Nikola Kutin
email: nikola.kutin@etu.univ-nantes.fr

Zakaria Moussa
email: zakaria.moussa@univ-nantes.fr

Thomas Vallée
email: thomas.vallee@univ-nantes.fr

Factors behind the Freight Rates in the Liner Shipping Industry

Keywords: Freight rates, Container trade, Shipping, Vector Autoregressive Model

This paper analyses the relationship between the China Containerized Freight Index (CCFI), Containership Earnings, Fleet Development, bunker price and Global Economic Activity. The Markov-switching Vector Autoregressive model has been applied by assuming the existence of two regimes. The first one is characterized by low and volatile freight rates, while the second one is more stable with high earnings and high freight rates. Three major cycles in the liner shipping industry have been identified. Moreover, by applying the Impulse Reponses Function, we have estimated the reaction of the freight rates following an increase of 1% of the other variables in the model.

Introduction

Maritime transport is of major importance for the world's economy as over 80% of the world's trade is carried by sea [START_REF] Unctad | Review of Maritime Transport[END_REF]. It is the cheapest and most efficient way to transport goods in large quantity. Since the 1960s, containerization has greatly reduced the expense of international trade and increased its speed, especially in consumer goods and commodities. It has also dramatically changed the ports' infrastructures and shipbuilding industry. Even though in 2016, containerized cargo accounted for only 16.7%1 of the total volume of transported goods, its value is more than 60%2 . According to data of Clarkson Research, since 1990, the containerized trade has increased by more than 600%.

However, the industry can be characterized by a high level of vulnerability and cyclic pattern. Following the Subprime crisis, in 2009 container prices fell by 14% [START_REF] Unctad | Review of Maritime Transport[END_REF] which was caused by the dramatic decrease of 10.8% of the cargo transported in containers. It was the first time since the invention of the containers that the World GDP grew faster than the World Seaborne Container Trade. Since then, the industry has been in an unstable state. In the last years, freight rates have been very low, ship values have plummeted and competition on the various trade routes has intensified [START_REF] Rex | Shipping market review, Danish ship finance[END_REF]. In February 2017, the 7 th biggest shipping company Hanjin Shipping declared bankruptcy. It is a clear sign that even the big players in the industry are threatened by the current situation. Prior to the subprime crisis, the price of oil was steadily increasing (Figure 1). On some major routes, shipping lines undertook different strategies with regard to the commercial speed of the fleet and the scale of the vessels [START_REF] Notteboom | The effect of high fuel costs on liner service configuration in container shipping[END_REF]. [START_REF] Notteboom | Fuel surcharge practices of container shipping lines: Is it about cost recovery or revenue making[END_REF] estimated that by increasing the speed by two knots on average the fuel cost increases by around 40%. The high fuel prices gave incentives to shipping companies to invest in bigger ships and increase the time of travelling by reducing the speed and deploying more ships on some maritime routes. After the world crisis and the slowdown of the world economy, the excessive capacity seemed to have been a major concern for the industry. As shown in Figure 1, global idle container ship capacity represented 11.4% in January, 2010 and despite the decreasing trend of the fleet growth, it has remained relatively high (5.64% in March, 2017).

Freight Rates in the containerized industry have a direct impact on the shipping lines decisions and world trade. They can be defined as the price for transporting a standard 20-foot container from one port of origin to a destination port. However, analyzing on a macro level the factors behind the freight rates is not an easy task. The complexity arises from the fact that a large number of parties, each one with different objectives, is involved in container shipping (that is consignor, a consignee, an ocean carrier, freight forwarders, inland carriers, banks, legal experts, insurance brokers, customs, port/terminal operators, and inland depot operators). [START_REF] Lee | Ocean container transport in global supply chains: Overview and research opportunities[END_REF] classified them into service users and service providers. In addition, Container rates are quoted as the price per box, "Freight all Kinds". This makes it even more complex to generalize how rates are structured spatially, because the tarification is not related to the product being transported as in some other trades [START_REF] Slack | Container freight rates and the role of surcharges[END_REF]).

Container freight rates can be broken down into three main components: Basic Ocean Freight (BAS), Mandatory surcharges and Value Added Services (VAS). BAS is mainly determined by the origin and destination of the cargo. Mandatory surcharges include a variety of charges that are added to BAS. For instance Bunker Adjustment Factor take into account the variation of the cost spent on fuel while Terminal Handling Charges are based on the cost of handling a container. Finally, VAS is related to the requirements of any particular client.

Examples of such services are container cleaning and cold treatment. According to Maersk Line, BAS accounts for 50%, Mandatory surcharges for 25% and VAS for 25%3 . [START_REF] Slack | Container freight rates and the role of surcharges[END_REF] found out that surcharges account for more than 50% in the total freight charged to customers.

The objective of this paper is to analyze the relationship between key variables in the liner shipping industry by taking into account its cyclical and nonlinear patterns. Firstly, we depict some key findings and gaps in the literature regarding the ocean freight rates. Secondly, the methodology of Markov-Switching Vector Autoregressive (MS-VAR) models and the Impulse Response function are presented along with detailed description of each one of the selected variables. Thirdly, a detailed explanation of the different regimes as well as their switching probability are explained. This section of the paper also describes the six economic cycles that have been identified and the impulse responses of China Containerized Freight Index (CCFI) following a shock of another variable in the model. Finally, a summary of the main findings and possibility of future research are provided.

Literature Review

Since the Subprime Crisis, freight rates in the liner shipping industry have been very volatile and many empirical studies have been conducted to understand its behavior. De Oliveira (2014) applied different methods of regressions, such as the random effects (RE) and fixed effects (FE), and the Hausman-Taylor (HT) estimator to analyze the freight rates. His study included 1,128 quotations for transporting a standard 20-foot container between six European countries and 47 partners. The independent variables in his model were the port-to-port distance, number of transshipments, economies of scale and Liner Shipping Connectivity Index. The results show that inward freight rates are on average, 23% higher than for outward ones, with the competition having a strong effect on the freight rates and surcharges appearing to be one of the main revenue-makers. [START_REF] Slack | Container freight rates and the role of surcharges[END_REF] analyzed the factors behind the freight rates in the case of 35 port destinations from Northern Europe. Their study shows that that distance is not a factor in explaining freight rates and the regional and temporal differences do not have any significant impact on the price of shipments. They also found that there is no correlation between the distance and BAF and no significant relationship between freight rates, imports and size of the large vessels. [START_REF] Behrens | Transportation, freight rates, and economic geography[END_REF] suggested that the impact of globalization as well as regional disparities on freight rates are overstated. The main reason for asymmetric freight rates is the presence of empty containers. [START_REF] Lun | Demand chain management in the container shipping service industry[END_REF]

used Path analysis in Structural Equation

Modeling (SEM) to analyze the causality between the following variables: container trade, total fleet, freight rate, new building vessel price, second-hand vessel price, and demolition vessel price. The data is weekly for a period of twelve years, collected from Clarkson Research. Results suggest that there is significant relationship between total Fleet size and seaborne trade and no link between freight rate and seaborne trade. [START_REF] Tran | An empirical study of fleet expansion and growth of ship size in container liner shipping[END_REF] applied multiple regression to analyze the costs and revenues of the top 25 Container Operators. They found that market freight rate has an effect on revenue and the capacity has an impact on the total cost of the operators. Ordinary Least Square method was used by [START_REF] Duru | Long term freight market index and inferences[END_REF] to investigate the factors behind the Long term composite Freight Index (LFI), created by them and the seaborne trade. Their study shows that life expectancy is significant for both the models while the effect of fleet size is very weak.

Considering the nonlinear relations between the different factors influencing freight rates and their interdependence, some authors have applied different autoregressive models. [START_REF] Veenstra | A co-integration approach to forecasting freight rates in the dry bulk shipping sector[END_REF] used the Multivariate Cointegration approach to analyze the relationship of six bulk freight rates related to different commodities and routes. A cointegration between five of the variables in the model was found and a VEC model was formulated. The stochastic trend of all variables show that an important part of the pattern of the freight rates cannot be predicted. [START_REF] Veenstra | The term structure of ocean freight rates[END_REF] used the VAR model to examine the relation between spot and period freight rates for the ocean dry bulk shipping market. The research show the evidence of a term structure in the industry and all deviations from the present value relation are of a transitory nature. [START_REF] Chi | Exchange rate and transport cost sensitivities of bilateral freight flows between the US and China[END_REF] applied the ARDL approach to build export and import models regarding the trade between China and US. Results reveal that gross domestic product is the key determinant of bilateral freight flows, while the real income is closely related to the bilateral freight flows. They also found that transport costs can be an important factor affecting the inflows of Chinese commodities to the US.

However the cyclic nature of the freight rates should also be considered. An appropriate approach is Markov Switching Vector Autoregressive model (MS-VAR) which allows us to analyze the relationship between the variables according to different states of the market. There are very few studies that have applied such a methodology in the context of ocean freight rates. [START_REF] Bildirici | Baltic Dry Index as a major economic policy indicator: the relationship with economic growth[END_REF] analyzed the relation between GDP growth in US and the Baltic Dry Index (BDI) by assuming the existence of three regimes. The first one is associated with the crisis, the second one is the moderate growth regime and the third one is the high growth regime. They found that under the high growth regime, BDI improvement effects economic growth positively.

As seen from the literature review, most of the authors have focused on the ocean bulk freight rates. We have not found any study applying VAR methods to investigate the freight rates in the container trade. VAR modeling is widely applied in macroeconomics because it does not require as much knowledge about the forces influencing a variable as structural models with simultaneous equations. Its main advantage is that all variables are treated as endogenous and each one is explained by its lagged values, the lagged values of the other variables in the model and error terms. Thus, we can analyze the linear interdependence among multiple time series by avoiding the problem of endogeneity (an explanatory variable is correlated with the error term).

A standard approach of VAR modelling is the estimation of the impulse responses function of each of the variables to one unit increase in the current value of one of the VAR errors [START_REF] Stock | Vector autoregressions[END_REF]. Considering the current volatile freight rates and the oversupply in the industry, it is also important to investigate the potential structural changes in the containerized trade market.

Our paper aims to fill in this gap in the literature and to provide empirical application of MS-VAR and impulse responses function in the context of containerized trade market. The objective of the research is to identify the different business cycles that have occurred in the liner shipping industry in the last 13 years and to analyze their nature as well as to evaluate the effects on the container freight rates following a surprising increase in the oil price, earnings, Fleet Development or Global Economic Activity.

MS-VAR provides accurate estimates regarding the duration of each regime as well as the switching probabilities. Moreover, it allows one to distinguish different business cycles occurring during the observed period. For each regime we have computed the impulse responses of CCFI following a positive shock of another variable in the model.

