Clement Delmas
email: delmas.clement@chutoulouse.fr

Hélène Barbosa

Charles-Henri David

Frédéric Bouisset

Pierre-Guillaume Piriou

François Roubille

Juergen Leick

Marin Pavlov

Guillaume Leurent

Evgenij V Potapov

Pierre- Guillaume Piriou

Axel Linke

Johannes Mierke

Pia Lanmüller

Norman Mangner

Impella for the Management of Ventricular Septal Defect Complicating Acute Myocardial Infarction: A European Multicenter Registry

Keywords: mechanical circulatory support, mortality, acute myocardial infarction, mechanical complication

Ventricular septal defect (VSD) is a rare but fatal complication of myocardial infarction (MI). Over the past decades, the VSD occurrence has gradually declined caused by the increased use of rapid revascularization. Nevertheless, VSD is estimated to complicate about 0.3% of ST elevation myocardial infarction (STEMI).1 Clinical presentation ranges from isolated murmur on thoracic auscultation to a severe form of cardiogenic shock (CS). Post-MI VSD is a serious disease with a high early mortality of 40-80% escalating to up to 94% at 30 days in case of medical management. Despite medical advances, this prognosis has remained unchanged for years, particularly in cases of hemodynamic instability and CS.2 Surgical septum repair is the mainstay treatment, mostly by VSD patch closure through a double ventriculotomy technique but remains a high-risk procedure. Percutaneous closure is increasingly undertaken alone or in association with surgical management with promising results in selected patients.3 However, VSD closure is not always technically feasible (necrotic edges) or patients are not operable caused by hemodynamic compromise and severe comorbidities including multiple organ failure associated with CS itself. [START_REF] Chioncel | Epidemiology, pathophysiology and contemporary management of cardiogenic shock-A position statement from the Heart Failure Association of the European Society of Cardiology[END_REF] Observational and retrospective data suggest that early repair (<7 days from MI) appears to be associated with higher mortality, but these results may reflect a patient selection bias in favor of late surgery because only stabilized patients could be operated.5 Clinical stabilization and evolution of the infarcted heart tissue theoretically allow for a better tissue suture substrate and more efficient repair.

Therefore, while awaiting surgery, treatment should focus on lowering left ventricle end diastolic pressure (LVEDP) to decrease the left-to-right shunt and myocardial oxygen consumption, while increasing anterograde cardiac output to maintain organ perfusion and prevent/treat secondary multiple organ failure. Although medical management using catecholamines may be considered, their effect may be limited in patients with CS. Temporary mechanical circulatory support (t-MCS) devices are a promising therapeutic option given their ability to improve hemodynamics and tissue perfusion.6 However, data on the intracardiac and systemic hemodynamic effects of t-MCS in post-MI VSD or based on hemodynamic simulation are scarce.7 Although the catheter-based axialflow pump Impella (Abiomed, Danvers, MA, USA) is not recommended in this clinical setting by the manufacturer, experienced centers used it in post-MI VSD patients. We collected the data through a European multicenter retrospective study to describe characteristics, associated management, and outcomes.

Materials and Methods

Study Design

We conducted a European retrospective multicenter registry that aimed to collect post-MI VSD patients managed with Impella device (2.5, CP or 5.0) during the last 12 years (2010-2021). The registry was proposed by email or phone to experts' European centers recognized for their experience with Impella in CS management.

Population

Patients age of greater than or equal to 18 years who received an Impella for hemodynamic support caused by a post-MI VSD admitted to the critical care unit of the participating center were included. The non-inclusion criteria were patient under measure of justice. Cardiogenic shock was defined by the combination of a systemic hypotension (systolic blood pressure [SBP] ≤ 90 mm Hg or need for vasopressors/inotropes) with presence of at least one sign of systemic hypoperfusion (eg, confusion, oliguria, marbles, or lactate elevation) and signs of pulmonary congestion (crackles or left ventricular filling pressures elevation based on noninvasive or invasive evaluation). The choice of the t-MCS, its timing but also any t-MCS combination was made locally by the team treating the patient in accordance with the contemporary local protocols of the VSD management.

Data Collection

Data were abstracted from the medical record to a standardized electronic case report form by the sites study coordinators. Information on the patient's demographic characteristics, medical history, clinical presentation, hemodynamic, echocardiographic, and angiographic characteristics, treatment received during hospitalization, hospital discharge status, and follow-up status were collected.

Follow-Up

At 24 hours, simplified acute physiology score (SAPS) II score, noninvasive and invasive hemodynamic, biology (lactate, creatinine, bilirubin, lactate deshydrogenase [LDH], hemoglobin, platelets), medication, and organ support were also collected.

In-hospital complications were collected based on medical file of the patients (ischemic and hemorrhagic stroke, acute renal failure requiring dialysis, major bleeding; Bleeding Academic Research Consortium classification [BARC] ≥ 3 or needing surgery), acute limb ischemia requiring interventional or surgical management, Impella device dysfunction needing its drop off, infection (insertion sites, septicemia, pneumonia), clinical hemolysis needing medical intervention (vascular filling, reduction of axial flow pump (AFP) flow or device replacement or drop off).

All-cause mortality was assessed at hospital discharge (primary end-point) and at 6 and 12 months by consultation of the patient's medical file, or telephone call of the patient, his attending physician, or relatives.

Ethics and Regulation

The study was consistent with the principles set out in the Helsinki Declaration. After evaluation and validation by the data protection officer and in accordance with the General Data Protection Regulation, this study has met all the criteria and is registered in the retrospective study register of the Toulouse University Hospital (number's register: RnIPH2021-101) and covered by the MR-004 (CNIL number: 2206723 v0). The individual anonymized data sharing were performed according to the permission of the Independant review board (IRB) of each center.

Statistical Analysis

Continuous variables are presented as medians (interquartile range) and categorical variables as counts (percentage). First, we compared patients according to in-hospital survival. Second, we compared patients according to the type of Impella used: surgical (mainly 5.0) versus percutaneous (exclusively 2.5 and CP). Third, patients were compared according to the associated VSD management (surgical versus percutaneous closure versus medical). Comparison between groups was made with Wilcoxon signed rank tests for continuous variable and with χ 2 tests or Fisher exact tests according to their respective conditions of applicability for categorical variables.

All tests were two-sided, and p values of less than 0.05 were considered statistically significant. Missing data were systematically reported as they were removed from the statistical analysis. All statistical analyses were performed with XLSTAT 2022.1 by Addinsoft (Paris, France).

Results

Population

Of 20 centers contacted, 17 (85%) answered, and 7 (41.2%) reported the use of Impella for a post-MI VSD in at least one case. Table S1, Supplemental Digital Content, http://links.lww.com/ASAIO/B131, describes the center that responded and number of VSD cases included. Over the study period (2010-2021), a total of 28 patients were included. Table 1 describes patients' characteristics at admission. Patients were predominantly male (n = 18, 64%) and were 66 (58.5-78.5) years old. Survivors were younger (61 vs. 74 years old, p = 0.02). Cardiovascular risk factors were frequent, especially hypertension (n = 17, 61%). At admission for acute coronary syndrome (ACS) (see Table S2, Supplemental Digital Content, http://links. lww.com/ASAIO/B131), patients were hemodynamically stable (median systolic blood pressure 110 (86-127) mm Hg with a median heart rate of 100 (86-106) beats/min). Nevertheless, 16 (57%) patients presented with left and 13 (46%) patients with right heart failure signs. Trans-thoracic echocardiography showed a left ventricle dysfunction with left ventricular ejection fraction (LVEF) of less than or equal to 40% for 14 (50%) patients and 15 (53%) patients had right ventricle dysfunction (tricuspid annular plane systolic excursion [TAPSE] < 17 mm). The right coronary artery was the main culprit lesion found for 20 patients (71%).

Median time between ACS occurrence (defined as chest pain time) and medical care was 3 days (interquantile range [IQR] = 1-6). Coronary revascularization was performed in 64% of the patients: 57% by percutaneous coronary intervention (n = 16) and 11% by coronary artery bypass graft (CABG) (n = 3).

Presentation at Ventricular Septal Defect Diagnosis

Baseline characteristics of the population at VSD diagnosis are presented in Table 2. Median time from pain onset to VSD diagnosis was 5 days (IQR 3-8). The median systolic blood pressure was 101 mm Hg with a diastolic blood pressure of 62 mm Hg and a heart rate of 106 beats/min. Eighteen (64%) and 17 (61%) patients presented with signs of left and right heart failure, respectively. Oliguria was present in 18 (64%), mottling in 15 (54%), and confusion in 10 (36%) patients. Trans-thoracic echocardiography showed an LVEF of less than or equal to 40% in 16 (57%) patients, and 17 (61%) patients had right ventricle dysfunction (TAPSE < 17 mm). The VSD was apical in 10 (36%) and inferior basal in 17 (61%) patients (localization not available for one patient).

Management

Table 3 describes the in-hospital management. Dobutamine support was used for 17 (55%) patients with a median duration of 3 days (IQR 1-7). Norepinephrine support was required for 27 (96%) patients with a median duration of 3 days (IQR 2-8). Twenty-five (89%) patients needed invasive mechanical ventilation and 14 (40%) renal replacement therapy.

The Impella was implanted early (on the day of vasopressor/ inotropes introduction) via a percutaneous or a surgical approach in 21 (75%) and seven (25%) patients, respectively. Five patients received a 5.0 device, 2 a surgical CP device, 18 a percutaneous CP device, and 1 a 2.5 device. The type of Impella device was not known for two patients. The median duration of support was 6 days (IQR 2-11). Additional t-MCS was used in 14 (50%) patients, mainly as an extracorporeal life support (ECLS) (n = 9, 32%) (Figure 1).

A surgical or a percutaneous VSD closure was attempted in 10 (36%) and six (21%) patients, respectively. Among these patients, 12 (75%) were supported by Impella before the VSD closure and four (25%) patients were implanted the same day. No combination of surgical and percutaneous closure was reported. Finally, an isolated medical management was selected for 12 (43%) patients.

In-Hospital Complications

The most common Impella-associated complication was a major bleeding (BARC ≥ 3) in 14 (50%) patients followed by a clinical hemolysis (n = 3, 11%) (Table 4). No significant difference between survivors and nonsurvivors was found. Figure S1, Supplemental Digital Content, http://links.lww.com/ASAIO/ B131, shows differences in Impella-associated complications according to the pump type (percutaneous versus surgical).

Outcomes and Associate Correlates

We observed no significant difference between survivors and non-survivors in clinical parameters or biomarkers within the first 24 hours from the time of Impella implantation (not shown).

In-hospital and 1 year mortality rates were both 75%. Only seven (25%) patients were discharged alive from hospital. No long-term ventricular assist device or heart transplantation was realized.

We found no difference in survival according to the type of Impella used (percutaneous versus surgical) (see Figure S2, Supplemental Digital Content, http://links.lww.com/ASAIO/ B131) nor according to the different combinations of t-MCS used (Figure 2). On the contrary, we found a significantly higher survival for patients with combined surgical closure and Impella support (Figure 3 and Table 3). There was no difference in terms of delay between VSD diagnosis and closure between survivors and non-survivors.

