Pierre H Émono
email: pierre.hemono@excent.fr

M'hammed Sahnoun

Ahmed Nait Chabane
email: anaitchabane@cesi.fr

Optimizing Resource Allocation in the Flexible Job Shop Problem: Assessing the Impact of Rest Breaks on Task Strenuousness Reduction

Keywords: Rest break, Strenuousness, Cobots, Optimization

The integration of collaborative robots (cobots) in production workshops aims to enhance productivity while minimizing physical strain for human operators. However, physical strain is often treated merely as a constraint rather than as an objective to address. To effectively model the production process and incorporate human factors, it is crucial to employ an appropriate index that accounts for precise ergonomic rules.

In this paper, we propose a method to organize induced waiting times in the context of Flexible Job Shop Scheduling (FJSP) with the primary goal of evaluating the effects of rest breaks on fatigue generated by physical efforts in various operations. Our method introduces a specific index, based on ergonomic standards, to assess the difficulty level of operations for both humans and robots. The index incorporates normalized strain values, ensuring its generalizability across diverse industries. Moreover, the method considers a wide range of work scenarios, encompassing both individual and collaborative settings, making it applicable to different industrial contexts. Furthermore, we analyze the accumulated fatigue experienced by human agents over several scenarios, both before and after the optimization process. By doing so, we not only gain insights into the production process but also facilitate the optimization of human-robot collaboration. Our findings indicate the influence of rest break duration on the recovery of human workers and offer strategies to reduce strenuousness during production planning involving cobots. This work contributes to creating a more efficient and ergonomic working environment for all parties involved, promoting human well-being and presence in the industrial workshop. By emphasizing the significance of considering physical strain and rest breaks as essential factors in the 5.0 factory, we pave the way for improved human-robot interaction and performance in manufacturing settings.

I. INTRODUCTION

Industry 5.0 is currently focused on re-establishing the human element at the core of the manufacturing system and prioritizing the well-being of individuals [START_REF] George | Industrial revolution 5.0: the transformation of the modern manufacturing process to enable man and machine to work hand in hand[END_REF]. As a result, a re-evaluation of risk management, including factors such as fatigue risk and synchronisation of different components and services in the workshop, has become essential [START_REF] Messaadia | Plm adoption in smes context[END_REF]. The workload of an agent is identified as one of the primary contributors to fatigue risk [START_REF] Lerman | Fatigue risk management in the workplace[END_REF]. The Fatigue Risk Management System (FRMS) [START_REF] Lerman | Fatigue risk management in the workplace[END_REF], derived from the Safety Management System (SMS), incorporates considerations of the impact of fatigue on work. A previous work [START_REF] Hémono | Multi-Objective Optimization of Active Human-Robot Collaboration: Case Study of the Aircraft Industry[END_REF] models the workshop as an environment where the task must be scheduled over several machines in this representation of works, the Flexible Job-Shop Scheduling Problem (FJSP) provides a mathematical framework for task allocation across different machines while adhering to a set of constraints. Although the FJSP model has been used primarily to optimize production costs [START_REF] Yaphiar | Mixed model assembly line balancing for human-robot shared tasks[END_REF], it has also been used to address the physical strain experienced by operators [START_REF] Falerni | A Framework for Human-Robot Collaboration Enhanced by Preference Learning and Ergonomics[END_REF]. The proposed scheduling aims to alleviate the physical strain on assembly line operators and optimize either the makespan or costs. Due to the scheduling's inherent precedence constraints, human workers frequently experience waiting periods during the production process. These waiting times are usually not long enough to restore the operators' energy levels, and they can prevent them from doing another task in a different process.

This study aims to regulate specific waiting times and their duration. We propose a tool that optimizes work schedules by maximizing rest break duration's for human agents and effectively managing strenuousness based on their physical workload, the impact on hardship is assessed with regard to the minimization of fatigue induces by maximized rest breaks duration's.

This paper is organized as follows: Section II provides a brief overview of the current state of the art in fatigue modeling and the impact of rest breaks on fatigue. Section III presents our approach. Section IV presents the obtained results and includes a discussion. Section V ends this paper with a conclusion and a set of perspectives.

II. STATE OF THE ART

In terms of management of injury risk and production issues, fatigue as been recognized as an essential element [START_REF] Xu | Fatigue, personnel scheduling and operations: Review and research opportunities[END_REF].