Methodology

MS-VAR model

Following the pioneer work of [START_REF] Sims | Macroeconomics and reality[END_REF], VAR modeling has become a useful tool in macroeconomics. Contrary to the structural models with simultaneous equations, VAR avoids the problems of over-identification (differentiating between correlation and causation by imposing restrictions). In VAR models, variables are endogenous. VAR provides a coherent and credible approach to data description, forecasting, structural inference, and policy analysis. In order to take into account the structural breaks in the time series and the different states that the containerized market operates, we combine the VAR approach with the Markov Chain process in the so-called Markov-switching Vector Autoregressive (MS-VAR) modelling.

Following [START_REF] Hamilton | A new approach to the economic analysis of nonstationary time series and the business cycle[END_REF], the general MS-VAR is depicted by equations (1) where it is assumed that a change in regime corresponds to an immediate one-time jump in the process mean. The description of MS-VAR in the current paper follows [START_REF] Ehrmann | Regime-dependent impulse response functions in a Markov-switching vector autoregression model[END_REF]. In our model, we consider that the mean would smoothly approach a new level after the transition from one regime to another. We do it in an extension of Hamilton's approach to a regime-switching VAR system [START_REF] Krolzig | Markov-switching vector autoregressions: Modelling, statistical inference, and application to business cycle analysis[END_REF]. (2)

For a given parametric specification of the model, probabilities are assigned to the unobserved regimes conditional on the available information set which constitutes an optimal inference on the latent state of the market [START_REF] Girardin | Quantitative easing works: Lessons from the unique experience in Japan 2001-2006[END_REF]. We thus obtain the probability of staying in a given regime when starting from that regime, as well as the probability of shifting to another regime. The classification of regimes and the dating algorithm used imply that every observation in the sample is assigned to one of the two regimes. We assign an observation to a specific regime when the smoothed probability of being in that regime is higher than 50%. The smoothed probability of being in a given regime is computed by using all the observations in the sample.

Impulse Response Function

In a MS-VAR model, with regime-dependence in the mean, variance and autoregressive parameters, a large number of parameters can switch between regimes. [START_REF] Ehrmann | Regime-dependent impulse response functions in a Markov-switching vector autoregression model[END_REF] suggest the use of regime-dependent impulse response functions which allow the investigation of how fundamental disturbances affect the variables in the model. For each regime, there is a set of impulse response functions. Each response function depends on the prevailing regime at the time of the shock. They facilitate the interpretation of switching parameters by providing a convenient way to summarize the information contained in the autoregressive parameters, variances and covariances of each regime [START_REF] Girardin | Quantitative easing works: Lessons from the unique experience in Japan 2001-2006[END_REF]. We follow the method of identification and estimation of impulse response proposed by [START_REF] Ehrmann | Regime-dependent impulse response functions in a Markov-switching vector autoregression model[END_REF].

There are regime-dependent impulse response functions which correspond to the reaction of variable to disturbances in regime .

( In order to make the precision of the impulse responses function more accurate, the bootstrapping technique was applied. This technique is used to create artificial observations for the variables and use them for the same estimation procedures as the original dataset. The artificial histories are created by replacing the parameters in the model with their estimated variance-covariance matrix, and then calculating the endogenous variables [START_REF] Ehrmann | Regime-dependent impulse response functions in a Markov-switching vector autoregression model[END_REF]. However, in MS-VAR, applying the bootstrapping technique is complicated due to the existence of a hidden Markov-chain process which determines the regimes. Therefore, we first create a history for the regimes, and then we use it to continue for the endogenous variables. Finally, the bootstrapped estimates of the response vectors are computed.

Data

The major international and domestic shipping companies for 11 major ports of departure in China.

The reliability of the data is ensured by a freight rate formulation committee composed by representatives of major domestic and foreign shipping companies. This is the key variable in the research and it is denoted as CCFI in our analyses.

-Clarksons Average Containership Earnings ($/Day)

Clarksons Average Containership Earnings is based on information provided by Clarksons brokers on a daily and weekly basis. This information is used to calculate the earnings in the liner shipping industry by taking into account the number of the fully cellular containerships. Clarkson Research compute earnings based on a single freight rate. This indicator is a proxy for the profitability in the container shipping industry. In our research, this variable is denoted as "Earnings". In broad terms, earnings for each route are calculated by taking the total revenue, deducting current bunker costs based on prices at representative regional bunker ports, estimated port costs (after currency adjustments) and total commission and then dividing the result by the number of voyage days (Clarkson Research, 2015).

-Total Containership Fleet Development (Thousand TEUs)

The size of all containerships is estimated in TEUs. This variable refers to the supply of the Liner Shipping. In the current paper we refer to this variable as "Fleet".

-West Texas Intermediate oil price ($/bbl) West Texas Intermediate, also known as Texas light sweet, is a grade of crude oil used as a benchmark in oil pricing. The bunker cost accounts for 70% of the operational cost of carriers and thus WTI could be used as a proxy for the operational costs of the carriers.

-Global Real Economic Activity (index)

The index of global real economic activity is computed from data collected from the monthly report on "Ship and Economics" published by Drewry Shipping Consultants Ltd. It is based on a global index of dry cargo single voyage freight rates of various bulk dry cargos such as grain, oilseeds, coal, iron ore, fertilizer, and scrap metal. The index can be traced back to 1968 and it is a direct measure of global economic activity which does not require exchange-rate weighting. It automatically aggregates real economic activity in all countries and incorporates shifting country weights, changes in the composition of real output, and changes in the propensity to import industrial commodities for a given unit of real output [START_REF] Kilian | Not all oil price shocks are alike: Disentangling demand and supply shocks in the crude oil market[END_REF].

Empirical Results

Descriptive statistics and specifications of MS-VAR

We use monthly time series from March 2003 to February 2017. In the MS-VAR model and impulse response functions, all variables were transformed into log and we use their growth rates for our analysis. Following the results of the Augmented Dickey-Fuller test, we also transform all variables into first difference, except Global Activity.

In the current paper we estimate an unrestricted MS-VAR model with intercept and without trend. We choose 2 lags based on the AIC, HQ and SC criteria. We also assume the existence of two regimes.

Business Cycles in the Liner shipping industry

As we have already emphasized, the MS-VAR model allows us to identify the different business cycles that have occurred from May 2003 to February 2017 (we have included two lags in the model). Figure 2 shows the transitional switching probabilities of regime 1 (the blue line) as well as the value of the CCFI (the red line). Both regimes are very persistent as shown in the matrix below. The regime switching probabilities for Regime 1 and Regime 2 are 18% and 17%, respectively. A major event that marks the switch of the regime 2 to regime 1 is the Subprime Crisis. The switch occurs in September 2008 and has a major impact on the variables in the model.

Throughout the observed period, Regime 1 and Regime 2 last 71 and 95 months, respectively. Table 3 andTable 4 show the descriptive statistics of the variables under each of the identified regimes. Regime 1 is characterized by lower average container freight rates and considerably higher volatility (Standard Deviation). Moreover, under the second regime the average Global Activity is negative (-15.2), while in regime one, its mean value is 29.32.

Earnings are also considerably higher under the second regime (15,908). On the other hand, the mean and the standard deviation of the Fleet are higher in the first regime. It suggests, that Regime 1 is characterized by a low level of economic activity, a high level of volatility of freight rates, a low level of earnings and a high growth rate of the containership capacity. The second regime is slightly more persistent than the first one and can be defined as a regime of a high level of economic activity, high and stable container freight rates, a high level of earnings and a low level of fleet capacity. The effect of the Subprime Crisis on Containerized trade started in September 2008, when we have a switch from Regime 2 to Regime 1. This marks the beginning of the second business cycle, which is characterized by a high level of uncertainty. This cycle continues until August 2014 (72 months in total). Throughout this period we have frequent switches between the two regimes. Even though the average CCFI has decreased by only 3% compared to the previous cycle, its volatility has increased by 102%. Earnings and Global Activity also had a dramatic decrease of 62% and 95%, respectively (Table 6). This period could be characterized as a postcrisis period where government took considerable measures to support the industry and to maintain economic growth. This business cycle lasts 72 months. From September 2014 to February 2017, we observe a business cycle which falls entirely under Regime 1 and we have named it "The New Normal" which lasts 30 months. During this period, the average Global Activity decreased by 39.52 points (Table 7). The average CCFI is 21% lower than the second business cycle and its volatility is 20% higher. Earnings became less volatile and decreased by 7% compare to the previous cycle. The Fleet has increased by 30% and its volatility is 58% lower. The proposed MS-VAR model tracks well different cycles that have occurred in the liner shipping industry. Both regimes are very persistent, which allows us to assume that the last business cycle, "The New Normal" might continue for at least a few more years. The growing number of conferences between carriers is a clear sign that shipping companies are trying to minimize their costs in order to deal with the low world demand and high level of containership capacity. We also observe a monopolization of the shipping market. According to Alphaliner, in 2008 Maersk had a share of around 6% of the world fleet, while in April 2017, it was more than 16%.

CCFI

CCFI

Impulse Reponses Functions

We use impulse response functions to show how the endogenous variables react to shocks of one standard deviation of the residuals of each equation (Crude Oil, Earnings, Fleet or Global Activity). The Impulse response functions are conditional on a given regime prevailing at the time of the disturbance and throughout the duration of the response [START_REF] Ehrmann | Regime-dependent impulse response functions in a Markov-switching vector autoregression model[END_REF].

In order to increase the precision of our estimation, we have applied the standard bootstrapping techniques. This method consists of increasing the number of observations by creating artificial histories for the variables. We have made 1000 bootstrap replications. Similarly to [START_REF] Uhlig | What are the effects of monetary policy on output? Results from an agnostic identification procedure[END_REF] and considering the findings of [START_REF] Inoue | Joint confidence sets for structural impulse responses[END_REF], we use a confidence interval of 68%. A positive shock of 1% of the crude oil has a significant effect on CCFI only in the second regime. This could be explained by the fact that most of the observations which fall under the second regime (stable growth) are before September 2008. During that time, the oil price had an increasing trend (see Figure 1) and thus the freight rates were strongly associated with the bunker price. On the other hand, throughout Regime 1 which is post crisis regime, the price of crude oil is quite low and has a decreasing trend.

The size of the world Fleet has substantial impact in both regimes. It is more pronounced in Regime 1, where a 1% increase of the size of the fleet leads to an immediate 3% decrease of the container freight rates. Then it takes around 7 months for CCFI to go to its equilibrium level.

In the second regime, the shock of the Fleet is also significant and CCFI decreases by approximately 1.4%. These results suggest that the supply in the shipping industry plays a major role in the container freight rate price formation. During a period of growth and low volatility, companies are able to better plan their strategies in terms of fleet development, while during a period of crisis, we have a substantial decrease in the demand for container transport which drives the freight rates down. Shipping companies are trying to optimize their fleet by renewing their fleets. According to the data of Clarkson research, between March 2003 and August 2008, the monthly average of scrapped vessels were 3,257 TEUs, while from September 2009 to March 2017, the monthly average was 16,138 TEUs, which is the equivalent of one Triple E containership. In January 2014, the equivalent of demolished ships was 84,740 TEUs, the highest level ever recorded. We should emphasize that many environmental and safety regulations also play a role in the increase of scrapping activities.