Survivors were more likely to have undergone CABG revascularization (43% vs. 0%, p = 0.011). Moreover, non-survivors were older (74 vs. 61 years old, p < 0.022) at admission. No other significant difference was found according to medical history, characteristics at admission or at VSD diagnosis, or according to management.

Discussion

We reported the first large multicentric series of patients with post-MI VSD supported by Impella device regardless of the type of implantation (surgical or percutaneous) and the associated VSD management (Figure S1, Supplemental Digital Content, http://links.lww.com/ASAIO/B131). This report shows feasibility of Impella support in compensating the interventricular shunt in post-MI VSD patients. Further, patients without surgical repair do not survive. Elucidating the place of Impella in the management of post-MI VSD will require larger cohorts and standardization of patient/VSD management, to assess whether these serious outcomes are patient-related, secondary to the device itself or timing of the implantation.

In the registry, we reported an excessive mortality of post-MI VSD patients (75% in-hospital and 1 year mortality). The cause of such a high mortality is difficult to pin down unambiguously however, the results must be interpreted considering several points and potential confounders. First, we included exclusively patients in CS (stage C-E from the Society for Cardiovascular Angiography and Interventions (SCAI) classification), whereas recent UK and Japanese nationwide registries reported only 50-60% of patients in CS.3,5 In our cohort, the shock was more profound, inotropes, and vasopressors were used more often (dobutamine 55% and norepinephrine 96% in our cohort vs. 26% of inotropes in the Japanese registry), as well as renal replacement therapy (40% in our population vs. 25% in the Japanese registry).5 In accordance to our hypothesis, CS at presentation was independently associated with in-hospital (adjusted hazard ratio [HR] = 2.17 [1.43-3.29], p < 0.001) and long-term mortality (adjusted HR = 1.97 [1.37-2.84], p < 0.001) in the recent and large UK registry.3 Second, we included patients regardless of whether VSD closure (surgical or percutaneous) was attempted for technical or ethical considerations. Conservative treatment was associated with extremely high early mortality in multiple studies (>90% in-hospital mortality). We aimed to report the Impella device use regardless of the associated treatment contrary to previous series focusing on the mode of closure (surgery or interventional cardiology) which eliminated patients managed medically. This may in part explain high mortality rate in our cohort since, as expected, none of the 12 patients managed medically survived. The higher inhospital survival (70%) was observed for AFP patients with associated surgical closure.

For European Society of Cardiology Guidelines, t-MCS may be used as a bridge-to-recovery or surgery in patients with VSD associated CS with a class IIa recommendation (level of evidence C) without specifying the type and timing of the support.8 To date, mainstay is still intra-aortic balloon pump (IABP) based on the old data suggesting an afterload reduction and a left-to right shunting decrease. Alternatively, the use of Tandem Heart (Cardiac Assist Inc., Pittsburgh, PA) was reported in a small series (11 post-MI VSD patients) with interesting results in case of early implantation before VSD surgical closure with only 25% mortality rate at 6 months.9 Limited availability and the need for high expertise for implantation may explain the low implementation rate. Recent simulated hemodynamic analysis comparing effects of different t-MCS in VSD model showed more efficient hemodynamic effect with Impella.7 Indeed, despite attractive and potentially advantageous hemodynamic effects of Impella over other devices, the data on its use alone or in combination with ECLS in the context of post-MI VSD are scarce.10-12

The potential positive effect of the Impella is based on a direct pumping of the blood from the left ventricule (LV) to the aorta, increase in total flow to the body (+1.1 to +1.6 L/ min of cardiac output according to its size (Impella CP vs. 5.0) versus +0. 5 L/min for IABP), and significant decrease in the left to right shunt through the VSD (decrease of the Qp: Qs from 3.0 to 2.3 and 2.1, respectively for Impella CP and 5.0).7

Consequently, direct LV unloading decreases LVEDP and increases coronary perfusion, suggesting higher potential LV recovery and preventing worsening of the right ventricule (RV) function. Small and retrospective design of our case series with a lack of hemodynamic data, and the absence of control group preclude the confirmation or rejection of our conclusions.

Potential benefit of direct LV unloading could be counterbalanced by the adverse effects associated with the device itself (hemolysis, bleedings, infection), the risk of producing rightto left shunting and the risk of increased VSD and systemic embolization because of its intraarterial/intraventricular position.13,14 In our population, the three most relevant inhospital complications were significant bleeding (50%), acute renal failure (46%), and pneumonia (36%). Bleeding is a well-known complication of t-MCS regardless of the type, a fact confirmed in studies with Impella support in 8-71% of cases according to the definition used and context of implantation and support.15 Indeed access site-related bleeding and associated transfusion are reported in up to 25% and 67% of Impella cases respectively with higher rate in case of CS and t-MCS combination.13,15,16 Numerous factors depending on 1) the patient: initial presentation (hemodynamic instability and multiorgan failure), comorbidities (peripheral artery disease), associated treatment (anticoagulation and antiplatelets regimens); 2) the device: implantation, management, and explanation (echo or fluoroscopic guidance, device fixation, fluids and inotropes management, hemostatic compression device); and 3) the possible development of acquired von Willebrand syndrome,17 could promote bleeding occurrence and may be preventable to a degree. [START_REF] Lüsebrink | Percutaneous transvalvular microaxial flow pump support in cardiology[END_REF] No deoxygenation caused by inversion of the shunt from right to left side was described contrary to previous case report. [START_REF] Sato | Right-to-left shunt depending on the support level of Impella for post-infarction ventricular septal defect[END_REF] Despite particular attention given to the risk of VSD size increase, systemic embolism, and stroke, we found no signal of harm in our small series. The risk of direct aspiration of VSD's debris by the impeller seems low and may probably be prevented by initial adequate positioning and close monitoring of the AFP. Moreover, the LV unloading may decrease shear forces delivered at the necrotic edges, thus lowering the risk of VSD enlargement and debris. Future dedicated and larger studies should pay particular attention to these issues.

Extracorporeal life support is possibly the most effective circulatory support and the best way to perfuse and oxygenate peripheral organs but several preclinical and clinical studies have shown the association with an increase LV afterload, increase in LVEDP and pulmonary capillary wedge pressure (PCWP).7, 19 LV overload correlated with the intensity of flow and associated with worst prognosis including mortality, especially in case of LV-failure.20, 21 In case of VSD, these effects potentially lead to a worsening of the left-to-right shunt with an increased risk of RV dysfunction and death.7 The combination of ECLS and Impella (association of ECLS and Impella device [ECMELLA]) could improve peripheral organ perfusion while unloading the LV and RV, however, it is associated with an increase in adverse event rates.22,23 The viable alternative is a single arterial access ECMELLA. [START_REF] Eulert-Grehn | ECMELLA 2.0: Single arterial access technique for a staged approach in cardiogenic shock[END_REF] In the absence of clear recommendation, the strategy should be chosen in a case-by-case discussion. The choice should depend on the available t-MCS, but also on the multidisciplinary expertise and practice of the center. Two important points seem to emerge from the literature: the early implantation of t-MCS aimed at preventing multiorgan failure,25 and the early LV unloading to preserve RV and LV functions. [START_REF] Schrage | Timing of active left ventricular unloading in patients on venoarterial extracorporeal membrane oxygenation therapy[END_REF] This justifies a multidisciplinary approach in expert medical surgical centers allowing an evolving personalized approach to each patient, whether in terms of medical management, type of device, and timing of implantation, but also in VSD closure strategy.

Limitations

First, and most importantly, the retrospective nonconsecutive study design precluded definitive conclusion with a possible selection bias despite the data coming from multiple centers especially as most patients came from one center. But as Impella use in post-MI VSD is scarce because of device availability, local protocols, and beliefs, this study reflects real-life practice and collection of all available cases was our main objective. We assume that our report is only an exploratory analysis to be used as a first description/step before performing larger studies. Second, no multivariate analysis was performed because of patient's heterogeneity and small number of patients. In particular, immortal time bias was not accounted for preventing any clear conclusion on the role of surgical VSD closure. Third, the international multicenter retrospective design necessitated a simplified data collection form with a limited number of variables to avoid missing data limiting analyses. Thus, the invasive hemodynamic variables before implantation or under support were available for only few patients-a fact that does not allow to present consistent hemodynamic effect of the Impella device in this context. Fourth, no consensual medical management protocol and indications for type of device and timing of implantation are known even though recommendations suggest urgent/emergent surgical repair under inotrope/vasopressor and t-MCS support.8 This is supported by data as ours, which showed that mortality associated with a conservative approach reaches 100%.

Conclusion

Impella use in patients with post-MI VSD seems feasible in this first exploratory analysis. Larger prospective registries with predefined variables and hemodynamic parameters are necessary to further elucidate potential benefits of LV unloading with this device in the setting of post-MI VSD.

Acknowledgment

Investigators: Bonello L (Hôpital Nord, AP-HM, Marseille, France), Schurtz G (Lille University Hospital, France), Gaudard P (CHU de Montpellier, Montpellier, France), Genet T (CHU de Tours, France), Lebreton G (La Pitié Salpétrière, AP-HP, Paris, France), Pappalardo F (Milan University Hospital, Italy), Hassager C (Copenhagen University Hospital, Danemark), Kindo M (Nouvel Hôpital Civil, Strasbourg, France), Pernot M (Hôpital Cardiologique, CHU Bordeaux, Pessac, France), Pozzi M, (Department of Cardiac Surgery, "Louis Pradel" Cardiologic Hospital, Lyon, France), Schibilsky D (Department of Cardiac Surgery, Freiburg University Hospital, Germany), Letocart V (Department of Cardiology, Nantes University Hospital, France), and Charbonneau E (Rennes University Hospital, France).

Conflict of interest statement

Disclosure:

C.D. has received personal fees from Abott and Abiomed outside the submitted work. G.L. has received proctor, orator and consulting fees from Abbott SA. E.V.P. has received institutional fees from Abott, Medtronic, and Abiomed outside the submitted work. J.L. has received research funds that were paid to the institution from Shockwave Medical Inc. and from Abiomed, and payment or honoraria from AstraZeneca and Boston Scientific, outside the submitted work. J.M. has received personal fees from Abiomed outside the submitted work. P.L. has received personal fees from Abiomed outside the submitted work. N.M. has received personal fees from Edwards Lifesciences, Medtronic, Biotronik, Novartis, Sanofi Genzyme, AstraZeneca, Pfizer, Bayer, Abbott, Abiomed, and Boston Scientific outside the submitted work. The other authors have no conflicts of interest to report.