In [START_REF] Eklund | Ergonomics, Quality and Continuous Improvement Conceptual and Empirical Relationships in an Industrial Context[END_REF] it has been observed that worker fatigue is linked to a progressive decline in quality over time, moreover the bad effects of fatigue not only affect the product quality but also reduce the well being of workers as shown in [START_REF] Tucker | The impact of rest breaks upon accident risk, fatigue and performance: a review[END_REF]. As a result, this decline leads to an elevated rate of defects in the production system. In [START_REF] Michalos | The effect of job rotation during assembly on the quality of final product[END_REF] it is emphasized that prolonged fatigue can contribute to an increased occurrence of human errors, which subsequently leads to a deterioration in product quality. According to [START_REF] Hémono | Multi-Objective Optimization of Active Human-Robot Collaboration: Case Study of the Aircraft Industry[END_REF], one of the primary causes of fatigue is the work schedule itself. Designing task planning and allocation based on ergonomic standards can help mitigate errors, accidents, and health problems [START_REF] Hémono | Multi-Objective Optimization of Active Human-Robot Collaboration: Case Study of the Aircraft Industry[END_REF]. Rest breaks form an integral part of ergonomic design, and effectively managing them can contribute to the prevention of musculoskeletal disorders. The implementation of rest break could allow an improvement of task performance as shown in [START_REF] Wendsche | The impact of supplementary short rest breaks on task performance: A meta-analysis[END_REF] where the makespan has been reduced with an increasing of quality and quantity of products. Several studies has shown that short rest breaks over the working days are beneficial for tasks performance and can compensate for a loss of productive working [START_REF] Tucker | The impact of rest breaks upon accident risk, fatigue and performance: a review[END_REF]. Recovery periods could be organized in the work shift (internal recovery) or between work shifts (external recovery) [START_REF] Geurts | Recovery as an explanatory mechanism in the relation between acute stress reactions and chronic health impairment[END_REF]. The optimal time for a rest break for internal recovery is defined between three and ten minutes according to [START_REF] Richter | Belastung und Beanspruchung: Stress, Ermüdung und Burnout im Arbeitsleben[END_REF].

In this study we identified several methods for fatigue modelling as shown in table I.

Reference

Model Parameters [START_REF] Glock | An integrated cost and worker fatigue evaluation model of a packaging process[END_REF] F s (t i ) = 1 -exp(θ i t a ) F s (t i ) is the fatigue accumulated by time t i ; θ i is the fatigue growth parameter when a worker completes a task i during time t i . [START_REF] Jaber | Modelling worker fatigue and recovery in dual-resource constrained systems[END_REF] F imax = M LC • f i • M ET i F imax is the maximum fatigue index for task i ; MLC is maximum load capability; M ET i represents the maximum endurance time for task i individual's maximum capability; [START_REF] Jaber | Incorporating human fatigue and recovery into the learning-forgetting process[END_REF] F (t) = 1 -exp(-λt) F (t) is the fatigue accumulated by time t; λ is the fatigue parameter. [START_REF] Sobhani | Incorporating human factors-related performance variation in optimizing a serial system[END_REF] F jt = 1 -exp(θ j t) F jt is the percentage of maximum fatigue developed by the injured operator; θ j is the workload level at the workplace. [START_REF] Perez | Discrete event simulation as an ergonomic tool to predict workload exposures during systems design[END_REF] F i = ti M ETi F i denotes fatigue contribution per task i, and M ET i denotes maximum endurance time for task i. [START_REF] Wang | Fatigue minimization work shift scheduling for air traffic controllers[END_REF] x i,j = x i,0 e T j l=1 ki,j

x i,j represents the fatigue state of the i-th person during the j-th interval of length T , and k i,j is the fatigue coefficient.

The rest breaks could be structured as shown in II R (τi) = F (t) e -µτi F (t) tracks the total fatigue accumulated until a given time t, while R (τi) represents the residual fatigue after a rest period of duration τ i (τ i ≥ 0). The recovery parameter µ governs the rate at which fatigue is restored. [START_REF] Calzavara | A model for rest allowance estimation to improve tasks assignment to operators[END_REF] RA = F tw e -µτ F tw represents the accumulated fatigue over time; tw denotes the duration of the working activity; τ represents the recovery time required for the operator to recover from the accumulated fatigue; µ is a physiological factor that alleviates the recovery process.

