain and Weak Optical Feedback on the Dynamics of Semiconductor Lasers[END_REF].

This stabilizing effect of gain saturation is due to the damping effect of the intrinsic nonlinear part of rate equation [START_REF] Lenstra | Coherence collapse in single mode semiconductor lasers due to optical feedback[END_REF]. As a consequence, reducing the value of ε will increase, in the ECSL dynamics, the driving action of G N,|E| E (t) relatively to the one of γe iω 0 τ E (t -τ ), blurring the timedelay signature compared to fast timescales emerging from the interplay of the laser intrinsic nonlinearities and the optical feedback. From a quantitative point of view, a decrease of the gain saturation has two simultaneous manifestations: first, the amplitude of the extremum located around the timedelay value decreases, second fast oscillations appearing in the vicinity of the time delay and leads to a blurred timedelay signature. These two combined effects also allow for an increase of the time separation between τ and τ RO that maintains the time-delay concealment. It also allows for the use of higher values of pumping current pJ th . Figure 11 presents these results: three different identification scenarios based on the ACF are considered for various choices of τ , τ RO and γ for a value of the gain saturation ε that progressively increases. Interestingly, the first row shows, at a fixed feedback rate, that for a large time separation between τ and τ RO , where it may be expected to result in a blurred signature [Fig. 11(d4)], a progressive tendency to the disappearance of the time-delay signature is observed [Fig. 11(d1)]. These results hold for the various parameters used in Fig. 11. In conclusion, small values of the gain saturation favors the masking of the time-delay by emphasis the contribution of the intrinsic nonlinearity in the intensity time-series. They furthermore allow for extending the range in which the timedelay identification is difficult both in terms of feedback rate and separation of time-scales.

C. Application to chaos-based communications

The security in terms of time-delay identification appears as a requirement for the security of chaos-based communication setups involving physical devices such as ECSLs. However, it needs to be compatible with a sufficient chaos complexity as well as good synchronization, to ensure secure communications at low bit-error rates (BER). Indeed, it is important to investigate if the signal generated, with a good time-delay concealment, using our set of parameters, can possibly lead to secure transmission that can be recovered at the receiving end using chaos synchronization. Typically, the ECSL unidirectional coupling is used to form a chaotic communication scheme [START_REF] Mirasso | Synchronization of chaotic semiconductor lasers: Application to encoded communications[END_REF]. Two types of synchronization exist in such a scheme: the so-called injection-locking synchronization (generalized synchronization) and complete synchronization [START_REF] Locquet | Synchronization regimes of optical-feedback-induced chaos in unidirectionally coupled semiconductor lasers[END_REF]. The first regime occurs when the master and slave ECSLs have identical optical feedback strengths (γ m = γ s ), whereas the second regime is achieved for a master optical feedback equal to the sum of the slave optical feedback with the coupling strength (γ m = γ s + γ c ), providing the condition that γ c > γ s . The measurement of the quality of synchronization is based on the cross-correlation function defined by

Γ X m ,X s (θ) = (X m (t) -µ Xm ) (X s (t + θ) -µ Xs ) σ Xm σ Xs , with µ X m,s = X m,s and σ X m,s = X m,s -µ X m,s 1/2 .
With the set of parameters of Fig. 3(b2)-(c2), a good timedelay concealment is achieved, but it leads to a poor quality of anticipating synchronization with max θ |Γ Xm,Xs (θ)| ≈ 0.2. This result was expected because of the use of weak feedback rate in the master ECSL. In order to achieve complete synchronization the coupling strength must be equal to the feedback rate and must therefore take a value less than 5 GHz.

In the case of injection-locking synchronization, by contrast, the coupling strength γ c is no longer limited by the value of the master optical feedback γ m . Numerical simulations show that as soon as the coupling strength satisfies γ c ≥ 25 GHz we achieve very good synchronization: max θ |Γ X m ,X s (θ)| ≥ 0.95. Hence, injection-locking synchronization can be used to transmit information securely, when the parameters are taken to ensure time-delay concealment, whereas complete synchronization is not adapted.