Three Major Learning Theories and their Applications in Educational Settings

Mehedi Hasan Anik a a Côte d'Azur University, Nice, France I have developed simple yet useful materials concerning how three major learning theoriesbehaviorism, cognitivism, and constructivism can impact our education, and identified the major key points relevant to educational application. I organized the key points under the themes namelymajor beliefs, aims of education, characteristics of the curriculum, teaching-learning strategies, teacher-student relationship, strengths, and weaknesses. Hopefully, the comic characters and jokes will make the theories interesting to the readers. At the end of this reading of the theories, readers/teachers will be able to -▪ Explain the key ideas of the three learning theories.

▪ Explain how these theories are linked to education.

▪ Identify appropriate T/L strategies for specific learning goals.


J D Ludwig

P-E Masson-Laborde

S Hüller

W Rozmus

S C Wilks

Enhancement and Control of Laser Wakefields via a Backward Raman Amplifier

Keywords: numbers: Valid appear here Raman Amplification, Laser Wakefield Acceleration, Particle-In-Cell

The Backward Raman Amplifier (BRA) is proposed as a possible scheme for improving laser driven plasma wakefields. One-and two-dimensional particle-in-cell code simulations and a 3-Wave coupling model are presented and compared to demonstrate how the BRA can be applied to the laser wakefield accelerator (LWFA) in the non-relativistic regime to counteract limitations such as pump depletion and diffraction. This article provides a discussion on optimal parameters for the combination of BRA and LWFA and a prescription for a BRA pump frequency chirp to ensure coupling beyond the particle dephasing limit. Simulation results demonstrate a reduction or alleviation of the effects of diffraction, an increase in wake amplitude and sustainability, and provide direct insight into new methods of controlling plasma wakes in LWFA and other applications.

I. INTRODUCTION

The ability to create large amplitude plasma waves traveling near the speed of light using a laser pulse has led to several scientific breakthroughs such as laser wakefield acceleration (LWFA) 1 , betatron x-ray sources [2][3][4] , and terahertz generation 5 . In general the amplitude and application of these plasma waves can be limited by laser diffraction, depletion, and particle dephasing in the LWFA. In the strongly nonlinear, blow-out regime of the LWFA, dephasing of the relativistic electrons from the accelerating fields, the small charge of the accelerated bunch, and inability to accelerate positive charge for uses such as electronpositron colliders are limitations of this blow-out regime. Several different approaches have been proposed to overcome these drawbacks such as combining the blow-out and the direct laser acceleration scheme 6 . Staged LWFA accelerators 7 involve a pre-accelerated electron beam from the plasma "bubble" and a long wake field wave in the second acceleration stage. The alternative scenarios of LWFAs usually involve several laser beams and often employ linear wakes over a long acceleration length 8 . Inevitably they result in an increase in the particle beam emittance and come at a cost of increased size and complexity of the accelerator.

In this article we propose another approach, using a Backward Raman Amplifier (BRA) to maintain the driving laser pulse and hence to enhance and control the wakefield generation. Backward Raman amplification and compression [9][10][11] has been proposed as a laser amplification scheme; it has, however, seen mixed success in experimental demonstrations [12][13][14] . The experimental efficiency of the BRA scheme is reported to be less than 10%, which is below various theoretical predictions 15 . Clearly theoretical understanding of the BRA and of related physical processes remains incomplete. The role of wakefield generation that is emphasized in our paper is one of these physical processes that have not been discussed before in the context of the BRA. This in spite of the fact that wake generation is an inevitable feature of short pulse propagation in a plasma, which as we will demonstrate, causes a frequency shift of the seed pulse, and thus affects the BRA coupling. While the overall goal for BRA up to now has been to maximize laser pulse amplification, we consider its use as a control mechanism during plasma wake generation. Specifically, we examine the first application of BRA to amplify and sustain a short seed laser pulse while simultaneously enhancing wakefield generation in a plasma. This article will review short pulse propagation in the context of LWFA (Section II A), followed by a theoretical model for the BRA (Section II B), and present the results of combining the two (Section III A). We will describe the wake's effects on the BRA (Section III B) and use a chirped pump in the BRA scheme in order to enhance the wakefield generation (Section III C). Finally, our results are summarized in the conclusion (Section IV).

II. INTRODUCTION OF METHODS AND BASIC PROCESSES

The two processes, wake generation and BRA, have been extensively studied on their own. Before examining their nonlinear coupling and interactions we will first review the relevant properties of the linear wake and the backward Raman amplifier. The summary of known results presented below will be useful in further developments of the BRA for the wake field enhancement scenario.

The primary simulation tool in our studies is the relativistic particle-in-cell (PIC) code SCPIC 16 which has been written on the basis of the code Mandor 17 and has already been used in many high intensity laser-plasma applications for electron acceleration and LWFA [18][19][20][21] . As in similar studies [22][23][24] , we employ a moving window with the speed of light c. Simulations presented in this article will be labeled with a case number, corresponding to the seed parameters listed in Table I. In all BRA cases the plane wave pump intensity is 1 × 10 14 W/cm 2 , and the wave length is 1.064µm. In addition to PIC simulations we will consider a reduced description of the BRA based on a three wave coupling model. The wave coupling equations provide a useful description of the BRA that can be apply to long time and large distances of laser pump and seed interactions in cases where the multidimensional effects, related for example to laser pulse diffraction, are not dominant.

A. Wake Generation

The focus of our work is on short nonrelativistic pulses with a pulse duration, t pulse (FWHM), comparable in spatial extent to the plasma wavelength ct pulse ∼ λ p (with λ p = 2π/k p , k p = ω pe /c, where ω pe is the plasma frequency and c is the speed of light). The primary effect of short pulses on the background plasma is wake generation in the form of a longitudinal plasma wave at wave length λ p and phase velocity v p ≈ c. We denote in the following such pulses as "seed" pulses with the field amplitude a 1 . In the linear approximation, the plasma electron density perturbation associated with the wake, δn w , is given by

1 δn w n 0 (ζ, y, z) = - c/v g1 4k p ∞ ζ dζ sin[k p (ζ -ζ )]∇ 2 ζ ,y,z |a 1 | 2 , (1)
where ζ = x -v g1 t denotes the coordinate in the frame where the seed pulse moves with the group velocity v g1 in x-direction, n 0 denotes the background plasma electron density, and where the seed electric field amplitude is given by a 1 = eE1 mecω1 = 8.55 × 10 -10 (I 1 λ 2 1 [W cm -2 µm 2 ]) 1/2 in terms of the intensity of the short seed laser pulse I 1 and its wavelength λ 1 (λ 1 =1.13µm when n e = 0.0035n c , where n c is the critical electron density for this laser frequency), and

k p = ω pe /v g1 . The operator ∇ 2 ζ ,y,z = ∂ 2 ζ 2 + ∂ 2 y 2 + ∂ 2 z 2
is taken along the axis. To illustrate the pertinence of this expression, Eq. ( 1), we compare it in Fig. 1a with results from SCPIC simulations for a seed pulse having initially a Gaussian envelope with full-half-width-maximum (FWHM) of t pulse =30fs in time (and l s = ct pulse ≈ 9µm in length) and the peak seed pulse of amplitude a 1 = 0.205 in a plasma with density n e /n c = 0.0035 cf. Ref. 25 . This simulation corresponds to case "1", see herefore Table I for all simulation cases discussed.

FIG. 1.

In the short pulse regime, case "1" (see Table I): (a) 1D cut in x at y = 0 of fields and density from 2D PIC and theory, (b) Longitudinal wakefield from PIC, (c) Transverse wakefield from PIC. In the self-modulated regime, case "2": (d) 1D cut of fields and density from PIC, (e) Longitudinal wakefield from PIC, (f) Transverse component of wakefield from PIC. In both cases the initial seed amplitude is a1 = 0.205, and initial seed duration is 30fs. Plasma densities are respectively ne/nc = 0.0035 (case "1") (a)-(c) and ne/nc = 0.015 (case "2") (d)-(f), taken at time t = 1.16ps in the short pulse regime, and t = 13.19ps in the self-modulated regime. Both PIC simulations were run with 30cells/µm resolution, and 9 particles/cell, the simulation domains were 150x300µm for case "1" and 200x200µm for case "2".

The generation of the wake is not a resonant process but Eq. ( 1) predicts maximum response for l s = λ p /2. Alternatively, longer pulses l s > λ p give rise to "self-modulated" (SM) solutions where the wake Langmuir wave is generated at the front of the laser pulse and couples resonantly to the seed via forward Raman instability.

The simulation shown in Figures 1d,e,f, corresponding to case "2" (see Table I) illustrates, the regime of a self-modulated seed pulse with a large-amplitude wake. Except for the higher plasma density, n e /n c =0.015, resulting in a shorter plasma wave length λ p , case "2" has the same conditions as case "1". FIG. 2. Wake amplitude vs Propagation Distance reconstructed from subsequent moving window frames of 2D PIC simulation case "4". The laser is focused to the start of the plasma at x = 100µm and propogates from left to right leaving a wake in the plasma. The theoretical spot size of the laser from is overlayed in black lines with a corresponding Rayleigh length ZR ≈ 800µm. After propagating 3ZR the wake amplitude is reduced to approximatly 1/4 its initial amplitude.

To obtain sufficiently high intensities for wakefield generation, laser pulses must first undergo optical focusing. As a result of focusing, the spot size radius of the laser in a vacuum evolves as can be described by: r(x) = r 0 1 + x 2 /Z 2 R , where r(x) is the radius of the laser spot size that depends on the propagation distance x and on the Rayleigh length Z R determined by Z R = πf 2 λ where f is the mirror's f-number and λ the laser wave length. The Rayleigh length indicates the distance of laser propagation where the laser spot size is doubled in area, reducing its intensity by a factor of 2/ √ 2/1 in 3D/2D/1D. Since the amplitude of the wake depends on a 2 1 in Eq. ( 1), optimum wake generation requires the driving pulse to be focused close to its diffraction limit r 0 over as long a distance as possible. This can be achieved through the use of large f-number optics; however, physical constraints on the focusing distance limit the maximum f-number available in laboratory and therefore pulse diffraction can be the significant limiting factor in non-relativistic wakefield acceleration.

Although electron acceleration in the wake fields will not be discussed at length, we consider a dephasing time and length as a characteristic physical constraint for the BRA coupling. Depending on the plasma density and laser wave length, particle dephasing can occur when particles are accelerated to velocities higher than the phase velocity of the wake. These particles eventually overshoot the accelerating phase and are decelerated. Particle dephasing places a length constraint on the region of acceleration. For a highly relativistic electron, in a wake of radial extent much greater than λ p , the linear dephasing length can be calculated using the relative velocity difference between the particle and wake, the particle beam moves at velocity approximately c, while the wake's phase velocity v ph = c 1 -n e /n c . This results in the dephasing constraint (c -v ph )t d = λ p /2. Defining the dephasing length as

L d = ct d , results in λ p /2 = L d [1-(1-λ 2 /λ 2 p ) 1/2 ] being ≈ L d λ 2 /2λ 2 p
for n e /n c 1, so that the dephasing length can be approximated as 1 , L d ≈ λ 3 p /λ 2 . For a given laser wavelength and plasma density the dephasing length is a constant barrier on particle acceleration that is underutilized due to the diffraction of low f-number lasers. To ensure the wakefield is maintained over the extent of the dephasing length we propose the use of a backward Raman amplifier which allows for corrections to be applied to the wake generating laser during its propagation in the plasma.