The authors [START_REF] Glock | An integrated cost and worker fatigue evaluation model of a packaging process[END_REF] and [START_REF] Sobhani | Incorporating human factors-related performance variation in optimizing a serial system[END_REF] proposed studies utilize exponential functions to model excessive fatigue associated with work. These studies incorporate additional factors such as workload variation, machine-related tasks, and dynamic activities in the packaging process to enhance the accuracy of fatigue estimation models. In [START_REF] Jaber | Modelling worker fatigue and recovery in dual-resource constrained systems[END_REF] a model that assumes linear fatigue progression over time is presented. This model is adjusted using new data and takes into account operator fatigue when designing dual-resource constrained systems. Similarly, in [START_REF] Jaber | Incorporating human fatigue and recovery into the learning-forgetting process[END_REF], research works focus on muscular fatigue associated with sustaining a particular posture or force level required for a task. They propose an exponential increase in fatigue over time based on the muscle's Maximum Voluntary Contraction (MVC). In [START_REF] Perez | Discrete event simulation as an ergonomic tool to predict workload exposures during systems design[END_REF], a comprehensive approach incorporating discrete event simulation (DES), bio-mechanical analysis, and static fatigue models is employed to estimate fatigue rates. This methodology enables designers to assess the ergonomic implications associated with various system design alternatives. In [START_REF] Wang | Fatigue minimization work shift scheduling for air traffic controllers[END_REF], researchers have proposed an exponential model to characterize fatigue levels specifically in air traffic controllers. Their findings emphasize the faster accumulation of fatigue in states of heightened exhaustion.

The studies have identified three primary approaches to defining rest scenarios. These include varying the number of breaks in the production process, for example, in [START_REF] Glock | An integrated cost and worker fatigue evaluation model of a packaging process[END_REF] a rest break between jobs is planned if the level of fatigue exceeds a certain threshold or in [START_REF] Bechtold | Maximization of labor productivity through optimal rest-break schedules[END_REF] optimizing rest break effects by adjusting break length, or combining both methods.

The methods including a fatigue parameter such as [START_REF] Jaber | Incorporating human fatigue and recovery into the learning-forgetting process[END_REF] and [START_REF] Glock | An integrated cost and worker fatigue evaluation model of a packaging process[END_REF] are well suited regarding to the FJSP model developed with a strenuousness fatigue. As we consider the fatigue can be modelled as an exponential law the rest should also been considerate as exponential as in the works of [START_REF] Calzavara | A model for rest allowance estimation to improve tasks assignment to operators[END_REF], we also consider that the recuperation speed is slower than the fatigue growth (half as fast).

Fatigue has also been extensively studied in the context of the Flexible Job Shop Scheduling Problem (FJSP). In [START_REF] Tan | A fatigue-conscious dual resource constrained flexible job shop scheduling problem by enhanced nsga-ii: An application from casting workshop[END_REF], a comprehensive analysis of a fatigue-conscious dual-resourceconstrained FJSP is presented. The authors proposed an innovative approach that utilizes exponential functions to model both fatigue and rest periods. This model takes into account the strenuousness of fatigue and assumes that fatigue and rest follow an exponential pattern. The study conducted in [START_REF] Tan | A fatigue-conscious dual resource constrained flexible job shop scheduling problem by enhanced nsga-ii: An application from casting workshop[END_REF] exemplifies the efficacy of incorporating fatigue-awareness in FJSP models using the exponential fatigue and rest framework. Their approach is solved using an enhanced version of the Non-dominated Sorting Genetic Algorithm II (NSGA-II). By integrating fatigue modeling into the scheduling problem, their methodology allows for a more realistic representation of operator well-being and performance during job shop operations. In [START_REF] Aribi | Multi-objective optimization of the dynamic and flexible job shop scheduling problem under workers fatigue constraints[END_REF], the Dynamic Flexible Job Shop Scheduling Problem under Worker Fatigue Constraint (DFJSP-WF) is adressed using the NSGA-II algorithm. They considered worker fatigue as a constraint, leading to tardiness in the production process. Their dynamic approach accounts for job cancellation, new job arrivals, and machine breakdown, ensuring efficient scheduling in real-world production settings while prioritizing worker well-being and minimizing tardiness. In [START_REF] Du | A hybrid metaheuristic algorithm with novel decoding methods for flexible flow shop scheduling considering human fatigue[END_REF], the authors solve the Dual Resource Constrained Flexible Flow Shop Scheduling Problem (DRC-FFSP) using a Mixed-Integer Programming (MIP) model. They aim to minimize makespan while considering constraints related to machines and heterogeneous human workers with varying skills and characteristics. To achieve efficient scheduling solutions, they propose two paradigms of a Hybrid Metaheuristic Algorithm (HMA).