Earnings in the shipping industry are significant only under the regime of growth and low volatility. A in 1% in the level of earnings would lead to a 0.08% increase of CCFI. Even though the impact on the container freight rates is not very great, it lasts for 12 months.

The last variable in the model, Index of Global Economic Activity (GREA), is significant only in the First Regime of the impulse response function of CCFI. However, the effect of this variable is not substantial and its duration is only 2 months. It is interesting to highlight the fact that under the second regime, the function has a decreasing trend.

In appendices 1, 2, and 4 are presented the impulse response functions of Earnings, Fleet, Global Activity and Crude Oil respectively. It is interesting to observe that CCFI has a very substantial negative effect on Earnings in the case of both regimes. To a lesser extent, CCFI has a positive impact on Global Activity and the price of crude oil.

Conclusion

This paper contributes to the existing literature by exploring the relationship between the China Containerized Freight Index (CCFI), Clarksons Average Containership Earnings, Fleet Development, the price of Crude Oil and Global Real Economic Activity by applying MS-VAR and Impulse response function approaches. We have applied the MS-VAR model without trend and intercept by assuming the existence of two regimes. The first one can be characterized as a regime of crisis, low freight rates and high volatility occurring after the Subprime crisis, while the second one is a state of steady growth and low volatility in the shipping industry. The identified regimes are very persistent and track well the trends and business cycles in the shipping industry between May 2003 and February 2017. We have identified 3 major business cycles. The first one is characterized by a period of long and steady growth which lasts until August 2008. The following cycle is a transitional one which lasts 6 years. The third cycle is characterized by very low freight rates, low earnings and substantial size of the world fleet.

The impulse responses methodology allows us to pin down the consistent and efficient estimates of the reaction of CCFI to different factors according to the two states of the industry (Regime 1 and Regime 2). The variable which has a profound negative impact on the container freight rates is the growth of the world fleet. Its effect is more pronounced under the regime of crisis. Earnings in the shipping market and the crude oil price are only significant in the second regime and both have a positive impact on CCFI. The Global Economic activity has a minor effect on CCFI in the regime of crisis.

Potential future research could include the use of another indicator as a proxy for freight rates instead of CCFI. In this way we could double check our findings. Moreover, a variable which depicts the market concentration in the liner shipping industry (for example, the share of the top 10 companies in terms of transported TEUs) could also be included in the model. Other variables which reflect the Economic activity worldwide such as the industrial production of OECD member states could be also be considered.

The last identified economic cycle ("The New Normal") is characterized by low container freight rates, low earnings, high fleet capacity and relatively low crude oil prices.

Shipping companies are dealing with this unstable prospect of the market by limiting their operational costs (that is slow steaming strategies and economies of scale) and sharing capacity (creation of alliances). As UNCTAD pointed out, the market has become more monopolized and small players are vulnerable. In addition, even the big companies, such as Hanjin Shipping, could face bankruptcy. When we take into account the regionalization of the trade (for example, ASEAN Economic Community, Trade agreement between the EU and Canada, etc.) and the slowdown of the Chinese economy, it is obvious that the current unstable situation of "New Normal" might last a few more years.

Figure 1

 1 Figure 1 Containership Fleet Growth and Idle Capacity: Source: Clarkson Research

  where refers to China Containerized Freight Composite Index (CCFI), West Texas Intermediate oil price, Clarksons Average Containership Earnings, Total Containership Fleet Development and Index of Global Economic Activity. These exogenous variables are explained by an intercept , autoregressive terms of order ρ and a residual . The number of regimes in the model is two, , and all parameters are allowed to switch between the regimes. Therefore, each of the two regimes is defined by an intercept , autoregressive term and matrix . After the change in regime there is thus an immediate one-time jump in the variance of errors. This model is based on the assumption of varying processes according to the economic cycle in the liner shipping industry controlled by the unobserved variable . Recall that in our model is assumed to follow the discrete time and discrete state stochastic process of a hidden Markov chain and governed by transition probabilities , and . The conditional probabilities are collected into a transition matrix P as follows:

)

  Equation 3 shows the mathematical representation of the impulse response function for regime . It reveals the expected changes in endogenous variables at time to one standard deviation shock to the -th fundamental disturbance at time in regime . A series of dimensional response vectors predict the response of the endogenous variables.

  Ehrmann et al. (2003) proposed the following five steps to apply bootstrapping technique in the context of MS-VAR. Firstly, a history for the hidden regime is created by replacing the exogenous transition matrix with its estimated value . At each time , a random number is drawn from a uniform [0,1] distribution and compared with the conditional transition probabilities to determine whether there is a switch of the regime. Secondly, a history for the endogenous variables is generated. All parameters in Equation 1 are replaced by their estimated values and new residuals are computed. Thirdly, we use the data of the artificial history to compute bootstrapped parameters for , the transition matrix , and the smooth probabilities Fourthly, a set of restrictions are imposed which provide bootstrapped estimates of the matrices .

  following data have been selected: China Containerized Freight Composite Index (CCFI), Clarksons Average Containership Earnings ($/Day), Total Containership Fleet Development (thousand TEUs), West Texas Intermediate oil price ($/bbl) and the updated version of the index of global real economic activity. In total, there are 168 observations. The first three variables were collected from Clarksons Shipping Intelligence Network. Statistics regarding the price of crude oil were provided by the Energy and Information Administration of US (https://www.eia.gov/). The Global Real Economic Activity index was computed by (Kilian, 2009). -China Containerized Freight Composite Index -CCFI (Index) China (Export) Containerized Freight Index (CCFI) is sponsored by the Ministry of Communications of China and formulated by the Shanghai Shipping Exchange. The index serves as a barometer of the liner shipping industry and it is computed based on data collected from 22

Figure 2

 2 Figure 2 Smoothed probability of Regime 1 (volatile and decreasing freight Rates) and CCFI index. Source: Author's drawing

Figure 3 :

 3 Figure 3: Response of CCFI reacting to shock to the other variables in the MS-VAR model according to the two regimes.

  Descriptive statisticsCorrelation coefficients for the whole sample are presented in Table1. It shows that all variables in the model have a positive correlation with CCFI except the Fleet. However the correlation coefficients are less than 50%. The strong and negative correlation between Earning and Fleet (-0.80) suggests a possible negative effect of the growing capacity in the shipping industry on the Earnings. There is also a negative correlation of 75% between Fleet and Global

	Activity.
		CCFI	Earnings Fleet	Crude Oil	Global Activity
	CCFI	1
	Earnings	0.4369	1
	Fleet	-0.4613	-0.7975	1
	Crude Oil	0.3906	-0.2051	0.2811	1
	Global Activity	0.4553	0.6475	-0.7538	0.0962	1
			Table 2 Correlation Matrix

Table 1 provides a summary of the descriptive statistics for all variables. The average values of all indicators are positive. Standard Deviation is very high for the Crude oil, Earnings, and Global activity.

According to the results of Skewness and Kurtosis, most of the variables are symmetrically distributed except CCFI and Global Activity. In addition, the Jarque-Bera test confirms these results and accepts the normality hypothesis. In order to check whether each variable follows a unit-root process, we applied the Dickey-Fuller test. The unit root test of Augmented Dickey Fuller test clearly rejects the null hypothesis of the stationarity of all variables except Global Activity. It means that all variables except Global Activity have a unit root and are generated by a stochastic process. Table 1

  By looking at Figure2we can distinguish three economic cycles. The first one is from May 2003 to August 2008 which falls under Regime 2. This period can be characterized by steady growth, very low volatility of CCFI, a high level of earnings, a low price of crude oil and a very high level of Global Activity (Table5). This is the longest period which falls uniquely under the second regime (64 months).

		CCFI	Earnings	Fleet	Crude Oil	Global Activity
	Min	995.21	14,315.05	6,257.82	28.11	17.80
	Max	1,194.93	28,650.86	11,824.90	133.88	66.78
	Average	1,094.43	20,011.35	8,506.10	62.41	41.45
	Std Dev.	59.38	3,879.29		1,682.30	25.75	14.42
	Median	1,098.37	18,893.90	8,169.19	59.35	42.99
			Table 5 : Business Cycle 1: Steady Growth
		Descriptive statistics (May 2003 -August 2008)
				Earnings	Fleet	Crude Oil	Global Activity
	Min	641.50		4,619.34	11,926.45	30.32	-133.13
	Max	1,263.72	1,5740.16	20,043.44	106.29	44.45
	Average	943.04		7,310.74	16,677.73	69.88	-15.42
	Std. Dev.	161.10		2,344.44	2,679.00	23.65	32.54
	Med	956.77		6,461.60	17,084.64	69.64	-15.28
	Kurtosis	-1.12		1.85		-1.18	-1.62	1.66
	Skewness	0.15		-1.47		0.32	-0.01	0.84
			Table 3 Descriptive Statistics of Regime 1
		CCFI	Earnings	Fleet	Crude Oil	Global Activity
	Min	946.37	4,395.78	6,257.82	28.11	-29.43
	Max	1,315.87	28,650.86	17,872.49	133.88	66.78
	Average	1,097.40	15,908.95	10,642.82	71.30	29.32
	Std. Dev.	74.32	6,906.49	3,524.77	25.72	25.35
	Med	1,097.94	16,205.72	6,501.78	30.34	42.01
	Kurtosis	0.50	-0.86	-0.92	-0.55	-0.22
	Skewness	0.48	-0.16	0.57	0.24	-0.80
			Table 4 Descriptive Statistics of Regime 2

Table 6 :

 6 Business Cycle 2: Uncertainty, Descriptive statistics(September 2008 -August 2014)

			Earnings	Fleet	Crude Oil	Global Activity
	Min	777.04	4,395.78	11,926.45	39.09	-41.27
	Max	1,315.87	15,740.16	17,872.49	109.53	44.45
	Average	1,057.41	7,504.49	14,884.93	86.40	2.19
	Std. Dev.	119.77	2,725.36	1,776.50	16.82	24.20
	Median	1,066.60	6,537.33	15,066.53	89.33	2.69

  In April 2016, CCFI registered its lowest monthly value ever recorded, 641.50 points, and in August 2016 the Hanjin Shipping, the seventh biggest shipping company, declared bankruptcy.