B. Backward Raman Amplifier

Backward Raman Amplification (BRA) involves the use of a long pump pulse (ω 0 , k 0 ) counterpropagating with respect to the short seed pulse (ω

1 = ω 0 -ω pe , k 1 = k p -k 0 ).
When the two lasers overlap they beat at the plasma frequency, exciting a Langmuir wave (ω L = ω pe , k L ≈ 2k 0 -k p ) that resonantly couples to the seed. Energy flows from pump to seed, resulting in amplification and compression of the seed.

BRA is typically modeled in 1D with a set of three coupled equations for the slowlyvarying field envelopes of the waves, the pump wave a 0 , the seed a 1 , and the plasma (Langmuir) wave a L . 10 We will compare in the following the results from PIC simulations of the BRA with solutions of this 3-wave coupling model defined by the following set of equations : 10 ,

[∂ τ -(|v g0 | + v g1 )∂ ζ ] a 0 = -γ 0 a L â1 , (2)
∂ τ â1 = γ 0 a 0 a * L , (3)
[∂ τ -v g1 ∂ ζ + ν] a L = γ 0 a 0 â * 1 , (4)
where a 0,1 ≡ eE 0 /(m e cω 0,1 ) are the normalized amplitudes of the electromagnetic fields E 0,1 of the pump (0) and the seed (1) wave, with â1 = (ω 0 /ω 1 ) 1/2 a 1 , and with (ω 0,1 , k 0,1 ) as the corresponding frequencies and wave vectors; a L = (ω L /ω 0 ) 1/2 eE L /(m e cω p ) denotes the normalized amplitude of the Langmuir wave electric field, E L , with (ω L , k L ) and the coupling constant reads

γ 0 = (k L c/4)ω p /(ω L ω 1 ) 1/2
. The equations are written in the stationary frame of the seed pulse, using the variables ζ = x -v g1 t, τ = t, where v g0 and v g1 are the pump and seed wave group velocities, and where the Langmuir wave group velocity has been neglected. The damping coefficient ν has been added to the Langmuir wave equation ( 4). Its functional form and magnitude can simply be the linear Landau damping ν L = (πω 3 pe /2k 2 )∂f M /∂v | v=ω L /k evaluated at the Maxwellian distribution function, f M . On the long time scale ν may be modified due to nonlinear evolution of Langmuir waves that may include electron trapping and wave coupling. Such long time effects do not usually arise during BRA coupling for short seed pulse durations as in the example of Fig. 3.

Figure 3 illustrates the comparison between PIC simulation and the 3-wave coupling model for parameters corresponding to case "3", leading to the π-pulse solution 10 of Stimulated Raman Scattering (SRS). The seed pulse amplitude at time t = 3.57ps, has already experienced amplification with respect to its initial amplitude a 1 = 0.011. Amplification is due to the pump laser -propagating from the right to the left -initially at a 0 = 0.011, through coupling with the Langmuir wave. The seed pulse duration t pulse is 100fs at FWHM, the plasma density and electron temperature are n e /n c = 0.05 and T e = 50eV, respectively. The example of case "3", and previous studies based on PIC simulation with similar parameters 22,24,27 , have been performed in order to validate 3-wave coupling models. Among these studies, Ref. 27 dealt with long time self-similar evolution of 3-wave solutions 10 albeit under idealized conditions of cold background plasma.

FIG. 3. Comparison of field amplitudes between PIC simulation (color and broadened lines as a result of enveloping the fields) and 3-wave coupling model (solid black lines): for the seed, line (2) (with amplitudes ×1/2); for the laser pump, line (1); and the Langmuir wave, line (3), at interaction time t = 3.57ps. Amplitudes are normalized to the incoming pump amplitude. The PIC simulation was run with a simulation domain of 400µm, 135cells/µm resolution, and 1024 particles/cell.

The good agreement between the 3-wave coupling model ( 2), ( 3), ( 4) and PIC simulation shown in Fig. 3 characterizes the initial short time evolution of BRA. As we will discuss below, plasma wake generation and subsequent seed pulse frequency shift, and self-modulation of the relatively long electromagnetic pulse are among nonlinear effects that limit application of a straightforward 3-wave coupling model. Such limitations motivate a need for PIC simulations when the reduced wave coupling models of BRA are derived.

The different modes in k-space are illustrated in Fig. 4 that shows the electric field spectrum with electrostatic components in orange and electromagnetic components in blue. The laser pump mode at k 0 , the wake at k p = ω pe /c, the seed at k 1 = k 0 -k p , and the Langmuir waves from Raman coupling k L ≈ 2k 0 -k p . The one dimensional Fourier transform is taken in space and is presented as "FFT(E y )" and "FFT(E x )" for the Fourier transform of E y and E x , respectively into k x .

III. BACKWARD RAMAN AMPLIFIER FOR LASER WAKEFIELD GENERATION

In the following section we will demonstrate the control and enhancement of the laser produced wakefield by combining it with the backward Raman amplifier scheme. For this purpose we present the results of simulations where we bring together both effects, BRA and wakefield, in order to optimize and sustain the laser generated wakefield. We first present the results of 2D PIC simulations for the plasma conditions corresponding to the case "1" from the Table I except for the plane wave limit of the seed and the pump pulses. The discussion of these results will introduce important physical processes related to BRA and wake coupling. Next we will consider BRA as the mechanism that overcomes diffraction of the seed pulse and extends wake generation to at least the timescales on the order of the particle dephasing length, Sec. III A. In the Sec. III B we will present results on timescales much greater than the dephasing length and consider the effects of wakefield generation on the resonance coupling of the BRA. We will also revise a theoretical model for both laser wakefield generation and backward Raman amplification and consider chirp of the pump, cf. Sec. III C, as the mechanism mitigating frequency and wavelength dephasing due to the wake generation.

The results below correspond to a 30 fs FWHM driving a seed laser pulse of intensity 5 × 10 16 W/cm 2 at best focus and a flat top plane wave pump of intensity 1 × 10 14 W/cm 2 , in a plasma of density n e = 0.0035n c . The parameters are chosen such that the seed efficiently excites plasma waves of the wake (ct seed = λ p /2) in the non-relativistic regime (a 1 = 0.205), and the pump intensity is sufficiently low such that it does not produce backscattered SRS from the particle noise before interaction with the seed pulse. The spatial SRS gain coefficient of a pump wave, G = L x γ 2 0 |a 0 | 2 /|ν L v g1 |, requires the length L x ≈ 70mm to reach G > 1 values for the parameters of this example. This estimate corresponds to the background electron temperature of 100 eV and the SRS Langmuir wave of the BRA at k L ≈ 0.46k D . Recent work 26 has preposed the use of a "flying focus" on the BRA pump to combat parasitic precursors, such a setup could possibly support larger pump intensities, and hence larger amplification of the seed.

While the low electron density of this example is an optimal choice for the wake generation by the 30 fs laser pulse, the electron temperature of 100 eV is consistent with the experimental conditions of gas jet plasmas 28 . The resulting large ratio k L /k D ≈ 0.46 of the BRA Langmuir wave characterizes the kinetic regime of the SRS 29,30 and of the Langmuir wave nonlinear evolution 31 . This leads to strong linear Landau damping, ν L ≈ 0.14ω pe , and in the nonlinear regime, to electron trapping and to trapped particle modulational instability [29][30][31] . However, the nonlinear physics of Langmuir wave evolution -clearly identifiable in simulation results below -becomes important only on time scales longer than the short seed pulse duration, and therefore it has negligible effect on the BRA and the wakefield generation. Consistently, simulations run at different temperatures (50eV, k L ≈ 0.32k D and 200eV, k L ≈ 0.65k D ) show only small variations in the resulting seed an wake amplitudes (see Fig. 9a,b).

FIG. 5. From 2D plane wave PIC simulation with conditions of case "1": (a) snapshot of the longitudinal electric field in the x -y-plane at t = 8.0ps (x: propagation direction; red/blue color corresponds to positive/negative field values, respectively); the superposed black line shows the transverse field of the seed pulse and pump from a cut at y = Ly/2; (b) spectrum of the electrostatic field in the kx-ky plane with subfigure showing a line-out at ky=0, both at the same time instant as in panel (a); (c) spectrum of electromagnetic field (summed over ky) as a function of time. The PIC simulation was run with a simulation domain of 10x200µm, 40cells/µm resolution, and 16 particles/cell.

Figure 5 shows the results of a 2D plane wave seed and pump simulation with parameters of case "1", the transverse dimension of 10µm and the transverse periodic boundary conditions. The three panels in Fig. 5 illustrate the main physical processes involved in the simultaneous BRA and the wakefield generation. Figure 5a displays longitudinal electric field of the Langmuir waves due to SRS at k L ≈ 0.46k D and the wake at k p = ω pe /c ≈ 0.014k D . Superimposed is the 1D cut at y = 5µm of the transverse electric field due to the seed and the pump lasers. At the time of Fig. 5, t = 8ps, the wake is well developed and extends hundreds of microns behind the seed. We have estimated the bounce frequency of trapped electrons in the SRS driven Langmuir wave. At the maximum amplitude of the Langmuir wave electric field such that eE L /mcω L ≈ 0.02 (see Fig. 6) the bounce frequency ω b = (k L eE L /m) 1/2 ≈ 9.3 × 10 13 1/s and therefore the characteristic trapping time τ b = 2π/ω b = 67.5fs is longer than the time of the seed pulse duration and the BRA coupling. On the other hand the freely propagating (to the left) SRS Langmuir waves experience effects of the trapped particles that result in the transverse modulations of their wavefronts on the longer time scale corresponding to the trapped particle modulational instability [29][30][31] . This is seen on the left of Fig. 5a, far behind the leading seed pulse, and in the two dimensional Fourier transform of the longitudinal fields in Fig. 5b. The two dimensional Fourier transforms are taken in space and are presented as "fft2(E x )" and "fft(E y )" for the Fourier transform of E x and E y , respectively into k x and k y , and k x respectively, (see figure label). The spectra are calculated over the whole length and width of the simulation window and in addition to the wake (k x = k p = 0.059k 0 , k y ≈ 0) and the SRS Langmuir wave (k x = k L = 1.94k 0 , k y ≈ 0) one can see broad continua of transverse components, resulting from the trapped particle modulational instability [29][30][31] and the second harmonic of the k L -wave. The detailed analysis of the spectra about k x ≈ 2k 0 has revealed additional components at k L ± nk p and ω L ± nω p (cf. insert in Fig. 5b) that result from the coupling between strong wakefield wave (k p , ω p ) and SRS driven Langmuir wave. Such coupling takes time and we have found no evidence that it actually affects the BRA interaction due to short seed pulse duration. These extra spectral components in the Langmuir wave fields can modify damping and contribute to frequency shifts, but again, only on the longer time scale than the Raman amplification process.