VI. CONCLUSION

In this paper, we have analyzed the security of an ECSL in terms of time-delay identification using standard time-delay estimators such as the autocorrelation function (ACF) and the delayed mutual information (DMI). The key role of the feedback rate γ and the pumping factor, as well as the choice of the time-delay (τ ) value in comparison with the relaxationoscillation period τ RO , has been underlined. It appears that the difficult identification scenarios occur for relatively weak feedback rates and weak pumping factors with close values of the two timescales τ and τ RO . These difficult identifications find their origin in the specific nonlinear dynamics and timescales appearing in the ECSL in its bifurcation cascade leading to chaos. Indeed, at weak feedback rates the chaos keeps reminiscence of the time-scales involved in the early stage of the laser dynamics, such as the undamped relaxation oscillation time-scale and possibly period-doubling and quasiperiodic dynamics. The time-delay estimators then exhibit complex modulated shapes showing these different laser dynamics timescales. When τ and τ RO are close to each other either, the true value τ is efficiently concealed thanks to a shift of the first Hopf frequency f H1 with respect to the relaxation oscillation frequency, such that the laser output at the Hopf frequency starts pulsating at a frequency which is neither close to τ nor to τ RO . The impact of additional laser internal parameters is also investigated. It appears that the decrease in gain saturation coefficient allows to use more distant values of τ and τ RO as well as to increase the pumping factor p while maintaining time-delay concealment. The robustness of our results has been checked with other signal processing techniques such as neural networks and filling-factor methods. We expect our results to be of interest for a proper design of a laser chaotic emitter that would allow for the best concealment of its system parameters, hence also improving security in laser chaos-based communications.

PLACE PHOTO HERE

Marc Sciamanna graduated in electrical engineering from the Facult Polytechnique de Mons, Belgium, in 2000. He received the Ph.D. degree from Facult Polytechnique de Mons, as a Research Fellow of the "Fonds National de la Recherche Scientifique" (FNRS). His Ph.D. dissertation was entitled "Nonlinear dynamics and polarization properties of externally driven semiconductor lasers." Currently, he is a permanent Researcher, Lecturer and faculty staff member of SUPELEC (Ecole Suprieure dElectricit), Metz, France, and LMOPS CNRS UMR-7132 Laboratory, Metz, France. His research interests include the physics and nonlinear dynamics of semiconductor lasers, the polarization properties of VCSELs, the study of dynamical instabilities related to optical feedback, optical injection, or large current modulation, synchronization and chaotic encryption using laser diodes. He is author of about 50 research papers in international journals and conference proceedings. He has given several invited talks in international conferences and has worked as a scientific program committee member in Germany (19921993), where he worked on exciton radiative decay in low-dimensional semiconductor structures. From 1993 to 1995, he was Center Fellow at the Center for Ultrafast Optical Science at the University of Michigan, Ann Arbor, where his work addressed ultrafast phenomena in quantum wells. He was an Assistant Professor of Physics at Washington State University from 1995 to 2001, at which time he joined the faculty at the Georgia Institute of Technology, Atlanta, where he is currently a Professor in the School of Electrical and Computer Engineering. His interests are in ultrahigh-speed devices, nanoscale engineering, and terahertz technology. The present focus of his group is the identification and evaluation of novel nonlinear and ultrafast optical processes in various materials and nanostructures for applications in optoelectronics and photonics. His group also works on photonic crystals and on terahertz sources and sensors. Dr. Citrin has served as an Associate Editor for the IEEE JOURNAL OF QUANTUM ELECTRONICS.

PLACE PHOTO HERE

Silvia Ortin Silvia Ortin graduated in electrical engineering from the University of Cantabria, Spain, in 2002. She is currently working toward the Ph.D. degree at the Institute of Physics of Cantabria (IFCA). Her research interests include optical chaos communications, chaotic time series analysis and nonlinear dynamics.

Fig. 1 .