Compared to other methods, our proposed model offers several key advantages. Firstly, its implementation is easier due to the use of a Mixed-Integer Linear Programming (MILP) formulation, which simplifies setup and solution compared to evolutionary algorithms like NSGA-II. Additionally, our model's offline fatigue calculation enhances efficiency and adaptability in scheduling scenarios. Moreover, the custom definition of strenuousness levels using a generalizable approach allows us to accurately capture individualized fatigue behavior, providing a realistic representation of operator wellbeing and performance [START_REF] Bouaziz | Modeling and simulation of human behavior impact on production throughput[END_REF]. Overall, our model presents a compelling solution for the Flexible Job Shop Scheduling Problem, combining ease of implementation, computational efficiency, and accurate fatigue modeling to optimize scheduling outcomes and promote worker health and productivity. Importantly, our model achieves these benefits without increasing the makespan, as it effectively decreases the fatigue level and improves the well-being of human agents.

III. PROBLEM STATEMENT

In this research paper, we focus on optimizing waiting times for human agents in the FJSP model. Production planning involving cobots sometimes results in waiting times for human agents due to resource allocation and precedence logic between operations. To achieve this, we propose a two-step approach. Firstly, we utilize the Gurobi solver [START_REF] Optimization | Gurobi Optimizer Reference Manual[END_REF] to find an initial solution for the scheduling problem. The Gurobi solver helps us efficiently determine an optimized schedule, minimizing the waiting times for human agents. The second step involves a manual analysis. In this step, we apply the "law of fatigue and rest" to the schedule obtained from the optimization process.

The waiting times are induces by the precedence constraints existing in the FJSP, theses times are often too short for consider resting for the agent [START_REF] Richter | Belastung und Beanspruchung: Stress, Ermüdung und Burnout im Arbeitsleben[END_REF]. In this research work we tried to organize the waiting time in order to obtain the better profit in terms of recuperation, the impact of operation's scheduling order and duration is analyzed. An extension of the model formulated by Özgüven & al [START_REF] Özgüven | Mathematical models for jobshop scheduling problems with routing and process plan flexibility[END_REF] is proposed.

This section presents our research work as an expansion of the classical FJSP Problem. We focus on a set of jobs denoted as I = 1, . . . , i, . . . , I. These jobs can be processed using a collection of m machines represented by M = {m r } m r-1 . Within this machine set, there are three types: Human agents (denoted as r = Ha), Robot agents (r = Ra), and Collaborative agents where humans and robots work together (r = Co).

In our model, both the robot and human agents have the option to work independently or collaborate to complete a job. Each job, denoted by i, is considered complete when all its constituent operations, denoted as I j1 , . . . , I jlj , are finished. The operation j of job i can be processed by any available resource and has a specific processing time denoted as pt j,i,r .

The assumptions underlying the classical FJSP can be stated as follows:

1) The processing times of operations are known and remain constant. 2) Setup times, which refer to the time required to prepare a machine for processing different operation, are not taken into account. 3) Transportation times, involving the movement of jobs or resources between machines, are not considered. 4) All jobs are available and ready for processing at the beginning (time 0). 5) Pre-emption, which refers to interrupting the processing of an operation to accommodate another, is not allowed. Each machine can handle only one operation at a time. 6) There exists a unique and feasible plan for each job, implying that there is only one valid sequence of operations for job completion. The variables and parameters of our study's model are extended from the works of [START_REF] Özgüven | Mathematical models for jobshop scheduling problems with routing and process plan flexibility[END_REF] and presented in Table . III and Table IV, respectively.

TABLE III: Decision variables

Variables Definition

X j,i,r 1 if the operation j of task i is realized by the resource r. 0 otherwise.

P j,i,j ′ ,i ′ ,r 1
If the operation j ′ , i ′ precedes the operation j, i. 0 Otherwise.

Cmax

The maximum completion time over all jobs (Makespan).

Hmax

Maximum hard working level. idle

The period of time between operations for an agent. C j,i,r

Completion time of operation j of task i on machine r. S j,i,r

Starting time of operation j of task i on machine r.

Cj i

Completion time of task i. Processing time for the human agent. pt j,i,Ra

Processing time for the robot agent. pt j,i,Co

Processing time for collaborative task.

H j,i,Ha

Hard working level of operation j of task i for human.

H j,i,Ra

Hard working level of operation j of task i for cobot.

H j,i,Co

Hard working level of operation j of task i for both agents. α

Modulates Cmax and idle importance (α ∈ [0, 1] ⊂ R). M A large number. T Strenuousness maximum level authorized.

Objective functions:

The multi-objective approach aims to minimize both the makespan (f 1 ) and waiting times (f 2 ) between operations. By utilizing the lexicographic method (lexM in) [START_REF] George | Industrial revolution 5.0: the transformation of the modern manufacturing process to enable man and machine to work hand in hand[END_REF], the optimization process first focuses on minimizing the makespan and then prioritizes reducing waiting times. This approach ensures longer rest breaks instead of multiple short waiting periods, resulting in improved scheduling efficiency. lexM in :

f 1 = (1 -α) × C max f 2 = α × idle (1)
This model is subject to a set of constraints:

r∈R (X j,i,r ) ≥ 1 ∀j ∈ J, ∀i ∈ I (2)
The constraint [START_REF] Messaadia | Plm adoption in smes context[END_REF] ensures that all the operations are distributed among the agents.