The focus of this work is on coupling between two nonlinear processes, wakefield generation and BRA. We will demonstrate that wakefield generation can be extended over large plasma distances due to the BRA enhancing the amplitude of the seed pulse. On the other hand, the presence of several spectral components in the electrostatic fields can lead to deleterious effects such as producing different components at k L ± nk p and ω L ± nω p or adding frequency shifts and instabilities due to trapped particles. These effects influence SRS Langmuir wave on the long time scale. We have also found that the time dependent density modifications due to wake produce k-vector and frequency shifts of the electromagnetic seed pulse. This effect is illustrated in Fig. 5c and will become central to our discussions in the following sections (cf. Fig. 13b). Despite our focus on short seed pulses (that extend half of a plasma wavelength) to maximize wakefield generation, the wake Langmuir waves are always created by high intensity femtosecond laser pulses, and therefore the BRA schemes should verify whether the wake will introduce frequency and wavelength shifts and account for these decoupling effects in the reduced models based on the three wave approximation.

The characteristic feature of the electrostatic field spectra is large separation between the wake at k p and the SRS Langmuir wave at k L (cf. spectral components 1 and 4 in Fig. 4 respectively). On the other hand the electromagnetic field components of the seed pulse (marked as 2 in Fig. 4) and of the pump (number 3), can overlap for lower densities than in case "2" (see table) of Fig. 4. This has been the condition of the current case "1" at n e /n c = 0.0035. Therefore in the comparison between the three wave coupling model, ( 2), ( 3), ( 4) and PIC simulations in Fig. 6 we have plotted the seed and pump wave amplitudes together. Note the remarkable agreement for the Langmuir wave amplitudes near the vicinity of the seed and discrepancy on the longer distance scale due to nonlinear effects including particle trapping and coupling to the wake. The three wave coupling model in Fig. 6 was solved with the Langmuir wave damping coefficient, ν = ν L = 0.14ω p . In general we have found weak dependence at the early time, i.e. until the first maximum in the Langmuir wave amplitude, on the damping coefficient ν (cf. also Ref. 30 ).

FIG. 6. From 2D plane wave PIC simulation with conditions of case "1": Comparison of field amplitude between PIC simulation (color and broadened lines as a result of enveloping the fields) and 3-wave coupling model (solid black line): for the pump+seed, line (1); and the Langmuir wave, line (2) (with amplitudes ×100), at interaction time t = 2.61ps. Amplitudes are normalized to the initial seed amplitude. The PIC simulation was run with a simulation domain of 10x200µm, 40cells/µm resolution, and 16 particles/cell.

For optimal BRA coupling the seed pulse duration should be long enough such that its spectral bandwidth is localized to the BRA resonance region; however, optimal wake generation requires the seed pulse duration to be of the order of half a plasma wavelength, ct pulse ∼ λ p /2, and the SRS growth rate increases with plasma density (shorter plasma wavelength). A chirped pump has previously been used to extend coupling in the BRA to all spectral components of a short seed pulse 32 . The effect of the broad seed pulse spectrum was examined in PIC simulations 33 showing Brilloun coupling 34 in the BRA amplification and compression scheme. Since our seed pulses are short and relatively low intensity, Brilloun coupling is expected to be unimportant. Our plasma density and seed pulse length parameters were chosen to be in a demonstrated region of optimal amplification 23 , to be experimentally relevant, and to balance the effects of density on BRA coupling. FIG. 7. Simulation results with f/15 at t=0.776ps. Laser polarization is in the y-direction and propagation is in the x-direction. (a) Ey with BRA, (b) Ey without BRA, (c) Difference in Ey between the subcases with and without BRA, (d) Ex with BRA, (e) Ex without BRA, (f) Difference in Ex between the subcases with and without BRA. Subplots (c)& (f) are obtained by ∆E x(y) = E x(y) (withoutBRA) -E x(y) (withBRA). Both PIC simulations of case "4" were run with a simulation domain of 150µm x300µm, 30cells/µm resolution, and 9 particles/cell.

A. Resonant BRA Figure 7 shows the layout of the f/15 simulations (case "4"), where the laser propagation is in the x-direction, and polarization is in the y-direction. In Figures 7 we compare the fields resulting from a simulation without BRA, E x in Fig. 7a and E y in Fig. 7b, to the simulation with BRA, Figs. 7d and e respectively, for case "4". Figures 7c and7f show the difference in each field component between both cases, namely ∆E x(y) = E x(y) (with BRA)-E x(y) (without BRA). From the quantity ∆E y in Fig. 7c, it can be seen that the seed pulse is enhanced as it passes through the plane wave pump (oscillation amplitude ∼ 9 × 10 -3 outside wake region), leaving a trail of depletion in the plane wave pump. Figure 7f shows enhancement to the wake due to the amplification of the seed. Figure 8 shows the time evolution of the seed's energy density with and without BRA. Without BRA the f/15 seed quickly diffracts resulting in much lower wake amplitude. With BRA the f/15 seed still diffracts, see Fig. 8b,however the energy density is nearly maintained by the BRA allowing the seed to continue to produce a large wake. In the f/30 case, Figs. 9a and9d, the amplifier operates faster than the pulse diffraction. This leads to significant enhancement of the pulse and wake until decoupling from the wake induced frequency shift. In the f/15 case, Figs. 9b and9e, the amplifier provides additional energy but at early times the pulse diffracts quicker than it can be replenished. In the f/15 case the competition between diffraction and amplification reaches a balance and the wake is maintained at nearly the same or higher amplitude than the unamplified pulse as shown in Fig. 9e. In the higher density plasma of case "2" the seed experiences modulation by its wake (Fig. 1d) which -despite diffraction -results in a focusing that increases the seed amplitude with or without the BRA. When a BRA is applied to case "2" there is improved BRA coupling due to the higher plasma density, this results in a large enhancement to both the seed and wake amplitude as shown in Figs. 9c and9f.

A simple model that is consistent with PIC simulations is proposed to describe competition between diffraction effects and the BRA. Assuming that the seed pulse is well approximated by the Gaussian solution to the wave equation during the BRA interaction, we can model the intensity evolution of the seed taking into account both the effects of diffraction and the BRA. Typical results of the BRA for the wake generation are illustrated in Figs. 9. Appart from case "3", all our BRA simulations are in the superradiant regime 9,35,36 (ω pe ≤ 2ω 0 √ a 0 a 1 ) where the laser pump is strongly depleted as it transfers energy to the seed. This can be clearly observed in Fig. 7c displaying depleted pump behind the right propagating seed pulse. With the parameter 0 ≤ α ≤ 1 being the efficiency of the energy flow from pump to seed, the incremental energy (dE) transfer from pump to seed during time dt, is dE 1 = -dE 0 = 2αI 0 A 1 (t)dt. Where A 1 (t) is the time dependant spot size area of the seed laser, and the factor of 2 arises due to the counterpropagation of the pump and seed. Integrating this results in order to obtain the seed intensity evolution gives

I 1 (t) = 2αc ls t 0 I 0 (t ) 1+ ct Z R 2 d-1 2 dt + I 1 (0) 1+ ct Z R 2 1-d 2 , ( 5
)
where d indicates the spatial dimension of the solution, i.e. d =2 or =3 for 2D or 3D respectively, l s = ct pulse , and Z R is the Rayleigh length of the seed pulse. The solution to Eq. ( 5) with α = 0.45 efficiency and constant pump intensity I 0 = 1 × 10 14 W/cm 2 shows good agreement with PIC simulation case "4" in Fig. 9b. We can estimate the energy transfer from the pump to the seed in order to balance intensity loss due to diffraction. To determine the pump intensity constraint one can set ∂I1 ∂t = 0 to get the solution:

I 0 (t) = I 1 (0) d -1 2 l s αZ R 1+ ct Z R 2 -1 ct Z R , (6)
One can verify this result by substituting Eq. ( 6) into Eq. ( 5) to confirm I 1 (t) = I 1 (0). Since diffraction results in a dynamic loss in seed intensity, the pump requirements I 0 (t) (6) are not constant in time, but can be evaluated near the seed's Rayleigh length (t ≈ Z R /c, where losses are strongest) to find a constant pump intensity that ensures diffraction is always overcome. As seen in PIC simulation case "4" (c.f. Fig. 9b) Eq. ( 6) (with α = 1) predicts a BRA pump of intensity 3 × 10 14 W/cm 2 while our BRA pump intensity is 1×10 14 W/cm 2 hence it is impossible for our pump to overcome diffraction near the Rayleigh length (simulation time, t ≈ 4ps) but our pump is more than sufficient to maintain the pulse at later times. FIG. 10. Limiting lengths for particle energy gain (in 2D) as a function of the seed pulse aperture (f-number) for a 30fs driving pulse in a plasma of density ne = 0.0035nc. LWFA without BRA is restricted to the green region; when a BRA is applied diffraction is overcome allowing access to the yellow region. Markers indicate results from PIC simulations (run until the dephasing time t≈17ps). The diffraction limit is defined to be the length at which diffraction causes the wake amplitude to be ≤75% of its maximum unamplified value. The 3D diffraction limit without BRA is shown by the dashed line; the dephasing length remains constant in 3D.

Figure 10 summarizes the results of 2D PIC simulations of BRA applied to LWFA. Efficient particle acceleration can be limited by diffraction of the pulse, but is ultimately dictated by the particle dephasing length. For a range of experimentally relevant f-numbers, Fig. 10 demonstrates that BRA applied to LWFA can help maintain the wake amplitude of diffracting lasers and allow for better or full use of the particle dephasing limit. Simulation results in our standard parameters (cases "1","4","5","6") show that when a BRA pump of intensity I 0 = 1 × 10 14 W/cm 2 is applied to seed lasers with best-focus intensity I 1 (0) = 5 × 10 16 W/cm 2 , the effects of diffraction are overcome for f-numbers of 15 or larger, but enhancement cannot continue indefinitely. In the next section we will discuss a new saturation mechanism of the BRA in context to wake generation.

B. Loss of the BRA resonance due to the wake effects

In this section we examine the BRA applied to LWFA at times exceeding particle dephasing. For our standard parameters (cases "1" & "4") particle dephasing happens at t ≈ 17ps (t ≈ 1.5ps for SM case "2"), however BRA and LWFA applications that are not concerned about dephasing (ex: betatron x-ray sources) may still take advantage of the enhancement via BRA.

As shown in Fig. 9a andd, at late times the BRA eventually decouples from the seed resulting in a loss of seed and wake amplitude. This decoupling can be attributed to the frequency and wavelength shifts in the seed pictured in Fig. 11. FIG. 11. Longitudinal wavenumbers of electromagnetic fields vs time from 2D PIC Simulations (a) case "1", (b) case "2", showing a decoupling between pump and seed attributed to the red-shifting of the seed. Line 1 is the pump at k0, line 2 shows the mean wavenumber of the redshifting seed.