 1 Fig. 1. ECSL intensity time series (1st column), Autocorrelation function (2nd column) and mutual information (3rd column) for increasing value of the feedback rate γ = 2GHz (1st row), γ = 5GHz (2nd row), γ = 10GHz (3rd row), γ = 15GHz (4th row) with a time-delay value τ = 5ns and τ RO = 0.75ns. The vertical dotted-dashed and dashed lines give the time location of τ RO and τ , respectively.

Fig. 2 .

 2 Fig. 2. Impact of the feedback variation and pumping factor on the amplitude and time location of the most significant peak in the vicinity W (τ ) = [4.5ns, 5.5ns] of the time delay τ = 5 ns. (a1)-(b1) gives respectively the amplitude and time location of max θ∈W (τ ) |Γ(θ)| (a2)-(b2) gives respectively the amplitude and time location of max θ∈W (τ ) |H(θ)|. The solid lines with triangle ( ), round (•), and square ( ) markers stands for p = 1.05, p = 1.26 and p = 1.72 , respectively. These three different values of p, correspond to the following relaxation oscillation periods: τ RO = 0.75 ns, τ RO = 0.33 ns and τ RO = 0.2 ns, respectively. In sub-figures (b1)-(b2), the dotted-dashed lines give the time location of the time-delay τ .

  (a2)]. The power spectrum shows the appearance of new frequencies [Fig. 5(b2)]. Amongst them, one strong frequency component appears separated from f H1 by ∆f = 0.19 GHz very close to the external-cavity frequency f EC = 1/τ = 0.2 GHz. This new frequency component induces a slow undamped periodic modulation of both the ACF and DMI [Fig. 5(c2)-(d2)]. Further increase of the feedback rate is accompanied by the appearance of numerous new frequencies [Fig. 5(b3)] that increase the attractor complexity [Fig. 5(a3)]. However, the strong frequency components are still located at the frequencies f H1 and f H1 -∆f , which guarantees the persistence of a global order of the time series on a long time-extension even if on a short time-extension the complexity (disorder) is increased. As a consequence, the ACF and DMI show a stronger modulation, comparing to Fig. 5(c2)-(d2). This is due to the local increase of complexity, although periodically the ACF still attains both maximum correlation and anticorrelation [Fig. 5(c3) and (d3)]. For still larger feedback rate the laser exhibits fully developed chaos [Fig. 5(a4)]. The intensity time series progressively looses correlation as time increases, which corresponds to damp the modulated shape of the ACF and DMI [Fig. 5(c4) and (d4)].

Fig. 3 .

 3 Fig. 3. ECSL intensity time series (1st column), autocorrelation function (2nd column) and mutual information (3rd column) for increasing value of the feedback rate γ = 2GHz (1st row), γ = 5GHz (2nd row), γ = 10GHz (3rd row), γ = 15GHz (4th row) with a time-delay value τ = 1.2ns and τ RO = 0.75ns. The vertical dotted-dashed and dashed lines give the time location of τ RO and τ , respectively.

Fig. 4 .

 4 Fig. 4. Bifurcation diagrams with the feedback rate γ taken as the bifurcation parameter. It shows various routes to chaos depending on the choice of τ and τ RO . (a) quasi-periodic (QP) route to chaos for τ = 5ns and τ RO = 0.75ns. (b) period-doubling (PD) route to chaos for τ = 1.2ns and τ RO = 0.75ns. (c) complex route to chaos for τ = 0.85ns and τ RO = 0.75ns

  (b1)] close to f RO . This time-periodic dynamics induces an oscillating behavior of the ACF [Fig. 6(c1)] and DMI [Fig. 6(d1)] similar to Fig. 5(c1) and (d1). When further increasing the feedback rate, the laser exhibits a period doubling bifurcation, which yields a new frequency at f H1 /2 [Fig. 6 (b2)]. The coexistence of these two timescales modulates the shape of the estimators [Fig. 6(c2)-(d2)], making 2τ H1 = 2/f H1 and its multiples the time-locations of the strongest contributions in the time-delay estimators. Further increase of the feedback rate leads to the appearance of many frequencies in the ECSL power spectrum, and has two effects on the estimators: the global decrease of the amplitude of the oscillations of the ACF and DMI, and a stronger modulation [Fig. 6(b3)-(c3)] due to similar reasons as the ones in Fig. 5(c3)-(d3). The appearance of the chaotic regime produces exponential damping of both the ACF and DMI [Fig. 6(c4)-(d4)]. Even in the chaotic regime the strongest frequency component is still found at the frequency f H1 = 1.34 GHz.