S j,i,r + C j,i,r ≤ (X j,i,r ).M ∀r ∈ R, ∀j ∈ J, ∀i ∈ I (3)
The constraint (3) sets the time of an operation to 0 if it is not allocated to an agent.

C j,i,r ≥ S j,i,r + pt j,i,r -(1 -X j,i,r ).M ∀j ∈ J, ∀i ∈ I, ∀r ∈ R (4)
The constraint (4) ensures that the completion time of an operation performed by any agent r is equal to it's starting time plus the duration of the operation. The second part of the equation guarantee that the completion time equal to 0 if the operation is not distributed to an agent.

S j,i,r ≥ C j ′ ,i ′ ,r -P j,i,j ′ ,i ′ ,r .M ∀j < j ′ , ∀i ∈ I, ∀i ′ ∈ I, ∀r ∈ R (5) S j ′ ,i ′ ,r ≥ C j,i,r -P j,i,j ′ ,i ′ ,r .M ∀j < j ′ , ∀i ∈ I, ∀i ′ ∈ I, ∀r ∈ R (6)
The equation ( 5) and ( 6) verify the precedence constraints between operations. P j,i,j ′ ,i ′ ,r + P j ′ ,i ′ ,j,i,r = 1 [START_REF] Xu | Fatigue, personnel scheduling and operations: Review and research opportunities[END_REF] The constraints [START_REF] Xu | Fatigue, personnel scheduling and operations: Review and research opportunities[END_REF] regulates the precedence order between operations.

C max ≥ C j,i,k ∀r ∈ R, ∀j ∈ J, ∀i ∈ I ( 8
)
Cj i ≥ C j,i,k ∀r ∈ R, ∀j ∈ J, ∀i ∈ I (9)
The constraints ( 8) and ( 9) ensure that the makespan and the completion time of the whole job are greater than or equal to the completion time of an operation.

The following constraints constitutes the extension of the model and our contribution.

r∈R (H j,i,r ) × X j,i,r ≤ H max ∀j ∈ J, ∀i ∈ I, ∀r ∈ R (10) H max ≤ T ∀j ∈ J, ∀i ∈ I, ∀r ∈ R (11)
The constraints [START_REF] Michalos | The effect of job rotation during assembly on the quality of final product[END_REF] ensure that the maximum strenuousness is more or equal to the sum of the strenuousness of the operations performed. The maximum strenuousness should not exceed a certain threshold [START_REF] Wendsche | The impact of supplementary short rest breaks on task performance: A meta-analysis[END_REF].

S j,i,r -C j ′ ,i ′ ,r × 1 -X j,i, ′ Ha ′ ≤ idle (12) ∀j ∈ J, ∀j ′ ∈ J, ∀i ∈ I, ∀i ′ ∈ I, ∀r ∈ Ha
The constraint labeled as [START_REF] Tucker | The impact of rest breaks upon accident risk, fatigue and performance: a review[END_REF] ensures that a human agent must start the next operation immediately after completing the previous operation, without any waiting time in between.

The laws of fatigue and recover are defined as in the works of [START_REF] Jaber | Incorporating human fatigue and recovery into the learning-forgetting process[END_REF]. The fatigue generated by an operation performed by human agent is added to the previous fatigue, then when a rest break occurs, the recover is subtracted from the previous fatigue.

F t = 1 -exp (-λ×µ×t)
• λ: The difficulty level.

• µ: The fatigue parameter.

• t: The operation's duration.

R (t) = F t × e -µ 2 ×τ
• F t : The previous fatigue.

• τ : The recovery parameter. In our study, we incorporate a difficulty level parameter to address fatigue, following the industrial ergonomic standard. This parameter represents the level of physical engagement an agent experiences during a task and is graded on a scale from 1 (least physically demanding) to 4 (most physically demanding) conditions. The fatigue parameter is defined arbitrarily. Additionally, we introduce a recovery parameter, set at half the fatigue parameter, to indicate that the rate of recovery is slower compared to the rate of fatigue buildup. Short waiting times, which do not qualify as proper rest breaks, can result in mental fatigue. This is because the operator doesn't have enough time to fully recover from the previous task, and the interruptions in the production process can be frustrating. To quantify mental workload, we define specific fatigue values that are half of the physical fatigue values.