		CCFI	Earnings	Fleet	Crude Oil	Global Activity
	Min	641.50	5,357.11	17,926.47	30.32	-133.13
	Max	1,107.11	10,935.99	20,043.44	93.21	2.18
	Average	834.42	6,978.76	19,302.69	50.69	-37.33
	Std Dev.	143.40	1,729.89	743.40	13.65	30.08
	Median	799.92	6,156.18	19,657.97	47.52	-33.49
			Table 7: Business Cycle 3: The new Normal
			Descriptive statistics (September 2008 -August 2014)

Estimated by the authors based on the data collected from Clarkson Research Shipping Intelligence and UNCATD official websites.

http://www.worldshipping.org/about-the-industry/global-trade

Freight shipping rates and container costs, Official Website of Maersk Line https://www.maerskline.com/news/2017/08/03/rates-and-pricing

Appendices


come

On-resonance material fast light

We theoretically revisit the problem of the propagation of coherent light pulses through a linear medium when the carrier frequency of the pulses coincides with the minimum of a narrow dip in the medium transmission. Considering realistic contrasts between the maximum and minimum transmission of the medium and incident pulses of strictly finite duration, we combine temporal and spectral approaches to obtain analytical expressions of the transmitted pulse that reproduce the main features of the exact numerical solutions derived by fast Fourier transform. A special attention is paid to the advance of the pulse maximum over that of a pulse covering the same distance in vacuum and the ratio of this advance to the pulse duration (fractional advance) is optimized.

I. INTRODUCTION

Fast light currently refers to situations in which a smooth pulse of coherent light is transmitted by a linear optical system in such a manner that the maximum of the output pulse anticipates that of a pulse having covered the same distance in vacuum. As extensively discussed in the literature (see, e.g., [START_REF] Chiao | Tunnelling times and superluminality[END_REF][START_REF] Boyd | Slow" and "Fast light[END_REF]), the phenomenon is not at odds with the relativistic causality. It appears in particular that the maximum of the output pulse is not as a direct reflection of that of the input pulse but results from the action of the system on the early part of the latter. A direct experimental evidence of this point is reported in [START_REF] Tomita | Direct Observation of a Pulse Peak Using a Peak-Removed Gaussian Optical Pulse in a Superluminal Medium[END_REF]. Fast light can be obtained when the system transmission displays a well-marked, narrow dip at the carrier frequency of the input pulse [START_REF] Macke | Optimal superluminal systems[END_REF]. Note however that this condition is not always sufficient. For optical systems that are not minimum-phase-shift [START_REF]We use the definitions, sign convention and results of the linear system theory[END_REF], in particular those involving mirrors [START_REF] Wang | Causal "all-pass" filters and Kramers-Kronig relations[END_REF][START_REF] Macke | Comment on: Gain-assisted superluminal light propagation through a Bose-Einstein condensate cavity system[END_REF] or polarizers [START_REF] Solli | Demonstration of Superluminal Effects in an Absorptionless, Nonreflective System[END_REF][START_REF] Brunner | Direct Measurement of Superluminal Group Velocity and Signal Velocity in an Optical Fiber[END_REF][START_REF] Macke | Simultaneous slow and fast light involving the Faraday effect[END_REF], it may even occur that a same transmission profile leads to either fast or slow light. In the case of purely propagative systems as considered in the following, fast light simply originates in the dispersive properties of the medium (material fast light) and the Kramers-Kronig relations apply in their common form. Even in this case a dip in the transmission profile does not always entail fast light. For instance when the dip corresponds to a minimum of transmission between two gain lines, there is a range of parameters for which there is no fast light whatever the dip depth is [START_REF] Macke | Propagation of light-pulses at a negative group-velocity[END_REF].

The simplest way to observe material fast light is to exploit the anomalous dispersion associated with an isolated absorption line of a passive medium. Experiments have been performed on various materials including semiconductor [START_REF] Chu | Linear pulse propagation in an absorbing medium[END_REF], molecular gas [START_REF] Ségard | Observation of negative velocity pulse propagation[END_REF], atomic vapour at room temperature [START_REF] Tanaka | Propagation of optical pulses in a resonantly absorbing medium: Observation of negative velocity in Rb vapour[END_REF], hot atomic vapour [START_REF] Keaveney | Maximal refraction and superluminal propagation in a gaseous nanolayer[END_REF] and clouds of cold atoms [START_REF] Jennewein | Propagation of light through small clouds of cold interacting atoms[END_REF], with propagation distances going from 390 nanometers [START_REF] Keaveney | Maximal refraction and superluminal propagation in a gaseous nanolayer[END_REF] to 24 meters [START_REF] Ségard | Observation of negative velocity pulse propagation[END_REF]. In order to control the pulse advance, the transmission dip is artificially created in other experiments by applying supplementary fields interacting non linearly with the medium and taking advantage of the phenomena of electromagnetically induced absorption [START_REF] Akulshin | Negative group velocity of a light pulse in cesium vapour[END_REF][START_REF] Godone | Slow light and superluminality in the coherent population trapping maser[END_REF][START_REF] Kim | Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line[END_REF][START_REF] Kang | Superluminal and slow light propagation in cold atoms[END_REF][START_REF] Mikhailov | Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas[END_REF][START_REF] Brown | Anomalous dispersion and negative group velocity in a coherence-free cold atomic medium[END_REF][START_REF] Akulshin | Fast light in atomic media[END_REF], Raman effect [START_REF] Wang | Gain-assisted superluminal light propagation[END_REF][START_REF] Stenner | The speed of light in a 'fast-light' optical medium[END_REF], Brillouin effect [26] and four wave mixing [START_REF] Glasser | Stimulated generation of superluminal light pulses via four-wave mixing[END_REF][START_REF] Swaim | Faster light with competing absorption and gain[END_REF].

In most experimental reports on material fast light, the emphasis is made on the absolute value of the advance of the pulse maximum over that of a pulse propagating in vacuum and on the corresponding value of the velocity often confused with the group velocity. In fact the true experimental challenge is to obtain advances which are significant compared to the pulse duration with acceptable pulse distortion. On the other hand, in the numerous theoretical papers on the subject, the proof of fast light is generally reduced to that of a superluminal group velocity without simulation evidencing a visible advance of the pulse maximum.

The present article is theoretical but devotes a special attention to actual or feasible experiments. Its purpose is to clarify some points about material fast light and to examine what are the practical limitations to the phenomenon, Mother Nature resisting to a violation of her principles even when this violation is only apparent. To do so, we come back to the basic system consisting in a passive medium with an isolated absorption line [START_REF] Chu | Linear pulse propagation in an absorbing medium[END_REF][START_REF] Ségard | Observation of negative velocity pulse propagation[END_REF][START_REF] Tanaka | Propagation of optical pulses in a resonantly absorbing medium: Observation of negative velocity in Rb vapour[END_REF][START_REF] Keaveney | Maximal refraction and superluminal propagation in a gaseous nanolayer[END_REF][START_REF] Jennewein | Propagation of light through small clouds of cold interacting atoms[END_REF]. The simplicity of this system, favourable from an experimental viewpoint, will enable us to obtain analytical results. Insofar as the performances of the medium are essentially determined by the contrast between its maximum and minimum of transmission [START_REF] Macke | Optimal superluminal systems[END_REF], it may be reasonably expected that these results have some generality. Anticipating a more extensive discussion, we note here that contrast exceeding 40 dB has been actually used in absorbing media [START_REF] Ségard | Observation of negative velocity pulse propagation[END_REF] whereas it seems difficult to use an equivalent optical gain without generating serious instabilities [START_REF] Steinberg | Dispersionless, highly superluminal propagation in a medium with a gain doublet[END_REF][START_REF] Stenner | Pump-beam-instability limits to Raman-gain doublet "fast light" pulse propagation[END_REF].

The arrangement of our paper is as follows. In Sec.II, we introduce the basic equations of the problem and analyse their physical meaning by privileging the temporal approach. Asymptotic solutions obtained in the short and long pulse limits are given in Sec.III and Sec.IV. Section V is devoted to the optimization of the fractional advance of the pulse maximum, ratio of this advance to the pulse duration. The results are discussed in Sec.VI and we conclude in Sec.VII by summarizing the main points.

II. GENERAL ANALYSIS

When coherent light pulses with a slowly varying envelope propagate in a linear medium of thickness ℓ, the envelope e(ℓ, t) of the output pulse can be deduced from the envelope e(0, t) of the incident pulse by the general relation:

e (ℓ, t) = h (ℓ, t) ⊗ e (0, t) (1)
where h (ℓ, t) is the impulse response of the medium and ⊗ designates a convolution product [START_REF]We use the definitions, sign convention and results of the linear system theory[END_REF]. We consider here a dilute medium with an isolated Lorentzian absorptionline at a frequency ω 0 coinciding with the carrier frequency ω c of the incident pulses. We have then [START_REF] Crisp | Propagation of Small-Area Pulses of Coherent Light through a Resonant Medium[END_REF]32]

h (ℓ, t) = δ (t) -Lγk (ℓ, t) u H (t) (2)
with

k (ℓ, t) = J 1 2 √ Lγt √ Lγt e -γt . (3)
In these expressions, δ (t), u H (t) and J 1 (x) respectively designate the delta function, the Heaviside unit step and the Bessel function of the first kind of index 1; t is the time retarded by the transit time ℓ/c at the velocity of light in vacuum (retarded time picture), γ is the halfwidth at half-maximum of the absorption line and L = αℓ/2 where α is the resonance absorption coefficient for the intensity. From Eq.(1), we finally get

e (ℓ, t) = e (0, t) -Lγ [k (ℓ, t) u H (t)] ⊗ [e (0, t)] . (4)
From a physical viewpoint, Eq.( 4) first shows that, if e (0, t) starts at a given time, it is the same for e (ℓ, t) at the corresponding retarded time in agreement with relativistic causality. Second, as noted by Feynman in a more general context [START_REF] Feynman | The Feynman Lectures on Physics[END_REF], the transmitted wave is the sum of the incident wave as if it had propagated in vacuum and of the wave reemitted by the medium. Fast light thus results from the interference between these two waves [34-37].