Fig. 6 .

 6 Fig.6. Projection of the attractor in (N, E) plane (1st column), power spectrum (FFT of |I(t)| 2 ) (2nd column) , autocorrelation function (3rd column) and mutual information (4th column) for increasing values of the feedback rate: γ = 0.6GHz (1st row), γ = 0.8GHz (2nd row), γ = 1.2GHz (3rd row), γ = 1.5GHz (4th row) with a time-delay value τ = 1.2ns and τ RO = 0.75ns. The vertical dotted-dashed and dashed lines give the time location of τ RO and τ , respectively.

Fig. 7 .

 7 Fig. 7. Evolution of the first Hopf Frequency f H1 as a function of the time delay τ , for τ RO = 0.75ns. The horizontal and vertical dotted-dashed lines represent respectively the relaxation oscillation frequency f RO = 1/τ RO , and multiples of the relaxation-oscillation period τ RO .

Fig. 8 .

 8 Fig. 8. Projection of the attractor in (N, E) plane (1st column), power spectrum (FFT of |I(t)| 2 ) (2nd column) , autocorrelation function (3rd column) and mutual information (4th column) for increasing values of the feedback rate: γ = 1GHz (1st row), γ = 1.2GHz (2nd row), γ = 1.4GHz (3rd row), γ = 1.6GHz (4th row) with a time-delay value τ = 0.85 ns and τ RO = 0.75 ns. The vertical dotted-dashed and dashed lines give the time location of τ RO and τ , respectively.

Fig. 9 .

 9 Fig. 9. Forecast error produced by the MNN on the intensity time series I(t) generated with the following parameters: τ = 1 ns, τ RO = 0.75 ns and γ = 2 GHz. The identification time-step is 5 ps. The vertical dotted-dashed and dashed lines give the time location of τ RO and τ , respectively.

Fig. 10 .Fig. 11 .

 1011 Fig.10. Influence of the rate of spontaneous emission β on the time delay identification. The first row corresponds to the case with the spontaneous emission noise β = 10 3 s -1 , the second row corresponds to the case β = 0 s -1 . Each column is associated to a given feedback rate. From left to right, the feedback rates are: γ = 2.5 GHz, γ = 5 GHz, γ = 10 GHz and γ = 15 GHz. The time delay and the relaxation period are equal to τ = 5 ns and τ RO = 0.75 ns. They are represented by the dotted and dotted-dashed lines, respectively.

  SPIE Photonics Europe 2006. He organized and cochaired the PHASE international workshop (PHysics and Applications of SEmiconductor LASERs), Supélec, Metz, France, in 2005; and acted as a Guest Editor of a special issue of Optical and Quantum Electronics related to the PHASE workshop. Dr. Sciamanna acted as a Board Member of the IEEE/LEOS Benelux Chapter, co-founded the IEEE/LEOS Benelux Student Chapter and co-chaired the IEEE/LEOS Workshop on Low-Cost Photonics (Mons, Belgium, June 13, 2003). He is Vice-Chairman of working group 2 "Physics of Devices" in European Action COST 288. He has been awarded with the Prize for Best Engineer in Electricity from Facult Polytechnique de Mons (2000), the OSA-Newfocus student travel grant award (2002), the IEEE/LEOS Graduate Student Fellowship Award (2002), and the SPIE F-MADE Scholarship Award in Optical Engineering (2003). David S. Citrin (M'93-SM'03) received the B.A. degree is in physics from Williams College, Williamstown, MA, in 1985, the M.S. and Ph.D. degrees in physics from the University of Illinois at Urbana-Champaign, in 1987 and 1991, respectively, where he worked on the optical properties of quantum wires. After receiving the Ph.D. degree, he was a Postdoctoral Research Fellow at the Max Planck Institute for Solid State Research, Stuttgart,

	PLACE
	PHOTO
	HERE

ACKNOWLEDGMENT

The authors would like to thank L. Pesquera for useful discussions and suggestions. The authors acknowledge the support of Conseil Régional de Lorraine and Fondation Supélec. The work of D. S. Citrin was supported in part by the National Science Foundation by grant ECCS 0523923.