It has been shown 37 that a short laser pulse that propagates in plasma with a density profile due to the wake's Langmuir wave, δn w /n 0 , will develop frequency shifts that can be described by :

δk 1 = - ω 2 p τ 2ω 1 n 0 ∂δn w ∂ζ , δω 1 = ω 2 p 2ω 1 δn w n 0 + δk 1 v g1 (7)
with

δk 1 = ∂ x Φ = ∂ ζ Φ and δω 1 = -∂ t Φ, where Φ(ζ, τ ) = -ω 2 p δn w τ /(2ω 1 n 0 )
. The characteristic linear dependence of the δk 1 in time, τ = t is clearly seen in SCPIC simulations and well reproduced by Eq.( 7). The resulting red shift in the wavelength and frequency follows from the geometry of the wake Langmuir wave, i.e. the seed pulse propagates together with the front of the wake experiencing constant and predominantly positive density gradient [cf. Eq. ( 7)]. The theoretical curve in Fig. 12b has been calculated using Eqs. ( 1) and ( 7), by taking the average of the wave-vector shift (7) over the seed pulse length,

δk 1 = l -1 s ls 0 dζ δk 1 (ζ ) = -k 0 ω p t 6π 2 n e n c |a 1 (0)| 2 , (8)
where we assumed k p l s = π and v g1 c.

FIG.

12. In the short pulse regime, case "1": (a) Fields and density from 1D PIC and theory at t = 4.9ps, theoretical curves overlaid in black. (b) Spectrum of transverse field from 1D PIC with theory from Eq. ( 8) shown in black. In the self-modulated regime, case "2": (c) fields and density from 1D PIC at t = 17.2ps and (d) spectrum of transverse field from 1D PIC. In both cases the initial seed amplitude is a1 = 0.205, and initial seed duration is 30fs. Plasma density are respectively ne/nc = 0.0035 and ne/nc = 0.015. Both PIC simulations were run with a simulation domain of 600µm, 60cells/µm resolution, and 128 particles/cell.

Note that the dramatic frequency cascading of the self-modulated pulse in Fig. 12d is entirely due to the coupling between the wake and subsequent stages of forward Raman scattering. This pulse never reaches intensities that would lead to relativistic self-focusing that is usually associated with the self-modulated regime 38 .

To model the frequency shifts in the seed, the system of the 3-wave coupling equations 10 , Eqs. (2-4) has to be modified by introducing in Eq. ( 3) the frequency shift term δω. Equation (3) in the system Eqs. (2-4) is hence replaced by

[∂ τ + iδω] â1 = γ 0 a 0 a * L . (9)
where δω = . This modification is necessary to account for the effects of the plasma wake on the BRA. For very short seed pulses as proposed in 9,39 the wake can have dramatic effects leading to the loss of resonance between BRA modes.

Stationary solutions to Eqs. ( 2), (9), and ( 4) are closely reproduce PIC simulations and are displayed in Fig. 13b to illustrate the need for including wake generation into theories involving short laser pulses such as BRA.

We will demonstrate how BRA can be applied to enhance the plasma wake when a chirped pump is employed. In absence of a frequency mismatch, i. e. δω = 0 in Eq. ( 3), the waves satisfy the usual matching conditions,

ω 0 = ω 1 + ω L , k 0 = k L + k 1 ( k 0 = -k 1 + k L for backscatter, with k a = | k a |, a = 0, 1, L).
When we describe the application of BRA to wake enhancement, we will consider short pulses on the order of the plasma wavelength λ p . Any application of the wave coupling model Eqs. (2-4) in description of BRA must include a nonlinearity of the wake where δn w ∼ |a 1 | 2 in the phase shift of Eq. (7).

Figure 13a shows the results of SCPIC simulations at t = 40ps for the seed pulse amplitude a 1 and the pump amplitude a 0 and for the normalized amplitude of the electrostatic field, a L , associated with the wake and Langmuir waves participating in the BRA that are characterized by short wavelength and are concentrated in the front part of the seed pulse. For this simulation with n e /n c = 0.0035, the initial seed pulse duration is τ L = 30fs measured as the FWHM of a Gaussian envelope, and the left-propagating laser pump amplitude a 0 = 0.0091 (case "1"). An important feature of our BRA application to enhance the plasma wake is the wavelength separation between two main Langmuir wave components, i.e. the wake and BRA excited perturbations. Our attempt to describe interactions, as seen in Figs. 13a andb, i.e. BRA and the wake generation, by means of the 3-wave coupling model, Eqs. ( 2), (3), and ( 4) can be well approximated by a function for the nonlinear shift in Eq. ( 3) of the following form,

δω = -g 1 n e /n c ω p |â 1 | 2 , ( 10
)
where the numerical constant g 1 = 1/(12π) was calculated for a sin model of the seed envelope. Fig. 13b shows the results of this model compared with SCPIC results using numerical solutions to the 3-wave coupling model (line 3), and integrating Eq. 10 with the seed amplitude obtained from PIC (line 2). SCPIC simulations show decoupling between the pump a 0 and the seed â1 due to the nonlinear evolution of the seed. This more complicated and nonuniform (within the pulse) frequency shift is absent in the 3-wave coupling model which shows further growth of the seed before saturation, leading to a larger frequency shifting rate. Figure 14 shows the wake amplitude versus seed propagation time from 1D simulations for different conditions. Figure 14a line 2 illustrates the enhancement of the wake amplitude due to BRA coupling until the decoupling at t = 40 ps due to nonlinear evolution of the seed.

C. Chirping the Pump

To overcome this decoupling due to the shift in the seed, a frequency-chirped pump can be used. To first order, the frequency of the seed follows Eq. (7). In this approximation a linearly chirped pump 32 can prolong coupling and increase wake production as shown in Fig. 14a line 3 illustrating the enhancement of the wake and delayed decoupling in this case. However, as the seed is amplified it produces a larger amplitude wake resulting in a non-linear change in frequency, and eventually a linearly chirped pump will not suffice to maintain coupling.

We can estimate this nonlinear frequency change by using Eq. ( 7). This expression can be easily modified to include the effects of Raman amplification through a linear amplitude growth rate 9,12 by substituting a 1 (0) with a 1 (t) = a 1 (0) (1 + bt), with b as a parameter. Substituting this into Eq. ( 8) and integrating leads to the following approximation of the seed's wavenumber shift:

δk 1 (t) = - k 3 p k 1 (0) v g1 6π a 2 1 (0)t + a 1 (0)b t 2 + 1 3 b 2 t 3 , (11)
The pump frequency satisfies the usual matching conditions for coupling when ω 0 (t) = ω 0 (0) + c δk 1 (t), corresponding to a chirped pump with ω 0 (0) as the unshifted frequency.

This result can also be obtained by applying similar steps to the theory given in Ref. 40 . Equation ( 11) underestimates the rate of frequency reduction as it does not account for time evolution of the critical density defined by the changing seed pulse; however, by adjusting the parameter b, it was used to model the spectral evolution of the seed (Fig. 13d line 2) and to model a pump that would ensure coupling (Fig. 13d line 1). Figs. 13c andd and 14a line 4 demonstrate such a case where the seed and pump maintain coupling throughout the non-linear shifting process with the value of b = 0.012ps -1 . One can observe the strong wake enhancement in Fig. 14a line 4 as the seed's frequency approaches the plasma frequency, and the seed is slowed down dramatically. Fig. 13c shows a snapshot of the pulse and its wake at t = 75ps giving a quantitative value for the electric fields, in this simulation E w ∼ 4.5 × 10 10 V/m. Strong electric fields such as these may be capable of reflecting, and accelerating ions 41 . FIG. 14. Wake amplitude versus seed propagation time from 1D simulations for (a) short Gaussian pulses with parameters of case "1" and (b) self modulated pulses with parameters of case "2". In (a), curves correspond to: 1 no pump, 2 constant pump, δk1 =0, 3 linearly chirped pump, δk1 from Eq. ( 11) with b =0 and 4 ideally chirped pump, δk1 from Eq. ( 11) with b =0.012 ps -1 . In (b) curves correspond to: 1 no pump, 2 constant pump and 3 linearly chirped pump.

To compare our result with realistic experimental conditions, we will consider the instantiations phase of the pump required BRA for coupling. We note that in the presence of a shift in the seed and frequency chirped pump, the Langmuir frequency ω L may deviate from resonance. However, since our parameters correspond to the superradiant regime 9,35,36 (ω pe ≤ 2ω 0 √ a 0 a 1 ) where there is increased bandwidth in the Langmuir waves, the interaction is still effective. To satisfy the wavenumber matching conditions k 0 = k L + k 1 , one may use the average shift of the seed wavenumber ( 8) (cf Fig. 12b) to determine the instantaneous phase of the pump to sustain the original resonance condition: Φ 0 (x, t) = ω 0 (t+x/c)+α(t+x/c) 2 /2 with α = -ω0ωpe 12π 2 ne nc |a 1 (0) 2 |, where α is the linear chirp coefficient, and has a value of -2.3x10 23 s -2 for our standard parameters ("1","4","5","6"). Linear chirp coefficients up to 4.47x10 23 s -2 have been obtained in experiments 32 and could be applied to maintain coupling.

When the effects of diffraction are completely overcome and the seed pulse amplitude grows in time, it is necessary to use the non-linear "ideal chirp" prescription (11) to maintain coupling. For early times (t ≤ 40ps) the "ideal chirp" prescription is well approximated by the linear shift (cf. figure 14a line 3), but for long durations the non-linear shift is unlikely to be achieved by current CPA lasers. Density tapering 42 could also potentially be used to extend the coupling duration between the pump and the seed. However, the increased plasma densities that are required also result in a greater red-shifting rate of the seed (Eq. 8), such that the seed's frequency evolution becomes non-linear and requires a very large, non-linear density gradient that is unlikely to be achieved in experiments.

IV. CONCLUSIONS

We have studied application of the BRA to the enhancement and control of the linear wake generation over a long plasma region. This has been achieved by energy transfer to a wake driving pulse via Raman coupling with a laser pump of low intensity and of long pulse duration. Using PIC simulations we have examined a wide range of background plasma conditions, as well as considered results of the previous BRA optimization studies 23 . Wake generation favors low background electron density, such as n e = 0.0035n c , where it is possible for a 30fs laser seed pulse to have a FWHM comparable with half of a plasma wavelength λ p /2. This optimal pulse length for wake generation at higher plasma densities would require shorter laser pulses and therefore will undermine the efficiency of the Raman amplification process. Also wakes at the lower densities are more effective for particle acceleration 1 as e.g. they correspond to a longer dephasing length. The physics of particle acceleration using wakes that were studied in this paper will be considered next in a separate publication. We have examined a range of electron temperatures and focused on the experimentally relevant 28 choice of T e =100 eV. By varying electron temperature we could alter properties of the resonant Langmuir wave driven by the SRS. At T e =100 eV and n e = 0.0035n c the k-vector of the plasma wave, k L ≈ 0.46k D and BRA operates in the kinetic regime of SRS [29][30][31] . However, at the short seed pulse durations kinetic effects related to trapped particle dynamics are only relevant on much longer time scales than BRA coupling. In addition the strong linear Landau damping limits the growth of SRS of the pump wave thus allowing for the long scale plasma to be considered. With the above choice of plasma parameters we have proceeded to examine the role of BRA in countering the limitations imposed by diffraction of the seed pulse and extending the wake generation to plasmas that are longer than the dephasing length. For a range of experimentally relevant f-numbers, Fig. 10 summarizes results of 2D PIC simulations and demonstrates that BRA applied to wake generation can help maintain the wake amplitude of diffracting laser pulses.