In the frequency domain, the counterpart of Eq.( 1) reads

E (ℓ, Ω) = H (ℓ, Ω) E (0, Ω) (5)
where E (ℓ, Ω), H (ℓ, Ω) and E (0, Ω) are, respectively, the Fourier transforms of e (ℓ, t), h (ℓ, t) and e (0, t). H (ℓ, Ω) is the transfer function of the medium [START_REF]We use the definitions, sign convention and results of the linear system theory[END_REF] which in the present case has the simple form [START_REF] Macke | Propagation of light-pulses at a negative group-velocity[END_REF]

H (ℓ, Ω) = exp - L 1 + iΩ/γ (6)
It yields a contrast C between the maximum and the minimum of the intensity transmission and a relative depth D of the transmission dip

C = H (ℓ, ∞) H (ℓ, 0) 2 = e 2L = e αℓ (7)
D = 1 -e -2L = 1 -e -αℓ (8)
We will restrict our analysis to contrasts below or equal to 50 dB [C dB = 10 log 10 (C)] , which seems an upper limit to the contrast that can be actually used in the experiments to avoid parasitic effects. This contrast is attained for L = 5.756 and the corresponding dip depth D is very close to 100 %. Note that, in most experimental demonstrations of fast light in atomic media, this depth does not exceed the value 86 % attained in absorbing media for the moderate optical thickness αℓ = 2 (L = 1, C ≈ 8.7 dB). General properties of the envelope of the transmitted field can be deduced from the transfer function. We note first that E (ℓ, 0) and E (0, 0) are, respectively, the areas of e (ℓ, t) and e (0, t). The former is thus equal to the latter multiplied by e -L . Second, the mean time of h (ℓ, t), i.e. the group delay τ g , equals the first cumulant κ 1 of H (ℓ, Ω) [START_REF] Dooges | Deterministic Methods in Systems Hydrology[END_REF]. Quite generally the cumulants κ n are given by the following expansion of ln [H (ℓ, Ω)] :

ln [H (ℓ, Ω)] = ln [H (ℓ, 0)] + ∞ n=1 κ n n! (-iΩ) n . (9)
In the present case, they take the simple form κ n = -n!L/γ n and the group advance a g = -τ g reads :

a g = L/γ (10)
Due the additivity property of the cumulants, a g is the advance of the centre-of-gravity of e (ℓ, t) over that of e (0, t), that is, in our retarded time picture, over that of the envelope of a pulse propagating without distortion at the velocity c. We emphasize that this result is general and holds even when the transmitted pulse is strongly distorted. As shows Eq.( 10), the advance a g is always positive. It should be carefully distinguished from the advance a of the pulse maximum and from that of the centre-of-gravity of the intensity profile |e (ℓ, t)| 2 which both may take negative values (delay instead of advance) [START_REF] Peatross | Average Energy Flow of Optical Pulses in Dispersive Media[END_REF][START_REF] Talukder | Superluminal to Subluminal Transition in the Pulse propagation in a Resonantly Absorbing Medium[END_REF][START_REF] Talukder | Measurement of Net Group and Reshaping Delays for Optical Pulses in Dispersive Media[END_REF][START_REF] Kohmoto | Nonadherence to the conventional group velocity for nanosecond light pulses in Rb vapour[END_REF][START_REF] Nanda | Why do superluminal pulses become subluminal once they go far enough?[END_REF].

In the following, we consider incident pulses of strictly finite duration, the envelopes of which are good approximations of the Gaussian envelope e G (0, t) = exp -t 2 /2σ 2 usually considered in the literature. We have retained

e 1 (0, t) = cos 2 Ω 1 t 2 Π Ω 1 t 2π = 1 + cos (Ω 1 t) 2 Π Ω 1 t 2π (11)
or, if necessary, the more speculative form

e 2 (0, t) = cos 4 Ω 2 t 2 Π Ω 2 t 2π = 1 + cos (Ω 2 t) 2 2 Π Ω 2 t 2π . (12)
In these expressions Π (x) is the rectangle function equal to 1 for -1/2 < x < 1/2 and 0 elsewhere. The envelopes e 1 (0, t), e G (0, t) and e 2 (0, t) are compared Fig. 1 also as the corresponding intensity profiles (nearly undistinguishable). The parameters Ω 1 , σ and Ω 2 are chosen such that the corresponding intensity profiles (see inset in Fig. 1) have the same full width at half-maximum τ p . This is achieved when Ω 1 τ p = 4 arccos 2 -1/4 , τ p = 2σ √ ln 2 and Ω 2 τ p = 4 arccos 2 -1/8 . The duration τ p will be taken as universal time unit in all the following. In this way the profile of the transmitted pulse is entirely determined by the two independent and dimensionless parameters γτ p and L (more generally the contrast C). As above mentioned the challenge in fast light experiments is to attain significant advance a of the pulse maximum compared to the pulse duration. From this viewpoint, a merit factor of the experiments is the fractional advance F = a/τ p . Insofar as the envelopes e i (0, t) (i = 1, 2 ) start at a finite time (t = -π/Ω i ), it is obvious that the fractional advance obtained with these envelopes will reproduce that obtained with Gaussian pulses if and only if its value is significantly less than π/ (Ω i τ p ), that is 1.37 for i = 1 and 1.91 for i = 2. As shown later e 1 (0, t) constitute a sufficient approximation of e G (0, t) when F ≤ 50 % whereas the better approximation provided by e 2 (0, t) is necessary for larger fractional advances. We finally men-tion that, beyond the advantage of a finite duration, both envelopes offer the possibility of a periodic continuation which will be useful to optimise the fractional advance (Sec. V). Except when contrary specified the envelope e 1 (0, t), leading to simpler calculations, is used in the following.

III. SHORT PULSE LIMIT

When the incident pulse is short enough, its spectrum mainly lies in the far wings of the line profile where the medium dispersion is normal [START_REF] Boyd | Slow" and "Fast light[END_REF]. A delay instead of an advance of the pulse maximum might be expected in such conditions. We show in this section that the opposite occurs. Simple analytical results are obtained when the duration τ p of the incident pulse is short compared to the duration of the pulse reemitted by the medium as considered in the discussion of Eq.( 4) that is when γτ p (1 + L/2) ≪ 1 [START_REF] Whatever | γ (1 + L/2)] -1 is a good estimate of the duration of k (ℓ, t) uH (t) at 1/e of its maximum amplitude[END_REF]. We have then

[k (ℓ, t) u H (t)] ⊗ [e (0, t)] ≈ k (ℓ, t) [u H (t) ⊗ e (0, t)] .
(13) For an incident pulse of envelope e 1 (0, t), it is easily shown that

[u H (t) ⊗ e (0, t)] = 1 2 t + sin (Ω 1 t) Ω 1 + π Ω 1 Π Ω 1 t 2π + π Ω 1 u H t - π Ω 1 (14)
and, according to Eq.( 4), the envelope of the transmitted field reads

e 1 (ℓ, t) ≈ e 1 (0, t) -Lγ [u H (t) ⊗ e (0, t)] J 1 2 √ Lγt √ Lγt e -γt .
(15) Figures 2 and3 are respectively obtained for large (L = 5.756) and moderate (L = 1) optical thickness. They show that the intensity profiles |e 1 (ℓ, t)| 2 determined by Eq.( 15) are in perfect agreement with the exact profiles derived by fast Fourier transform (FFT). Quite generally the fractional advance F = a/τ p of the pulse maximum has the approximate form

F ≈ 2Lγ Ω 2 1 τ p = 2Lγτ p Ω 2 1 τ 2 p ≈ 0.38Lγτ p (16)
whereas a/τ p = L/γτ p and thus a/a g = 0.38 (γτ p ) 2 . This means that, in the short pulse limit, the advance of the pulse maximum will be always much smaller than the group advance. This apparent paradox is explained by the fact that the envelope e 1 (ℓ, t) has a long tail which is essentially negative and artificially increases the advance a g of its centre of gravity. This tail has the general form The solid red line (the dashed blue line) is the exact solution obtained by FFT (the analytical solution obtained in the short pulse limit). As in all the following figures, the intensity profile of the incident pulse (dotted black line) is given for reference. Its full width at half maximum τp is taken as time unit. Inset: comparison of the envelopes of the transmitted pulse obtained for e(0, t) = e1(0, t) (solid red line) and e(0, t) = eG(0, t) (dashed blue line). The pulse tail displays a first zero for t/τp ≈ j 2 1,1 / (4Lγτp) ≈ 10.6 (see text). For large optical thickness, it oscillates with successive lobes of rapidly decreasing amplitude and zeroes at the times t n = j 2 1,n / (4Lγ) where j 1,n designates the corresponding zeroes of J 1 (x). This behaviour is illustrated in the inset of Fig. 2. Note that, for the parameters considered, the amplitude of the secondary lobes appearing for t > t 1 (not shown) is very weak, below 0.02 for the first and largest one. For moderate optical thickness, the decrease of k (ℓ, t) is mainly determined by the term exp (-γt) and the tail of the envelope is reduced to a simple exponential of the form -1.37Lγτ p exp [-γ (1 + L/2) t] as shown in the inset of Fig. 3 obtained for L = 1. For a five times smaller optical thickness (L = 0.2), the intensity of the tail would be 25 times weaker and the fractional advance of the maximum 5 times smaller. The normalized intensity profile of the transmitted pulse is then nearly undistinguishable from that of the incident pulse although the depth of the dip in the transmission profile is significant (D = 33 %).

e 1 (ℓ, t) ≈ - πLγ Ω 1 k (ℓ, t) ≈ -1.37Lγτ p k (ℓ, t) (17)
Similar calculations can be performed for Gaussian pulses. We only give below the expression of the envelope e G (ℓ, t) of the transmitted pulse:

e G (ℓ, t) ≈ exp - t 2 2σ 2 -Lγσ π 2 1 + erf t σ √ 2 k (ℓ, t) (18
) where erf (x) designates the error function. There is again a perfect agreement with the exact numerical result obtained by FFT. The fractional advance has now the approximate form

F ≈ Lγσ 2 τ p = σ τ p 2 Lγτ p ≈ 0.36Lγτ p (19)
The envelopes e G (ℓ, t) and e 1 (ℓ, t) are shown in the insets of Fig. 2 and Fig. 3. As expected, they are quite comparable.

IV. LONG PULSE OR ADIABATIC LIMIT

We consider in this section the case opposite to the previous one. We assume that the envelope e(0, t) of the incident pulse is everywhere slowly varying compared to k (ℓ, t) u H (t). This implies in particular that the pulse duration τ p is long compared to 1/γ. The function Lγk (ℓ, t) u H (t) can then be approximated by A k δ (t -τ kg ) where A k and τ kg are, respectively, its area and its centre of gravity. Since Lγk (ℓ, t) u H (t) is the inverse Fourier transform of K (ℓ, Ω) = 1 -H (ℓ, Ω), we have A k = 1 -H (ℓ, 0) = 1 -e -L and, by an expansion of ln [K (ℓ, Ω)] in cumulants [START_REF] Dooges | Deterministic Methods in Systems Hydrology[END_REF], τ kg = L/ γ e L -1 > 0. Injecting these results in Eq.( 4), we get

e (ℓ, t) ≈ e (0, t) -A k e (0, t -τ kg ) (20)
The envelope of the output field appears equal to that of the input field minus its copy reduced in amplitude and slightly retarded [34,[START_REF] Macke | Two-pulse interference and superluminality[END_REF][START_REF] Chamarro-Posada | Superluminal propagation in resonant dissipative media[END_REF]. Assuming that τ kg ≪ τ p , Eq.( 20) takes the form

e (ℓ, t) ≈ e -L [e (0, t) + a g ė (0, t)] (21)
where the dot designates a time-derivative and where we have taken into account the relations 1 -A k = e -L and A k τ kg / (1 -A k ) = a g . In the limit where a g /τ p ≪ 1, we retrieve the classical result e (ℓ, t) ≈ e -L e (0, t + a g )

In this limit, the envelope of the output pulse equals that of the input pulse attenuated by the steady state transmission e -L for the field and advanced by a g (no distortion). It should be however kept in mind that this ) and Ω1 = γ/4 (τp ≈ 9.15/γ). The solid red line (the dashed blue line) is the exact solution obtained by FFT (the analytical solution obtained in the long pulse or adiabatic limit). Inset: comparison of the envelopes of the transmitted pulse obtained for e(0, t) = e1(0, t) (solid red line) and e(0, t) = eG(0, t) (dashed blue line). asymptotic result only holds when a g /τ p ≪ 1 and, consequently, that the corresponding fractional delay will be limited.