D Rontani
email: damien.rontani@supelec.fr

A Locquet

M Sciamanna

D S Citrin

Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback

We investigate theoretically the possibility of retrieving the value of the time delay of a semiconductor laser with an external optical feedback from the analysis of its intensity time series. When the feedback rate is moderate and the injection current set such that the laser relaxation oscillation period is close to the delay, then the time-delay identification becomes extremely difficult, thus improving the security of chaos-based communications using external-cavity lasers. c 2007 Optical Society of America OCIS codes: 140. 1540, 190.3100 Chaotic cryptosystems exploit the noise-like appearance of a chaotic carrier to conceal an information-bearing message and the property of chaos synchronization [1] to decrypt 1 this message at the receiving end. The security of a chaotic cryptosystem is twofold: first, it should be difficult to identify the emitter parameters and decipher the message by analyzing the transmitted signal, and second, even knowing the parameters, the physical realization of a synchronizing receiver should be arduous.

Semiconductor-based external-cavity lasers (ECLs) have proven to be a particularly interesting source of optical chaos [2]. Indeed, the delay introduced in the system can lead to the generation of high-dimensional chaos [3], which makes it computationally difficult to identify the system with traditional time-series-analysis techniques. Recently, some techniques tailored for delay systems have been developed [4,5] which lead to computationally feasible system identification by working in a low-dimensional projection of the infinite-dimensional phase space. If the delay is unknown, however, it is impossible to perform such a projection and therefore to identify the system. Previous works on time-delay identification in ECLs have shown that the delay was easily retrievable from the autocorrelation function of the laser light output [6,7]. The use of additional cavities has later been suggested [7] as a way to complicate the retrieval of information about the time delay; however, this makes practical implementation difficult.

In this Letter, we consider a single-mode semiconductor laser with a single external optical feedback. With this simple system, we show that for a careful choice of the operational parameters, it becomes extremely difficult to identify the time delay with the autocorrelationfunction and mutual-information techniques, both calculated from the intensity time series.

The presence of two timescales, namely the relaxation-oscillation period τ RO and the external-cavity round-trip time τ , combined with appropriate values of the feedback rate γ, allows this masking. Our theoretical analysis is based on the Lang-Kobayashi (LK) rate-equation model [8],

Ė = 1 2 (1 + iα) G - 1 τ p E + γe iω 0 τ E (t -τ ) , (1)
Ṅ = pJ th - N τ s -G |E| 2 , ( 2
)
where E is the slowly-varying complex electric field, N is the average carrier density in the active region, α is the linewidth-enhancement factor,

G = G N (N -N 0 )(1 -ε|E| 2 )
is the optical gain with ε the saturation coefficient, N 0 is the carrier density at transparency, ω 0 is the frequency of the solitary laser, γ is the feedback rate, τ p is the photon lifetime, τ s is the carrier lifetime, J th is the threshold current density, p is the pumping factor, and τ is the delay. The relaxation oscillation period is an intrinsic damping time of the free running laser and is defined by

τ RO = (1/τ s τ p (μ -1) -μ 2 /4τ 2 s ) -1/2 , with μ = G N τ s τ p (pJ th -N 0 /τ s ).
We consider the following parameter values: α = 5, ω 0 τ = 0,

τ p = 2 ps, τ s = 2 ns, G N = 7.5 × 10 -13 m 3 s -1 , N 0 = 3 × 10 24 m -3 , ε = 2.5 × 10 -23 m 3 , J th = 1.83 × 10 33 m -3 s -1 , τ RO = 0.
75ns . For a system modelled by a delay-differential equation, the information about the delay value can typically be retrieved using either the autocorrelation function or the mutual information [9]. The autocorrelation function is a measure of how well a signal matches a time-shifted version of itself and is defined by

ρ I (θ) = (I (t + θ) -I (t) ) (I (t) -I (t) ) I (t) -I (t) 2 I (t + θ) -I (t) 2 1/2
where I = |E| 2 denotes the emitted laser intensity.