In this study, we have undertaken an analysis of optimization in two scenarios. These scenarios were chosen based on the potential for rearranging waiting times. The values of alpha were selected based on the variation of collaborative task planned, with two major steps observed at α = 0.3 and α = 0.5. A table depicting the resource allocation across different alpha values is presented in Table V. During the evolution of α, two transitions were observed that led to an increase in the number of collaborative operations. Among these transitions, two situations stand out as strong candidates: α values of 0.3 and 0.5.

IV. RESULTS AND DISCUSSION

In a scenario involving agents with heterogeneous capacity such as human and cobot, working alone or together to the completion of a job, the constraints are inducing waiting time for agents. If they are not affecting the robotic agent, theses waiting times impact human agents by adding fatigue and hardship caused by waiting time. By studying these waiting times we observed that a first optimization without regulation for break length generate erratic behaviors for scheduling of the operations subject to precedence constraints as shown in Figure 1. In order to extend rest break durations, the minimization of idle time in Equation ( 2) was used to reschedule operations. Initially, there were four breaks with durations of 0.11, 0.11, 3.81, and 2.45 minutes (Figure 1). After optimization, these breaks were consolidated into a single break of 6.86 minutes (Figure 2). Breaks with durations less than three minutes are not considered rest breaks, as mentioned in [START_REF] Geurts | Recovery as an explanatory mechanism in the relation between acute stress reactions and chronic health impairment[END_REF]. These short waiting times can lead to supplementary strenuousness because the human worker is unable to do anything else but wait. The exponential nature of fatigue and recovery laws means that longer durations result in lower fatigue indices.

Figure 3 and figure 4 illustrates the reduction in fatigue over time for two scenarios: before and after optimization, considering collaborative tasks. The new set of constraints and objectives allows a reduction in fatigue, approximately 2.5% and 4.5% lower, respectively. Importantly, this reduction was achieved without changing the order of operations, but by adjusting their scheduling to minimize idle time between them. While optimizing the scheduling process and reducing fatigue for human operators is crucial, it is equally vital to address the ethical implications of integrating collaborative robots (cobots) into the workplace. The increasing adoption of cobots may lead to concerns about job displacement, as workers may worry that automation will replace certain tasks, resulting in potential job losses. Therefore, it is imperative to implement cobots in a manner that complements human abilities and enhances overall productivity without replacing human workers.

Optimizing rest breaks and scheduling not only improves ergonomics and reduces fatigue for human agents but also promotes their overall well-being in the workplace. Minimizing strenuousness and providing adequate rest can lead to higher job satisfaction and better mental health among workers. Moreover, implementing collaborative robots in a manner that prioritizes human well-being fosters increased acceptance of cobots by human operators. By ensuring a fair distribution of tasks based on the custom strenuousness index, our model enhances job satisfaction and supports a harmonious human-robot collaboration in the production process.

V. CONCLUSION In this research, we have examined how rest breaks impact the level of strenuousness. Through the integration of con-straints and the utilization of a lexicographic multi-objective modeling approach, we have successfully minimized fatigue while preserving the original makespan. Our findings underscore the importance of considering the unique characteristics of each operation during scheduling, rather than solely focusing on their sequence, to optimize the ergonomic design of the process.

This study revolves around analyzing a real case of scheduling in an industrial workshop, subject to a comprehensive evaluation in real working conditions. The main focus is on enhancing the ergonomics of human-robot collaboration and the planning process, with the ultimate goal of increasing the acceptance of collaborative robots (cobots) by human operators and the well-being of human agent itself. Although mathematical modeling provides satisfying results, to effectively measure the impact of these improvements, the process will need to be implemented using the new model considering rest breaks duration maximization in order to observe the assessment of rest breaks on strenuousness, allowing for a direct comparison with the previous model. Notably, operator feedback will play a pivotal role in the later stages of the study, providing valuable insights into the scheduling process and ultimately aiming to foster improved acceptance of cobots in the production process while enhancing the well-being of human agents.

Moving forward, it would be advantageous to explore predictive models for determining rest break requirements to avoid surpassing pre-established fatigue thresholds. Additionally, dynamically adjusting the difficulty level based on the order of operations could offer a more ergonomic scheduling approach.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Gantt diagram before optimization of waiting times

Fig. 3 : 3 Fig. 4 :

 334 Fig. 3: Evolution of the fatigue over time before and after optimization α = 0.3

TABLE I :

 I References on Fatigue Quantification.