A great deal of an initial theoretical insight and motivations for experimental measurements of BRA have been based on the solutions to the simple three wave coupling model [9][10][11] . Such reduced models are useful in the understanding of BRA in large scale plasma experiments, provided diffraction of the laser pulses is not a factor and the effects of wake generation are included. We have demonstrated good agreement between PIC simulations and the reduced theoretical description at the early times of the BRA coupling. We have extended validity of the wave coupling models by accounting for the impact of the wake on the SRS coupling. This has been accomplished by including frequency and wavelength shifts in the electromagnetic seed equation that are caused by the time dependent, wake related, plasma inhomogeneities. The loss of the resonance coupling in the BRA that follows has been mitigated by introducing frequency chirp in the laser pump. Theory for the pump frequency chirp, in agreement with long time PIC simulations, showed the possibility of trapping the amplified laser seed pulse in a plasma. This results in a large increase in the wake amplitude and the generation of a shock like structure in the electrostatic field that will be further studied for the enhancement of particle acceleration.

FIG. 4 .

 4 FIG. 4. Electric field power spectrum vs wavenumber, showing electrostatic components in orange and electromagnetic components in blue. (1) Wake at the plasma wavnumber kp = ωpe/c. (2) Seed at k1 downshifted by kp from the pump. (3) Pump defined to be k0. (4) Langmuir waves from Raman coupling kL ≈ 2k0 -kp. Simulation parameters correspond to case "2".

FIG. 8 .

 8 FIG. 8. Evolution of the energy density of a f/15 seed in a plasma vs propagation time and transverse space. a) With no amplifier. b) With a Raman amplifier. The laser is at maximum focus at t = 0.2ps. Simulation parameters correspond to those of case "4".

FIG. 9 .

 9 FIG. 9. Comparison of seed and wake amplitudes with (red) and without (blue) BRA. (a) Seed amplitude comparison f/30, (b) Seed amplitude comparison f/15 with theory from Eq. (5) shown by the dashed black line, (c) Seed amplitude comparison SM f/30, (d) Wake amplitude comparison f/30, (e) Wake amplitude comparison f/15, (f) Wake amplitude comparison SM f/30. Simulation parameters correspond to those of case "1" for (a) & (d), case "4" for (b) & (e), and case "2" for (c) & (f).

FIG. 13 .

 13 FIG. 13. For the case "1" with constant pump: (a) fields â1 and aL from PIC at t = 40ps; (b) spectrum of transverse electric field (containing both pump and seed components) from PIC and from theory; line 1: for constant pump (line at k = k0), line 2: solution to Eq. (10) obtained from the amplitude of the seed in PIC, and line 3: solution to Eq.(10) obtained from a numerical solution to the 3-wave coupling model. For the ideally chirped pump: (c) fields â1 and aL from PIC at t = 75ps; (d) Spectrum of transverse electric field from PIC; line (1) : pump field spectrum prescribed by matching condition, solid line (2): seed spectrum obtained from Eq. (11). Both PIC simulations were run with a simulation domain of 600µm, 60cells/µm resolution, and 128 particles/cell.

TABLE I .

 I List of all simulations cases.

	Case	1	2	3	4	5	6
	Density ne/nc 0.0035 0.015 0.05 0.0035 0.0035 0.0035
	Te [eV]	100	100	50	100	100	100
	f-number	30	30	40	15	13	10
	ZR [µm]	3204 3462 7193 801	602	356
	t pulse [fs]	30	30	100	30	30	30
	λp/2c [fs]	30	14.5	8	30	30	30
	γ0/ωpe	0.016 0.0126 0.009 0.016 0.016 0.016
	a0	0.0091 0.0091 0.011 0.0091 0.0091 0.0091
	a1	0.205 0.205 0.011 0.205 0.205 0.205

ACKNOWLEDGMENTS

WR would like to thank Dr. P. Alves for useful discussions. JL and SW would like to thank Arthur Pak for useful discussions. JL is grateful for support from LLNL through the summer scholar program. JL and WR would like to acknowledge partial support from NSERC. SH is grateful to the CPHT computer support team.


Didier Louis
email: didier.louis@univ-nantes.fr

Cindy Lombart
email: clombart@audencia.com

Impact of CSR communication on brand business and social values

Keywords: CSR, CSR communication, sustainable consumer behavior, well-being, adverting effectiveness

Research on CSR has largely focused on the effects of companies' CSR actions on their performance or business value creation, but obtains contrasting results due to understudied mediating effects, i.e. indirect relationships between CSR and company performance. Consequently, the present study examines the direct, but also indirect, links between a company's perceived CSR, following a CSR communication, by introducing variables more specifically related to the social value the firm can deliver, upstream of its business value.

Using a sample of 414 consumers, results of structural equation modeling show the direct and indirect ways in which a brand's perceived CSR, following exposure to a CSR communication, can improve its market value in terms of behavioral intentions. It also underscores that the perceived CSR of a brand increases the social value of this brand by encouraging consumers to adopt sustainable behaviors in order to increase their well-being.

Introduction

Most companies carry out responsible actions and communicate about these actions in order to raise their profile with consumers and improve their perceived Corporate Social Responsibility (CSR) (e.g., [START_REF] Du | Maximizing business returns to Corporate Social Responsibility (CSR): The Role of CSR communication[END_REF][START_REF] Lee | What makes CSR communication lead to CSR participation? Testing the mediating effects of CSR associations, CSR credibility, and organization -public relationships[END_REF]. By doing this, they want to position themselves as responsible companies in the minds of these consumers, to differentiate themselves from their competitors (e.g., Chakrabory and Jha, 2019;[START_REF] Melé | Influence of sustainability practices and green image on the re-visit intention of small and medium-size towns[END_REF], and to encourage consumers to behave favorably toward them (via purchases/repurchases and word-of-mouth behavior) (e.g., [START_REF] Elg | CSR: Retailer activities vs consumer buying decisions[END_REF][START_REF] Aramburu | The effects of corporate social responsibility on customer loyalty: The mediating effect of reputation in cooperative banks versus commercial banks in Basque country[END_REF][START_REF] Latif | Corporate Social Responsibility (CSR) and customer loyalty in the hotel industry: A cross-country study[END_REF]. CSR communication is indeed an effective and relevant marketing tool for firms (e.g., [START_REF] Hsu | The advertising effects of corporate social responsibility on corporate reputation and brand equity: Evidence from the life insurance industry in Taiwan[END_REF][START_REF] Chakraborty | Corporate social responsibility in marketing: A review of the state-of-the-art literature[END_REF]. Some companies also prompt consumers to engage in more responsible practices [START_REF] White | How to SHIFT consumer behaviors to be more sustainable: A literature review and guiding framework[END_REF]. CSR can increase not only the business value but also the social value of a company [START_REF] Dang | Retailer corporate social responsibility and consumer citizenship behavior: The mediating roles of perceived consumer effectiveness and consumer trust[END_REF]. However, research on CSR has largely focused on the effects of companies' CSR actions on their business value or corporate performance (e.g., reputation, brand equity, sales, through consumer recruitment, trust, and loyalty ...). Scant research has focused on how a company's CSR actions generate social value (e.g., raising public awareness of social or environmental issues, facilitating behavioral changes beneficial to individuals and society in general …), upstream of its business value [START_REF] Lee | What makes CSR communication lead to CSR participation? Testing the mediating effects of CSR associations, CSR credibility, and organization -public relationships[END_REF][START_REF] Dang | Retailer corporate social responsibility and consumer citizenship behavior: The mediating roles of perceived consumer effectiveness and consumer trust[END_REF].

Moreover, research on CSR obtained contrasting results [START_REF] Ailawadi | Does retailer CSR enhance behavioral loyalty? A case for benefit segmentation[END_REF][START_REF] Anser | Moderating effect of innovation on corporate social responsibility and firm performance in realm of sustainable development[END_REF]. While some studies found that CSR positively influences corporate performance (e.g., [START_REF] Surroca | Corporate responsibility and financial performance: The role of intangible resources[END_REF][START_REF] Saeidi | How does corporate social responsibility contribute to firm financial performance? The mediating role of competitive advantage, reputation, and customer satisfaction[END_REF], others found that CSR may damage a company's business value (e.g., Lima Crisóstomo et al., 2011;[START_REF] Orlitzky | Strategic corporate social responsibility and environmental sustainability[END_REF]. Thus, managers and researchers are not fully convinced of the relationships between CSR and corporate performance (e.g., [START_REF] Miller | CSR reputation and firm performance: A dynamic approach[END_REF][START_REF] Muflih | The link between corporate social responsibility and customer loyalty: Empirical evidence from the Islamic banking industry[END_REF][START_REF] Swaen | Impact of customers' perceptions regarding corporate social responsibility and irresponsibility in the grocery retailing industry: The role of corporate reputation[END_REF]. Some scholars maintain that these inconsistent findings may be due to mediating effects, i.e. indirect relationships between CSR and company performance, which have been ignored by prior studies [START_REF] Saeidi | How does corporate social responsibility contribute to firm financial performance? The mediating role of competitive advantage, reputation, and customer satisfaction[END_REF][START_REF] Dang | Retailer corporate social responsibility and consumer citizenship behavior: The mediating roles of perceived consumer effectiveness and consumer trust[END_REF][START_REF] Wang | Corporate social responsibility on customer behaviour: The mediating role of corporate image and customer satisfaction[END_REF]. Therefore, several researchers recommend investigating the mediating mechanisms between a company's CSR and its outcomes [START_REF] Saeidi | How does corporate social responsibility contribute to firm financial performance? The mediating role of competitive advantage, reputation, and customer satisfaction[END_REF][START_REF] Dang | Retailer corporate social responsibility and consumer citizenship behavior: The mediating roles of perceived consumer effectiveness and consumer trust[END_REF][START_REF] Wang | Corporate social responsibility on customer behaviour: The mediating role of corporate image and customer satisfaction[END_REF].

Given the abovementioned research gaps, the objective of this study is to examine the direct and indirect links between a company's perceived CSR, following a CSR communication, and the business value generated. It also introduces variables related to the social value the firm can deliver. According to [START_REF] Lee | What makes CSR communication lead to CSR participation? Testing the mediating effects of CSR associations, CSR credibility, and organization -public relationships[END_REF], business value will be measured in this research by consumers' behavioral intentions (i.e., to purchase and to recommend) and social value by consumers' sustainable behaviour and well-being.