Figure 4 shows an example of situation for which Eq.( 22) provides a good approximation of the exact result. It is obtained for e(0, t) = e 1 (0, t) and the moderate optical thickness already considered in Fig. 3 (L = 1 leading to C ≈ 8.7 dB and D ≈ 86 %) but a much longer pulse duration (Ω 1 /γ = 1/4 that is γτ p ≈ 9.1). The fractional advance F = a/τ p of the pulse maximum is equal to 10.3 % whereas a/a g = 0.94, actually close to unity.

The asymptotic solution given by Eq.( 22) remains acceptable for long pulses even when the condition a g ≪ τ p is poorly satisfied. Figure 5 illustrates this point. It is also obtained for γτ p ≈ 9.1 but for L = 5.756 (C = 50 dB). We have now a fractional advance F = a/τ p ≈ 59 % and, despite that the condition of validity of Eq.( 22) is far from being fulfilled (a g /τ p ≈ 0.63), a and a g remain close. A new feature of Fig. 5 compared to Fig. 4 is the appearance of transients at the beginning (precursor) and the end (postcursor) of the envelope e 1 (0, t), that is for t/τ p = ±π/Ω 1 τ p ≈ ±1.37. They obviously originate in the discontinuity of the second derivative of e 1 (0, t) at the corresponding times, are absent in the case of Gaussian pulses (see inset of Fig. 5) and are not taken into account by the adiabatic approximation that assumes that the envelope of the incident pulse is everywhere continuous with continuous derivatives. By a cavalier extrapolation of results obtained in the study of the Sommerfeld precursors [START_REF] Macke | Simple asymptotic forms for Sommerfeld and Brillouin precursors[END_REF], we have found that the peak amplitude of the precursor roughly scales as

A p = Ω 1 γL 2 (23)
We have numerically checked that this law provides a good estimate of the exact result as soon as the precursor is visible without dominating excessively the main pulse.

In the conditions of Fig. 5, we get A p = 1.9 × 10 -3 in satisfactory agreement with the amplitude (≈ 2.5×10 -3 ) of the precursor shown in the inset of this figure. The order of magnitude given by Eq.( 23) also applies to the postcursor but there is now interference of the postcursor with the tail of the main field and it is then difficult to determine what originates in the postcursor alone.

V. OPTIMIZATION OF THE FRACTIONAL PULSE-ADVANCE

The convolution approach used in the previous sections provides both a physical insight on the fast light phenomenon and analytical results in the short and long pulse limits. Unfortunately the corresponding fractional advances F = a/τ p may be weak. Larger fractional advances are expected for intermediate pulse durations. The convolution approach can be extended to this case when the optical thickness is moderate, say for L ≤ 1. The function k (ℓ, t) appearing in the impulse response may then be approximated by exp [-γt (1 + L/2)] and the convolution product can be explicitly calculated. We will not develop this method here. Indeed more general and simpler results on e(ℓ, t), can be derived from the envelope e(ℓ, t) of the transmitted field obtained when the envelope e(0, t) of the incident pulse is replaced by its periodic continuation e(0, t) (see the thin lines in Fig. 1).

When e(0, t) = e 1 (0, t), its periodic continuation e 1 (0, t) = [1 + cos (Ω 1 t)] /2 contains only three frequencies (0, ±Ω 1 ). Taking into account that H (ℓ, Ω) and H (ℓ, -Ω) are complex conjugates, we get

e 1 (ℓ, t) = exp (-L) 2 + |H (ℓ, Ω 1 )| 2 cos [Ω 1 t + Φ (ℓ, Ω 1 )] (24)
where

Φ (ℓ, Ω) = arg [H (ℓ, Ω)] = LΩ/γ 1 + Ω 2 /γ 2 . ( 25
)
The advance a 1 of the maximum of e 1 (ℓ, t) on that of e 1 (0, t) is thus

a 1 = Φ (ℓ, Ω 1 ) Ω 1 = L/γ 1 + Ω 2 1 /γ 2 = a g 1 + Ω 2 1 /γ 2 . ( 26
)
So long as the responses to the successive pulses of e 1 (0, t) do not significantly overlap and as the precursor and postcursor have a negligible amplitude, e 1 (ℓ, t) is expected to be a good approximation of e 1 (ℓ, t) in their common domain of existence. We have then a ≈ a 1 and the fractional advance reads 27) Equation (26) shows that the advance is the largest and equal to the group advance a g when Ω 2 1 /γ 2 ≪ 1, that is for long incident pulses. Unfortunately, it results from Eq.( 27) that the corresponding fractional advance F is very weak. F attains its maximum value F m ≈ 0.22L when Ω 1 = γ (γτ p ≈ 2.29). The corresponding advance is half of the group advance (a ≈ a g /2). Besides the main pulse is followed by a second one, the relative intensity of which reads R = tanh 2 (L/4) .

F = a τ p ≈ Ω 1 a 1 Ω 1 τ p ≈ 0.44Ω 1 a 1 = 0.44 LΩ 1 /γ 1 + Ω 2 1 /γ 2 . (
Figure 6 obtained for L = 1 and Ω 1 = γ, shows that the exact intensity profile |e 1 (l, t)| 2 is actually quasi-confused with its periodic approximation | e 1 (l, t)| 2 . F and R are in perfect agreement with those given by Eq.( 27) and Eq.( 28), namely F = F m = 22 % and R = 6 %. A simulation (not shown) made for Ω 1 = γ and L = 2 (C ≈ 17 dB) evidences that | e 1 (l, t)|

2 is still a very good approximation of |e 1 (l, t)| 2 . We get then F ≈ 42 % and R = 22 %, very close to the values F ≈ 44 % and R = 21 % predicted by Eq.( 27) and Eq. [START_REF] Swaim | Faster light with competing absorption and gain[END_REF]. Note that the relative intensity of the second maximum of intensity is significantly larger than that obtained for L = 1.

For large optical thickness, problems arise from the overlapping of the responses to the successive pulses of the periodically continued input envelope and from the precursor and postcursor that can have significant amplitudes. Both problems are solved by replacing e 1 (0, t) considered above by e 2 (0, t) which is a better approximation of the Gaussian pulse. Indeed the successive pulses of e 2 (0, t) are better separated than those of e 1 (0, t) (see Fig. 1) and the higher order initial and final discontinuities of e 2 (0, t) ensure negligible amplitude of the precursor and postcursor. e(0, t) reads ) of the exact intensity profile of the transmitted pulse (solid red line) with the analytical profile obtained by periodically continuing the incident pulse e(0, t) = e1(0, t) (thin blue line). The pulse duration is that maximizing the fractional advance (Ω1 = γ, τp ≈ 2.29/γ) . It leads to a = ag/2 and to F = L/2γτp ≈ 22% . Inset: comparison of the envelopes of the transmitted pulse obtained for e(0, t) = e1(0, t) (solid red line) and e(0, t) = eG(0, t) (dashed blue line).

e 2 (0, t) = cos 4 Ω 2 t 2 = 3 8 + cos (Ω 2 t) 2 + cos (2Ω 2 t) 8 . (29)
It contains five frequencies (0, ±Ω 2 and ±2Ω 2 ) and leads to

e 2 (ℓ, t) = 3 8 exp (-L) + |H (ℓ, Ω 2 )| 2 cos [Ω 2 t + Φ (ℓ, Ω 2 )] + |H (ℓ, 2Ω 2 )| 8 cos [2Ω 2 t + Φ (ℓ, 2Ω 2 )] . (30)
The advance a 2 of the maximum of e 2 (ℓ, t) is obtained by solving the transcendent equation derived from Eq.( 30)

sin [Φ (ℓ, 2Ω 2 ) -2Ω 2 a 2 ] sin [Φ (ℓ, Ω 2 ) -Ω 2 a 2 ] = -2 H (ℓ, Ω 2 ) H (ℓ, 2Ω 2 ) . (31)
When L = 3 (C ≈ 26 dB), we find that F = a 2 /τ p takes its largest value F m ≈ 0.55 for Ω 2 ≈ 0.644γ (γτ p ≈ 2.55). Figure 7, obtained in those conditions, shows that the profile | e 2 (ℓ, t)| 2 perfectly fits the exact intensity profile |e 2 (ℓ, t)| 2 with F ≈ F m ≈ 55 %, a/a g ≈ 0.45 and a relative intensity of the second maximum R ≈ 33 %. We have added on Fig. 7 the profile |e 1 (ℓ, t)| 2 which would be obtained for the same duration of the incident pulse. For this optical thickness, it does not deviate too strongly from the previous one with F ≈ 61 %, a/a g ≈ 0.52 and R ≈ 35 %. The situation dramatically changes for the largest transmission-contrast and optical thickness considered in this article (C = 50 dB, L = 5.756). For e(0, t) = e 1 (0, t), the transmitted pulse is then strongly damaged by the precursor and postcursor and the periodic solution e 1 (l, t) fails to reproduce the exact result (even very approximately). On the other hand, the periodic solution given by Eq.( 30) remains a very good approximation of the exact result when e(0, t) = e 2 (0, t). The pulse duration is that maximizing the fractional advance, that is τp ≈ 3.18/γ leading to F ≈ 94% and a = 0.52 ag. Inset: comparison of the envelopes of the transmitted pulse obtained for e(0, t) = e2(0, t) (solid red line) and for e(0, t) = eG(0, t) (dashed blue line). Note that the slow light regime is attained for the Gaussian pulse (see text).

Figure 8 shows the intensity profiles obtained for the pulse duration γτ p ≈ 3.18 (Ω 2 ≈ 0.517γ) leading to the maximum fractional advance F ≈ F m ≈ 94 % with an absolute advance a ≈ 0.52a g . The only difference between the exact intensity profile |e 2 (ℓ, t)| 2 and its analytical periodic counterpart | e 2 (ℓ, t)| 2 lies in the second lobe, the relative intensity (R ≈ 79 %) of which is very slightly underestimated in | e 2 (ℓ, t)| 2 . A main result of the above study is that the pulse following the main one becomes invading when the trans- The inset shows that e2(0, t) and eG(0, t) both lead to fast light with very close pulse advances. mission contrast C increases. For C ≥ 25 dB (L ≥ 2.9), there is a range of pulse durations for which the peak intensity of the second pulse exceeds that of the first one. There is then a transition from fast to slow light [START_REF] Talukder | Superluminal to Subluminal Transition in the Pulse propagation in a Resonantly Absorbing Medium[END_REF][START_REF] Talukder | Measurement of Net Group and Reshaping Delays for Optical Pulses in Dispersive Media[END_REF][START_REF] Kohmoto | Nonadherence to the conventional group velocity for nanosecond light pulses in Rb vapour[END_REF][START_REF] Nanda | Why do superluminal pulses become subluminal once they go far enough?[END_REF]. When C = 50 dB and e(0, t) = e 2 (0, t), this occurs for 0.185 < γτ p < 2.95. For Gaussian incident pulses, the upper limit increases up to γτ p ≈ 3.34 and the inset of Fig. 8 (obtained for γτ p ≈ 3.18) shows that the slow light regime is actually attained. We however remark that fast light is always obtained when the pulse duration is sufficiently short or long.