TABLE II :

 II References on Recovery Quantification

	Reference	Model	Parameters
	[17]

TABLE IV :

 IV Parameters

	Parameters Definition
	dP j,i,j

′ ,i ′ 1 if the operation O j,i precedes the operation O j ′ ,i ′ . 0 otherwise. pt j,i,Ha

TABLE V :

 V Operations allocation over α Ha: Human agent; Ra: Robot agent; Co: Collaborative Ha Ha Ha Ha Ha Ha Ha Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra OP12 Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha OP13 Ha Ha Ha Ha Ha Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra OP14 Ha Ha Ha Ha Ha Co Co Co Co Co Co Co Co Co Co Co Co Co Co OP15 Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra OP21 Ra Ra Ra Ra Ra Ra Ra Ra Ra Co Co Co Co Co Co Co Co Co Co OP22 Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha OP23 Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha OP24 Ra Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha OP25 Ha Ha Ha Ha Ha Ha Ha Ha Ha Co Co Co Co Co Co Co Co Co Co OP26 Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha OP27 Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha OP31 Ha Ha Ha Ha Ha Co Co Co Co Co Co Co Co Co Co Co Co Co Co OP32 Ra Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha OP33 Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha Ha OP34 Ha Ha Ha Ha Ha Ha Ha Ha Ha Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra OP35 Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra Ra

	OP/α 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
	OP11 Ha Ha

ACKNOWLEDGMENT

The authors express their heartfelt gratitude to eXcent company and the National Research and Technology Agency (ANRT) for their invaluable financial support in this project.


D B A Tran

) O Lopez

M Manceau

A Goncharov

) M Abgrall

H Alvarez-Martinez

R Le Targat

E Cantin

P.-E Pottie

A Amy-Klein

B Darquié

Near-to mid-IR spectral purity transfer with a tunable frequency comb: methanol frequency metrology over a record frequency span

We report the development and operation of a frequency-comb-assisted high-resolution mid-infrared molecular spectrometer combining high spectral purity, SI-traceability, wide tunability and high sensitivity. An optical frequency comb is used to transfer the spectral purity of a SI-traceable 1.54 µm metrology-grade frequency reference to a 10.3 µm quantum cascade laser (QCL). The near-infrared reference is operated at the French time/frequency metrology institute, calibrated there to primary frequency standards, and transferred to Laboratoire de Physique des Lasers via the REFIMEVE fiber network. The QCL exhibits a sub-10 -15 frequency stability from 0.1 to 10 s and its frequency is traceable to the SI with a total uncertainty better than 4 × 10 -14 after 1-s averaging time. We have developed the instrumentation allowing comb modes to be continuously tuned over 9 GHz resulting in a QCL of record spectral purity uninterruptedly tunable at the precision of the reference over an unprecedented span of 1.4 GHz. We have used our apparatus to conduct sub-Doppler spectroscopy of methanol in a multi-pass cell, demonstrating state-of-art frequency uncertainties down to the few kilohertz level. We have observed weak intensity resonances unreported so far, resolved subtle doublets never seen before and brought to light discrepancies with the HITRAN database. This demonstrates the potential of our apparatus for probing subtle internal molecular processes, building accurate spectroscopic models of polyatomic molecules of atmospheric or astrophysical interest, and carrying out precise spectroscopic tests of fundamental physics.

I. INTRODUCTION

High precision molecular spectroscopy offers exciting perspectives ranging from fundamental physics [START_REF] Hinds | Testing time reversal symmetry using molecules[END_REF][START_REF] Roussy | An improved bound on the electron's electric dipole moment[END_REF][START_REF] Fiechter | Toward detection of the molecular parity violation in chiral Ru(acac) 3 and Os(acac) 3[END_REF][START_REF] Pastor | Testing the validity of Bose-Einstein statistics in molecules[END_REF][START_REF] Patra | Proton-electron mass ratio from laser spectroscopy of HD + at the part-per-trillion level[END_REF][START_REF] Diouf | Lambdips and Lamb-peaks in the saturation spectrum of HD[END_REF][START_REF] Barontini | Measuring the stability of fundamental constants with a network of clocks[END_REF][START_REF] Muller | A study of submillimeter methanol absorption toward PKS1830 -211: Excitation, invariance of the proton-electron mass ratio, and systematics[END_REF][START_REF] Segal | Studying fundamental physics using quantum enabled technologies with trapped molecular ions[END_REF][START_REF] Mudiayi | Linear probing of molecules at micrometric distances from a surface with sub-doppler frequency resolution[END_REF] and metrology [START_REF] Benabid | Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres[END_REF][START_REF] Castrillo | Optical determination of thermodynamic temperatures from a C 2 H 2 linedoublet in the near infrared[END_REF][START_REF] Mejri | Measuring the Boltzmann constant by mid-infrared laser spectroscopy of ammonia[END_REF] to astrochemistry [START_REF] Herbst | Complex organic interstellar molecules[END_REF] , remote sensing and Earth sciences [START_REF] Galli | Spectroscopic detection of radiocarbon dioxide at parts-per-quadrillion sensitivity[END_REF][START_REF] Harrison | Spectroscopic requirements for ACCURATE, a microwave and infrared-laser occultation satellite mission[END_REF][START_REF] Guinet | Laser spectroscopic study of ozone in the 100 ← 000 band for the SWIFT instrument[END_REF] . Experiments in these domains are often based on frequency measurements of molecular vibrations in the mid-infrared (mid-IR) molecular fingerprint region, therefore generating the need for mid-IR laser sources that are spectrally pure, accurate and widely-tunable.