3

The mutual information measures the information shared by two variables, namely, the intensities at times t and t + θ, averaged over t. It measures how much knowing one of these variables reduces our uncertainty about the other. We consider here the mutual information I(θ) between I(t) and I(t -θ) which is defined by

I(θ) = I(t),I(t-θ) p (I(t), I(t -θ)) log p (I(t), I(t -θ)) p (I(t)) p (I(t -θ))
where p (I(t), I(t -θ)) is the joint probability, and p (I(t)) and p (I(t -θ)) are the marginal probability densities.

We now consider scenarios with various values of τ all leading to a chaotic behavior and we investigate circumstances under which the time delay is effectively concealed in the c1) have the general shape described in the literature [7]: a central peak located at θ = 0 with a small temporal extension, and significant peaks at the delay and its multiples. These peaks are surrounded by small troughs at multiples of ±τ RO /2. The decrease of γ induces a decrease of the peaks amplitude at the delay and its multiples; meanwhile, a significant trough at τ RO /2 arises. For a smaller value of γ, we observe a decrease of the amplitude of the peaks at the delay and a significant peak appears at τ RO (Fig. 1.(b1)). For an even smaller value of the feedback rate (Fig. 1.(a1)), peaks appear at multiples of τ RO , revealing the importance of the relaxations oscillations in the dynamics. The presence of a second timescale due to the delayed feedback is revealed by the increase of the peaks value around τ and its multiples. In this latter case, delay identification is quite difficult due to the numerous satellite peaks present around τ . Additionally, we expect that the delay will be significantly overestimated as the largest peak around τ corresponds to the multiple of τ RO closest to the delay rather than to the delay itself.

1(a1)-(d1) and unveils the existence of a minimum for a given feedback rate γ.

These observations reveal the increasing reliability of τ W with increasing values of γ, because the choice of the peak in the autocorrelation becomes more obvious and simultaneously the delay overestimation decreases. Consequently, the system becomes less and less secure, regarding system identification. It is also known that increasing γ leads to an increase of the Kaplan-Yorke dimension in a LK system [3]. Our result shows that the high dimensionality of the chaos is not a sufficient condition for security, as delay identification becomes easier when the feedback rate, and thus the dimension, is increased.

From the first two figures, the relative importance of the two time scales in the autocorrelation function, controlled by the value of γ, has proven to be of key importance for the delay retrieval. The signatures of the delay and of the relaxation-oscillations period are always present in the autocorrelation function as illustrated in Fig. 1. At a moderate feedback rate (Fig. 1.(a)), there are numerous large peaks in the vicinity of θ = 0, corresponding to small multiples of τ RO /2, and similarly in the vicinity of τ , higher multiples of τ RO /2 are present. The presence of peaks at multiples of τ RO perturbs the delay identification but the significant separation between the values of τ and τ RO still allows a relatively clear identification of the delay value, as we observe a clear increase around the value of τ of peaks at high-order multiples of τ RO . We find that reducing the separation between τ and τ RO makes the identification problematic. The only significant peaks or troughs being located around τ RO /2, an eavesdropper would have an exceedingly difficult time to identify the delay. Finally Figs. 3.(a1) and (a2), which correspond to a feedback rate of 2.5 GHz, also show a case in which it is seems impossible to identify the delay as only a minor trough or peak is visible around the delay value and as an eavesdropper would have no indication that these correspond to the correct delay value.

A systematic numerical study shows that the delay is well concealed when τ belongs to the interval [τ RO + 0.15 ns, τ RO + 0.35 ns].