As mentioned in our introduction the main challenge of fast light experiments is to obtain a significant fractional advance with moderate pulse distortion. The first objective is attained in the conditions of Fig. 8 but not the latter owing to the presence of large secondary lobes in the intensity profile of the transmitted pulse. In order to reduce their relative intensity to a few percents, the incident pulse should be lengthened and this obviously reduces the fractional advance. Figure 9 is obtained in the conditions of Fig. 8 for γτ p = 7.5 instead of 3.18. The relative intensity of the second maximum (the fractional advance) then falls to R ≈ 2 % (F ≈ 67 %) whereas a ≈ 0.88a g . Note that the intensity profile of the transmitted pulse is perfectly fitted by the periodic solution | e 2 (ℓ, t)|

2 . This result is general. As long as the periodic solution works for the pulse duration maximizing the fractional delay, it works better for longer pulses. In particular, the approximation e (ℓ, t) ≈ e 1 (ℓ, t), very good for L = 1 and Ω 1 = γ (Fig. 6), is excellent when L = 1 and Ω 1 = γ/4 (Fig. 4), leading in this latter case to an advance which is exact with four significant figures.

When the intensity of the secondary lobes is weak enough, the distortion of the transmitted pulse mainly consists in a narrowing. This simply results from the negative value of the second cumulant of H (ℓ, Ω) and the additivity of the cumulants [START_REF] Dooges | Deterministic Methods in Systems Hydrology[END_REF]. The narrowing of the pulse originates a difference between the advances a ↑ at its rise and a ↓ at its fall. The distortion can then conveniently characterized by the narrowing indicator ξ = (a ↓ -a ↑ ) / (a ↓ + a ↑ ) where the advances a ↑ and a ↓ are measured at half-maximum intensity [START_REF] Macke | Propagation of light-pulses at a negative group-velocity[END_REF]. In the conditions of Fig. 4 [Fig. 9], we get ξ ≈ 8 % [ξ ≈ 13 %] in agreement with the value derived from Eq.( 24) [Eq. [START_REF] Stenner | Pump-beam-instability limits to Raman-gain doublet "fast light" pulse propagation[END_REF]].

The procedure of periodic continuation used in this section to maximize the fractional pulse advance can also be used to determine the effect of a detuning of the carrier frequency ω c of the pulses from the resonance frequency ω 0 . In a frame rotating at ω c , the transfer function reads

H (ℓ, Ω) = exp - L 1 + i (Ω + ∆) /γ ( 32
)
where ∆ is the detuning. When e(0, t) = e 2 (0, t), we get for e 2 (ℓ, t) the analytical expression

e 2 (ℓ, t) = 3H (ℓ, 0) 8 + H (ℓ, Ω 2 ) 4 exp (iΩ 2 t) + H (ℓ, -Ω 2 ) 4 exp (-iΩ 2 t) + H (ℓ, 2Ω 2 ) 16 exp (2iΩ 2 t) + H (ℓ, -2Ω 2 ) 16 exp (-2iΩ 2 t) (33)
which is reduced to Eq.( 30) in the resonant case. The envelope e 2 (ℓ, t) is now complex and the corresponding intensity | e 2 (ℓ, t)| 2 does not fall to zero in its main part. This behavior is illustrated on Fig. 10 obtained in the conditions of Fig. 7 for ∆/γ = ±0.6, ±0.5, ±0.3 and 0 (from top to bottom). As expected, the advance (the pulse amplitude) decreases (increases) as a function of the detuning, the fractional advance falling from 55 % for ∆ = 0 to 29 % for ∆ = ±0.6γ. Note also the disappearance of the secondary lobes and the strong asymmetry of the transmitted pulse when the detuning exceeds 0.4γ.

As expected the analytical periodic solution | e 2 (ℓ, t)| 2 perfectly fits the response |e 2 (ℓ, t)| 2 to the pulse of envelope e 2 (0, t) in the whole domain where e 2 (0, t) = 0 (|t| ≤ π/Ω 2 ≈ 1.9τ p ). Similar results are obtained when C = 50 dB . Taking ∆ = γ/5 and τ p = 6/γ, it is in particular possible to reproduce the fractional advance (F = 68 %) and the shape (including its asymmetry) of the probe pulse which have been recently evidenced in a four-wave mixing experiment [START_REF] Swaim | Faster light with competing absorption and gain[END_REF]. Again the intensity profile is very well fitted by the analytical periodic solution. We incidentally mention that, for the same contrast and the same detuning, slow light is obtained for shorter durations of the incident pulse. Otherwise said, a small detuning accelerates the transition from fast to slow light. This phenomenon, easily explained by a spectral analysis, is general and an asymmetry of the absorption line has a comparable effect [START_REF] Talukder | Superluminal to Subluminal Transition in the Pulse propagation in a Resonantly Absorbing Medium[END_REF][START_REF] Nanda | Why do superluminal pulses become subluminal once they go far enough?[END_REF].

Results obtained on resonance with a symmetric absorption line can be extended to the case where the absorption is exactly compensated by a gain which is also on resonance at the carrier frequency of the incident pulses. The medium is then transparent at this frequency and the transfer function of this twofold resonant medium reads

H (ℓ, Ω) = exp L 1 + iβΩ/γ - L 1 + iΩ/γ (34)
where L and γ refer to the absorption line as previously and β is the ratio of the widths of the absorption and gain lines (0 < β < 1). The transmittance |H (ℓ, Ω)| 2 actually equals 1 for Ω = 0 (transparent medium). It is maximum for Ω = ±γ/ √ β, yielding an intensity contrast

C = exp 2L 1 -β 1 + β . ( 35
)
For arbitrary β, the phase [START_REF]We use the definitions, sign convention and results of the linear system theory[END_REF] and the group advance read

Φ (ℓ, Ω) = LΩ/γ 1 + Ω 2 /γ 2 - βLΩ/γ 1 + β 2 Ω 2 /γ 2 (36)
a g = (1 -β) L γ = (1 + β) ln C 2γ (37)
In the limit β → 0 (infinitely broad gain-profile), these expressions are obviously identical to those obtained with a single absorption line. The second form of a g in Eq.( 37) might lead to expect that, for a same contrast C, the systems with β = 0 are more efficient that the reference medium. This is true for the absolute group advance but not for the fractional advance. An examination of the phase Φ (ℓ, Ω) as a function of Ω [5] yields a qualitative explanation of this apparent paradox. Figure 11, obtained for a same value of C when β = 0 and β = 1/2, shows that the slope dΦ/dΩ at Ω = 0 is actually larger for β = 0 . On the other hand the spectral domain where this slope is positive is narrower and, for a same distortion of the transmitted pulse, the incident pulse should be lengthened to the prejudice of the fractional advance.

In fact the maximum fractional advance does not significantly depend on β and is even independent of it in the range of validity of the approximation e (ℓ, t) ≈ e 1 (ℓ, t).

We then get

F ≈ Φ (ℓ, Ω 1 ) Ω 1 τ p ≈ 0.44L Ω 1 /γ 1 + Ω 2 1 /γ 2 - βΩ 1 /γ 1 + β 2 Ω 2 1 /γ 2 . ( 38
) F is maximum for 2β 2 Ω 2 1 /γ 2 = β 2 + 4β + 1 - (β + 1) β 2 + 6β + 1. This maximum reads F m ≈ 0.22L 1 -β 1 + β = 0.11 ln C (39)
and is actually independent of β for a given contrast. Moreover a simulation made for L = 3 and β = 1/2 (ln C = 2) leads to a normalized intensity profile that perfectly fits that obtained with a single absorption line for a same contrast, that is for L = 1 (Fig. 6).

VI. DISCUSION

In the numerous theoretical articles on fast light published in the last eighteen years, fast light in absorbing media is generally discarded in favor of gain-assisted fast light (renamed superluminal light). We point out on the contrary that, for a same fractional advance and a same distortion of the transmitted pulse, absorbing media are preferable to media with gain. In the most cited experimental report on fast light [START_REF] Wang | Gain-assisted superluminal light propagation[END_REF], the authors exploit the anomalous dispersion associated with a minimum of gain between two gain lines created by Raman pumping. As remarked in a premonitory article [START_REF] Steinberg | Dispersionless, highly superluminal propagation in a medium with a gain doublet[END_REF], the use of an amplifying medium raises some difficulties due in particular to the phenomenon of amplified spontaneous emission and a risk of lasing on the gain lines. Cross modulation instability also limits the gain that can be used [START_REF] Stenner | Pump-beam-instability limits to Raman-gain doublet "fast light" pulse propagation[END_REF]. Another point is that, for a given contrast, the gain-doublet arrangement yields weaker pulse advances than those obtained with a single absorption line [START_REF] Macke | Optimal superluminal systems[END_REF][START_REF] Macke | Propagation of light-pulses at a negative group-velocity[END_REF]. The fractional advance evidenced in [START_REF] Wang | Gain-assisted superluminal light propagation[END_REF] is actually very small (below 3 %) and, as noted in [START_REF] Ringmacher | Comment on "Gain-assisted Superluminal Light Propagation[END_REF], the advances at the rise and at the fall of the pulse are significantly different with a narrowing indicator ξ = 20 %. Moreover it should be remarked that, contrary to the title given in a subsequent article [START_REF] Dogariu | Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group delay[END_REF], the medium is far from being transparent with a transmittance of only 40 % and that the experimental results reported in [START_REF] Wang | Gain-assisted superluminal light propagation[END_REF][START_REF] Dogariu | Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group delay[END_REF] can be reproduced with the single absorption-line arrangement with a transmittance better than 66 %. See Fig. 1 in [START_REF] Macke | Propagation of light-pulses at a negative group-velocity[END_REF]. More convincing experiments using the gain-doublet arrangement are reported in [START_REF] Stenner | The speed of light in a 'fast-light' optical medium[END_REF]. The effects of the cross modulation instability [START_REF] Stenner | Pump-beam-instability limits to Raman-gain doublet "fast light" pulse propagation[END_REF] are here overcome by using two separate spatial regions for the two Raman pump beams [START_REF]pumping regions is used in the experiments on an optical fiber reported in[END_REF]. The fractional advance is now F ≈ 10 % with a narrowing indicator ξ ≈ 13 %. We however remark that the same fractional advance can be obtained in the single absorption-line arrangement with a fairly good transmittance (14 %) and a weaker pulse narrowing (ξ ≈ 8 %). See Fig. 4.