Research model and hypotheses

We draw on four major theories to develop our research hypotheses and model: signaling theory (Spence, 2002), advertising effectiveness [START_REF] Mackenzie | An empirical examination of the structural antecedents of attitude towards the ad in an advertising pretesting context[END_REF], social identity theory [START_REF] Tajfel | The social identity theory of intergroup behavior[END_REF] and the SHIFT framework [START_REF] White | How to SHIFT consumer behaviors to be more sustainable: A literature review and guiding framework[END_REF].

The research model proposed (Fig. 1) posits the effectiveness of a grocery brand's CSR communication in influencing consumers to purchase and recommend that brand's products. It then postulates the direct and indirect ways in which a grocery brand's perceived CSR, following exposure to CSR communication, can improve consumers' behavioral intentions. It also assumes that the perceived CSR of a grocery brand also increases the social value of this brand by encouraging consumers to adopt sustainable behaviors to increase their well-being.

Methodology

Within a panel of alumni from one of the authors' institutions, 414 consumers participated in our study. These consumers answered an on-line questionnaire after reading an explicit CSR communication [START_REF] Morsing | The 'catch 22' of communicating CSR: Findings from a Danish study[END_REF], a printed advertisement for a water brand sold in grocery stores, published in the national press. [START_REF] Zeithaml | The behavioral consequences of service quality[END_REF]. Consumers' well-being (WB) was measured by three items inspired by the work of [START_REF] Guevarra | To have in order to do: Exploring the effects of consuming experiential products on well-being[END_REF]. For these six measurement scales, consumers were asked to specify their degree of agreement or disagreement with each item (or level of probability for purchasing intention) on a six-point Likert scale ranging from "disagree completely" to "agree completely" (or "very unlikely" to "very likely").

Results

Partial least squares structural equation modelling (PLS-SEM) with a bootstrap method with 5,000 replications has been used to analyse the data. First, this research shows (Fig. 2) that the attitude toward CSR communication has a significant positive influence on consumers' sustainable behavior (Path Coefficient (PC)=0.356; p<0.01), their attitude toward the grocery brand (PC=0.436; p<0.01)) and the perceived CSR of the grocery brand (PC=0.456; p<0.01). Hypotheses H7, H2 and H1 are therefore supported by our data. This research validates and extends previous work in the field of advertising effectiveness to CSR communication [START_REF] Mackenzie | An empirical examination of the structural antecedents of attitude towards the ad in an advertising pretesting context[END_REF]. Further, it empirically confirms recent work by [START_REF] Zhao | Perception of corporate hypocrisy in china: The roles of corporate social responsibility implementation and communication[END_REF] who, within the framework of signaling theory (Spence, 2002), had postulated that consumers' reception of a company's CSR communication would have a positive impact on their perceptions of its CSR.

The perceived CSR of a grocery brand has a significant positive influence on the attitude toward this brand (PC=0.291; p< 0.01) and on consumers' behavioral intentions toward it (PC=0.158; p< 0.01). Hypotheses H4 and H5 are therefore affirmed. Attitude toward the grocery brand also has a significant positive influence on behavioral intentions toward this brand (PC=0.558; p<0.01). Hypothesis H3 is also supported. Attitude toward the grocery brand is therefore a partial mediator of the relationship between perceived grocery brand CSR and behavioral intentions toward this brand. Perceived grocery brand CSR also has a significant positive influence on consumers' sustainable behavior (PC=0.144; p<0.01). H6 is therefore supported by our data. This research thus confirms previous studies that have shown, based on social identity theory [START_REF] Tajfel | The social identity theory of intergroup behavior[END_REF], that a company adopting responsible practices may improve consumers' perceptions and increase firm profitability [START_REF] Luo | Corporate social responsibility, customer satisfaction, and market value[END_REF][START_REF] Olsen | Green claims and message frames: How green new products change brand attitude[END_REF]. It also empirically validates the previously suggested link between a firm's perceived CSR and consumers' SCB [START_REF] Durif | Socially responsible consumers: Profile and implications for marketing strategy[END_REF][START_REF] Iglesias | Co-creation: A key link between corporate social responsibility, customer trust, and customer loyalty[END_REF][START_REF] Dang | Retailer corporate social responsibility and consumer citizenship behavior: The mediating roles of perceived consumer effectiveness and consumer trust[END_REF].

Finally, consumers' sustainable behavior has a significant positive impact on their well-being (PC=0.388; p<0.01), which in turn influences their behavioral intentions (PC=0.154; p<0.01). Hypotheses H8 and H9 are therefore supported. Consumers' sustainable behavior and well-being therefore partially mediate the relationship between perceived grocery brand CSR and consumers' behavioral intentions toward this brand. The model proposed and tested explains 46.1% of the variance in consumers' behavioral intentions toward the grocery brand. This research thus empirically confirms, as suggested by the SHIFT framework [START_REF] White | How to SHIFT consumer behaviors to be more sustainable: A literature review and guiding framework[END_REF], that prompts can positively affect sustainable behaviors and that these sustainable behaviors may improve consumers' well-being (e.g., [START_REF] Dunn | If money doesn't make you happy, then you probably aren't spending it right[END_REF][START_REF] Minton | Greening up because the god: The relations among religion, sustainable consumption, and subjective well-being[END_REF].

Path

Conclusion

There is currently a debate in the literature about whether or not CSR directly influences consumers' behavioral intentions. Some research concludes that there is a direct link between these two variables (e.g., [START_REF] Anselmsson | Corporate social responsibility and the positioning of grocery brands: An exploratory study of retailer and manufacturer brands at point of purchase[END_REF][START_REF] Stanaland | Consumer perceptions of the antecedents and consequences of corporate social responsibility[END_REF][START_REF] Dang | Retailer corporate social responsibility and consumer citizenship behavior: The mediating roles of perceived consumer effectiveness and consumer trust[END_REF], but other studies find that this direct link is non-significant (e.g. [START_REF] Lombart | A study of the impact of corporate social responsibility and price image on retailer personality and consumers' reactions (satisfaction, trust and loyalty to the retailer)[END_REF][START_REF] Öberseder | Consumers' perceptions of corporate social responsibility: Scale development and validation[END_REF][START_REF] Wang | Corporate social responsibility on customer behaviour: The mediating role of corporate image and customer satisfaction[END_REF]. Several researchers have suggested that this link may be mediated by other variables, and called for future investigations of these relationships [START_REF] Saeidi | How does corporate social responsibility contribute to firm financial performance? The mediating role of competitive advantage, reputation, and customer satisfaction[END_REF][START_REF] Dang | Retailer corporate social responsibility and consumer citizenship behavior: The mediating roles of perceived consumer effectiveness and consumer trust[END_REF][START_REF] Wang | Corporate social responsibility on customer behaviour: The mediating role of corporate image and customer satisfaction[END_REF].

The present study enriches this debate by showing the existence of both direct and indirect links between perceived CSR, following exposure to a CSR communication, sustainable consumer behavior and consumer behavioral intentions. It thus responds to the current calls for additional research to shed light on mediating effects [START_REF] Saeidi | How does corporate social responsibility contribute to firm financial performance? The mediating role of competitive advantage, reputation, and customer satisfaction[END_REF][START_REF] Dang | Retailer corporate social responsibility and consumer citizenship behavior: The mediating roles of perceived consumer effectiveness and consumer trust[END_REF][START_REF] Wang | Corporate social responsibility on customer behaviour: The mediating role of corporate image and customer satisfaction[END_REF] between a grocery brand's perceived CSR and its business value but also social value [START_REF] Lee | What makes CSR communication lead to CSR participation? Testing the mediating effects of CSR associations, CSR credibility, and organization -public relationships[END_REF][START_REF] Dang | Retailer corporate social responsibility and consumer citizenship behavior: The mediating roles of perceived consumer effectiveness and consumer trust[END_REF]. To do so, it drew on and unified within one integrating model different theories in the field of marketing: advertising effectiveness (Mac-Kenzie and Lutz, 1989), signaling theory (Spence, 2002), social identity theory [START_REF] Tajfel | The social identity theory of intergroup behavior[END_REF] and the SHIFT framework [START_REF] White | How to SHIFT consumer behaviors to be more sustainable: A literature review and guiding framework[END_REF].

This research also shows that the attitude toward a grocery brand's CSR communication is pertinent because it reinforces both the image of the grocery brand as responsible and its positive evaluation by consumers, captured in this research through the attitude toward the grocery brand. Moreover, exposure to CSR communication, via the attitude it generates, allows consumers to adopt sustainable behaviors, notably recycling, and thus indirectly increases their well-being.

However, this research has limitations that provide avenues for future research. Different grocery brands and products could be studied to increase external validity: other water brands or other consumer products (national brands or private labels). Other business (satisfaction, trust, etc.) or social (consumer citizenship, etc.) outcomes could also be integrated into the model. Future experimental research could also disentangle the different components of CSR communication to determine those that most influence the perception of the brand's CSR and consumers' sustainable behavior. Finally, actions other than recycling could be examined, such as local consumption, animal protection, composting or environmental protection in the broad sense.

Fig. 1 :

 1 Fig.1: Research model proposed

  This ad was chosen because it highlights the grocery brand's CSR (concern for the environment and for consumers) and encourages consumers to adopt sustainable behaviors (through recycling and environmental protection).Consumers' attitude toward the CSR communication (ACC) was measured by three items taken from the work of[START_REF] Zhang | The effect of self-construals on the effectiveness of comparative advertising[END_REF]. Perceived corporate social responsibility (CSR) was measured by six items inspired by the critical review of[START_REF] Latif | Measuring corporate social responsibility: A critical review of survey instruments[END_REF]. Consumers' attitude toward the grocery brand (AB) was measured by four items taken from the work of[START_REF] Charton-Vachet | Impact of the link between individuals and their region on the customer-regional brand relationship[END_REF]. Sustainable consumer behavior (SCB) was measured by six items adapted from the work of[START_REF] Quoquab | A multiple-item scale for measuring 'sustainable consumption behaviour' construct: Development and psychometric evaluation[END_REF]. Consumers' behavioral intentions (FBI) was measured by three items drawn from the work of

	This advertisement features a plastic bottle of water. The label on the bottle men-
	tions the grocery brand and that it is a "100% recycled material bottle and always
	100% recyclable. Keep sorting in mind!". The advertisement also includes a text:
	"Today it is possible. Your brand is launching its first 100% recycled plastic water
	bottle and it's all thanks to your sorting your bottles! So let's keep it going togeth-
	er."
			Attitude toward
			brand in CSR
			communication (AB)	H3
	H2		H4
	Attitude toward CSR communication (ACC)	H1	Perceived (CSR) responsibility corporate social	H5	(FBI) communication brand in CSR Behavioural intentions toward
	H7		H6
					H9
		Sustainable
		consumer		Well-being
		behaviour (SCB)	H8	(WB)