In conclusion, we have demonstrated, using autocorrelation and mutual-information techniques, the key role of the feedback rate γ and of the separation between the time delay τ and the period of the relaxation oscillation τ RO on the retrieval of the time delay in an ECL. Two scenarios have been identified. First, for low feedback rates and significant separation between τ and τ RO , the presence of peaks or troughs at multiples of τ RO /2 makes the identification difficult. Second, when the feedback rate is low and the values of τ and τ RO are close, delay identification is very difficult because no significant peak appears close to the delay value. These two situations can prevent an eavesdropper to access the true measure of the time delay.

  autocorrelation and mutual information. Usually, in delay systems, the signature of the time delay shows up in the autocorrelation function ρ I and the mutual information I because of the non-local time dependence induced by the delayed feedback term. Therefore, the information shared by I(t) and I(t -τ ) is intimately related to γ. Figures 1 and 2 analyze this influence.

Figure 1

 1 Figure 1 gives the shape of the autocorrelation function (left column) and the associated

Figure 2

 2 Figure 2 analyzes the evolution with γ of the location (τ W ) and the amplitude (ρ W ) of

Figure 3 3 .

 33 illustrates such a case using the autocorrelation function (left column) and mutual information (right column) considering τ = 1 ns. The ECL behavior is chaotic for the four increasing values of γ considered in Fig. Figures 3.(d1) and 3.(d2) confirm the results obtained in Fig. 1: sharp peaks are located at the delay and its multiples. As γ decreases from 15 to 10 GHz , the peak close to the delay value in both ρ I and I decreases (Figs. 3.(c1)-(c2)). A further decrease of γ to 5 GHz reveals very interesting plots of the autocorrelation function and the mutual information. In Figs.3.(b1) and 3.(b2), we do not observe any clearly identifiable peaks around the delay value.

Fig. 2 .

 2 Fig. 2. Evolution of the position and amplitude of the maximum autocorrelation peak for

Fig. 3 .

 3 Fig. 3. Autocorrelation function and mutual information for τ = 1 ns and τ RO = 0.75 ns. (a1)-

The authors acknowledge the support of Région Lorraine and fondation Supélec. D. S.

Citrin was supported in part by the NSF, by grant ECCS 0523923.


Jean-Philippe Jenny
email: jean-philippe.jenny@inrae.fr

C Niemann

P Francus

A Noren

N Carvalhais

Carvalhais A Meta

A meta-analysis of paleolimnological records reveals the sensitivity of lacustrine carbon burial rates to carbon sources and preservation conditions during the Anthropocene

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.

A meta-analysis of paleolimnological records reveals the sensitivity of lacustrine carbon burial rates to carbon sources and preservation conditions during the Anthropocene

Inland waters are sites of active carbon (C) processing and transport along the land to ocean aquatic continuum (LOAC) that need to be accounted for closing the global C budget1,2 . However, monitoring data are lacking and do not extend back as far as few decades, limiting our comprehension of the LOAC C cycle for the last centuries. Lake sediments provide a key archive for assessing C transport and transformation that occurs in lake catchments. Here, the analysis of large numbers of samples was performed on 420 lakes sediment records of the world to assess C burial rates and sources during the last 300 years. C and N (organic and mineral forms) on discrete samples were analyzed using a Variomax elemental analyzer to assess total C sequestration by lakes, C sources, and long-term changes in the contribution of allochtonous vs autochtonous sources to C transfers in lake-watersheds. Continuous sediment records were generated using core scanners (i.e. micro-XRF) and computed tomography to provide nearannual trends on terrigenous elements, here used as proxies of allochtonous sources (e.g. Al, Ti, K, Fe), and Mn:Fe ratio to infer past oxygen conditions3 . Our results suggest that establishing a morphology-relevant lake typology that better characterises the types and distribution of oxygen conditions and terrigenous supplies across our sites is therefore the first step in providing a more robust evidence base for explaining the spatial-temporal variation in lake C burial rates.

INRA, UMR CARRTEL, Université Savoie Mont Mont Blanc, Thonon les Bains, France,

Max Planck Institute for Biogeochemistry, Jena, Germany,

Centre Eau Terre Environnement, INRS, G1K9A9 Québec (Qc), Canada and GEOTOP Research Center, Montréal, (Qc), Canada,

National Lacustrine Core Facility (LacCore), Minneapolis, USA