Independently of the abovementioned reasons that restrict the use of gain media and of eventual problems of detection in the case of absorbing media, one may wonder why fractional advances exceeding 40 % with moderate distortion have not been evidenced in linear media. The response is obviously that the large contrasts C required to attain this purpose entail a dramatic sensitivity of the transmitted pulse to defects of the incident one. As evidenced in the experiment reported in [START_REF] Ségard | Observation of negative velocity pulse propagation[END_REF] with C ≈ 45 dB, imperceptible quasi-discontinuities of the envelope of the incident pulse or of its derivatives originate transients (wiggles) in the envelope of the transmitted pulse which become invading when C is enhanced (even slightly). Simulations also show that the effect of a chirping of the incident pulse may become dramatic for large contrast. Although this point is beyond the scope of the present article (it is a subject per se), it is also probable that the effect of a partial light-incoherence, moderate for small contrast [START_REF] Wang | Effect of coherence on the superluminal propagation of light pulses through anomalously dispersive media with gain[END_REF], will be destructive when this contrast increases.

Compared to other arrangements used to evidence fast light, the single absorption-line arrangement has obviously the advantage of its simplicity, from both experimental and theoretical viewpoints. We however mention that the contrast required to attain a given fractional delay can be slightly reduced by using a doublet of absorption lines [START_REF] Macke | Propagation of light-pulses at a negative group-velocity[END_REF], the experimental implementation of which seems quite feasible. When the lines are separated by 2γ/ √ 3, the lowest order contribution to the pulse distortion cancels and the medium transmittance is flattened around Ω = 0. We then get ln C = 3L and

Φ (ℓ, Ω) = 2L (Ω/γ) 2/3 + Ω 2 /γ 2 (4/3 + Ω 2 /γ 2 ) 2 -4Ω 2 /3γ 2 . ( 40
)
In the range of validity of the approximation e (ℓ, t) ≈ e 1 (ℓ, t), the fractional advance reads F ≈ Φ (ℓ, Ω 1 ) / (Ω 1 τ p ) ≈ 0.44Φ (ℓ, Ω 1 ) and attains its maximum F m ≈ 0.39L for Ω 1 ≈ 1.39γ (τ p ≈ 1.65/γ). We have thus F m ≈ 0.13 ln C instead of 0.11 ln C with a single absorption-line. This means in particular that the fractional delay obtained for C ≈ 8.7 dB with a single absorption-line (Fig. 6) would be obtained for C ≈ 7.3 dB with a doublet. A simulation shows that the corresponding intensity profiles are very close, both being perfectly fitted by the analytical periodic solution of Eq.( 24). Similar results are obtained for large contrasts. An intensity profile comparable to that obtained Fig. 9 for C ≈ 50 dB with a single absorption-line can be obtained for C ≈ 47 dB , both profiles being now well fitted by the analytical periodic solution of Eq.( 30).

VII. CONCLUSION

The observation of significant fast-light effects requires the use of systems with a large contrast C between maximum and minimum transmissions. We have studied in detail the reference case of a dilute medium with a narrow absorption line, the frequency of which coincides with the carrier frequency of the incident pulse. The impulse response h (ℓ, t) relating the envelope e (ℓ, t) of the transmitted pulse to that e (0, t) of the incident pulse is then real. In a retarded time picture (time delayed by the transit time in vacuum), the group advance a g is positive no matter the propagation distance and can be identified to the advance of the center-of-gravity of e (ℓ, t) over that of e (0, t). This advance generally differs from the advance a of the pulse maximum and, quite generally, a large group advance is not a sufficient condition to observe significant fast light effects.

By convoluting h (ℓ, t) and e (0, t), fast light appears as resulting from the interference of the incident wave as if it had propagated in vacuum with the wave reemitted by the medium. Explicit analytical expressions of the convolution product are obtained in the short and long pulse limits. In the first case, we get a ≪ a g with moderate attenuation of the peak intensity. The experimental evidence of this behaviour seems to be an open challenge. In the second case, a tends to a g and the peak intensity is reduced by the factor C. Most experimental results have been obtained in conditions more or less approaching these ones. In both cases the fractional advance F = a/τ p is not optimum.

The observation of significant fractional advances leads to use incident pulses of intermediate duration. Analytical results are then obtained by a method of periodic continuation of the incident pulses. When the contrast C is below 20 dB as in most experiments actually performed in optics, the duration τ p maximising F has an explicit form and the corresponding advance a is half of the group advance a g . This latter result remains a good approximation when C is increased but the main transmitted pulse is then followed by large secondary pulses. Their lowering demands a lengthening of the pulses that reduces F below its maximum value.

The range of validity of the method of periodic continuation is very broad. It enables one to reproduce analytically the asymmetry of the transmitted pulses when the carrier frequency of the incident pulses is detuned from resonance. In the case of a resonant transparent medium, it shows that, for a same contrast, the maximum fractional advance is roughly equal to that of our reference medium.

Insofar as the fractional advance is mainly determined by the transmission contrast, the use of amplifying media seems unsuitable to evidence significant fast-light effects. Large gains indeed originate optical instabilities, obviously absent in absorbing media. Anyway, the use of large contrasts requires an exceptional quality of the incident pulse, the smallest imperfections of which can lead to considerable distortion of the transmitted pulse. That is probably the main practical limitation to fast light. Improvement of the quality of the incident pulses combined with the use of a doublet of absorption lines instead of a single line appears to be one way to evidence fast-light effects beyond those observed up to now.

Bruno

  Macke and Bernard Ségard * Laboratoire de Physique des Lasers, Atomes et Molécules , CNRS et Université de Lille, 59655 Villeneuve d'Ascq, France (Dated: July 3, 2018)

Figure 2 :

 2 Figure2: Normalized intensity profiles of the transmitted pulse obtained for e(0, t) = e1(0, t) , L = 5.756 (transmission contrast C = 50 dB ) and τp = 0.06/γ (Ω1 = 38γ ). The solid red line (the dashed blue line) is the exact solution obtained by FFT (the analytical solution obtained in the short pulse limit). As in all the following figures, the intensity profile of the incident pulse (dotted black line) is given for reference. Its full width at half maximum τp is taken as time unit. Inset: comparison of the envelopes of the transmitted pulse obtained for e(0, t) = e1(0, t) (solid red line) and e(0, t) = eG(0, t) (dashed blue line). The pulse tail displays a first zero for t/τp ≈ j 2 1,1 / (4Lγτp) ≈ 10.6 (see text).

Figure 3 :

 3 Figure 3: Same as Fig 2 when L = 1 (C ≈ 8.7 dB, dip depth D ≈ 86%) and τp = 0.2/γ (Ω1 = 11.4γ ). The tail of the envelopes (inset) has now an exponential decrease (see text).

Figure 4 :

 4 Figure4: Normalized intensity profiles of the transmitted pulse obtained for e(0, t) = e1(0, t) , L = 1 (C = 8.7 dB, dip depth D ≈ 86% ) and Ω1 = γ/4 (τp ≈ 9.15/γ). The solid red line (the dashed blue line) is the exact solution obtained by FFT (the analytical solution obtained in the long pulse or adiabatic limit). Inset: comparison of the envelopes of the transmitted pulse obtained for e(0, t) = e1(0, t) (solid red line) and e(0, t) = eG(0, t) (dashed blue line).

Figure 5 :

 5 Figure 5: Same as Fig 4 when L = 5.756 (C = 50 dB ) instead of L = 1. Note the presence of a precursor and a postcursor which had negligible amplitudes for L = 1 and are obviously absent in the case of a Gaussian incident pulse (see inset).

Figure 6 :

 6 Figure6: Comparison for L = 1 (C = 8.7 dB, dip depth D ≈ 86%) of the exact intensity profile of the transmitted pulse (solid red line) with the analytical profile obtained by periodically continuing the incident pulse e(0, t) = e1(0, t) (thin blue line). The pulse duration is that maximizing the fractional advance (Ω1 = γ, τp ≈ 2.29/γ) . It leads to a = ag/2 and to F = L/2γτp ≈ 22% . Inset: comparison of the envelopes of the transmitted pulse obtained for e(0, t) = e1(0, t) (solid red line) and e(0, t) = eG(0, t) (dashed blue line).

Figure 7 :

 7 Figure 7: Comparison for L = 3 (C = 26 dB) of the exact intensity profiles |e2(l, t)| 2 (solid red line) and |e1(l, t)| 2 (dashed blue line) with the analytical periodic profile | e2(l, t)| 2 (thin blue line). The pulse duration (τp ≈ 2.55/γ) is that maximizing the fractional advance of | e2(l, t)| 2. Inset: comparison of the envelopes of the transmitted pulse obtained for e(0, t) = e2(0, t) (solid red line) and for e(0, t) = eG(0, t) (dashed blue line).

Figure 8 :

 8 Figure 8: Comparison for L = 5.756 (C = 50 dB ) of the exact intensity profiles s |e2(l, t)| 2 (solid red line) with the analytical periodic profile | e2(l, t)| 2 (thin blue line).The pulse duration is that maximizing the fractional advance, that is τp ≈ 3.18/γ leading to F ≈ 94% and a = 0.52 ag. Inset: comparison of the envelopes of the transmitted pulse obtained for e(0, t) = e2(0, t) (solid red line) and for e(0, t) = eG(0, t) (dashed blue line). Note that the slow light regime is attained for the Gaussian pulse (see text).

Figure 9 :

 9 Figure 9: Same as Fig 8 for τp ≈ 7.5/γ instead of 3.18/γ . The relative intensity of the second maximum is so reduced to R = 2% whereas the fractional advance falls to F ≈ 67%.The inset shows that e2(0, t) and eG(0, t) both lead to fast light with very close pulse advances.

Figure 10 :

 10 Figure 10: Effect on the transmitted intensity of a frequency detuning of the incident pulse for e(0, t) = e2(0, t). Parameters L = 3, (C = 26 dB), τp ≈ 2.55/γ (as in Fig 7) and, from top to bottom, ∆/γ = ±0.6, ±0.5, ±0.3 and 0. The solid red lines (the thin blue lines) are the exact solutions (the analytical periodic solutions).

Figure 11 :

 11 Figure 11: Phase of transparent resonant media as a function of the detuning. The solid red line (the dashed blue line) is obtained for β = 1/2 (β = 0). The phase (the detuning) is expressed in units of ln C (γ).

ACKNOWLEDGEMENTS

This work has been partially supported by the Ministère de l'Enseignement Supérieur, de la Recherche et de l'Innovation, the Conseil Régional des Hauts de France and the European Regional Development Fund (ERDF) through the Contrat de Projets État-Région (CPER) 2015-2020, as well as by the Agence Nationale de la Recherche through the LABEX CEMPI project (ANR-11-LABX-0007).