Here we report on the development and operation of a quantum cascade laser (QCL) based spectrometer that provides a unique combination of sensitivity, frequency resolution and tunability of any mid-IR spectrometer to date. An Erbiumdoped fiber mode-locked optical frequency comb is used to transfer the spectral purity of a metrology-grade ultrastable frequency reference traceable to the International System of Units (SI) from the near-IR to the mid-IR. In addition, we have developed the instrumentation allowing comb modes to be continuously tuned over 9 GHz. This results in a QCL a) Current address: Time and Frequency Department, National Physical Laboratory, Teddington, United Kingdom b) Permanent address: Institute of Laser Physics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia c) Current address: Laboratoire de Physique des Lasers, Université Sorbonne Paris Nord, CNRS, Villetaneuse, France of record spectral purity that can be continuously tuned at the precision of the frequency reference over an unprecedented frequency span of 1.4 GHz. These developments constitute enabling technologies for driving the next generation of ultra-high resolution molecular spectroscopy apparatus in the molecular fingerprint region. High spectral purity is key to reaching the resolutions required for resolving the subtlest structures, clusters of blended lines which constitute unique probes of fundamental processes in molecules. Together with SI-traceability and high sensitivity, it allows systematic effects to be unraveled in order to minimize line center uncertainties. Adding finally wide continuous tuning capabilities to the toolbox of present and future mid-IR photonics is essential for finding/resolving new unreported lines, in particular weak transitions, as demonstrated here, providing in-depth insight into the internal molecular dynamics.

Distributed feedback (DFB) quantum cascade lasers (QCLs) are available over wide ranges of the mid-IR (from 3 to 25 µm), can be tuned over several gigahertz and have continuous-wave (cw) power levels in the milliwatt to watt range near room-temperature. They however show substantial frequency noise [START_REF] Myers | Free-running frequency stability of mid-infrared quantum cascade lasers[END_REF][START_REF] Bartalini | Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow-Townes limit[END_REF][START_REF] Tombez | Frequency noise of free-running 4.6 µm distributed feedback quantum cascade lasers near room temperature[END_REF][START_REF] Bartalini | Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser[END_REF][START_REF] Tombez | Linewidth of a quantumcascade laser assessed from its frequency noise spectrum and impact of the current driver[END_REF][START_REF] Mills | Coherent phase lock of a 9 µm quantum cascade laser to a 2 µm thulium optical frequency comb[END_REF][START_REF] Galli | Comb-assisted subkilohertz linewidth quantum cascade laser for high-precision mid-infrared spectroscopy[END_REF][START_REF] Sow | A widely tunable 10-µm quantum cascade laser phase-locked to a state-ofthe-art mid-infrared reference for precision molecular spectroscopy[END_REF][START_REF] Argence | Quantum cascade laser frequency stabilization at the sub-Hz level[END_REF][START_REF] Cappelli | Intrinsic linewidth of quantum cascade laser frequency combs[END_REF][START_REF] Chomet | Highly coherent phase-lock of an 8.1 µm quantum cascade laser to a turn-key mid-IR frequency comb[END_REF] . For the most precise measurements considered here, high-spectral purity and traceability to a frequency standard are both required. This can be achieved by phase-locking to: (i) the mid-IR secondary frequency standard [START_REF] Sow | A widely tunable 10-µm quantum cascade laser phase-locked to a state-ofthe-art mid-infrared reference for precision molecular spectroscopy[END_REF] (a CO 2 laser stabilized to a saturated absorption line of OsO 4 [START_REF] Acef | Metrological properties of CO 2 OsO 4 optical frequency standard[END_REF] ), resulting in a 10 Hz line width, 1 Hz stability at 1 s and accuracy of a few tens of hertz; (ii) a near-IR metrology-grade frequency reference traceable to primary frequency standards, resulting in ultimate sub-Hz stabilities and accuracies, and linewidth narrowing down to the 0.1 Hz level [START_REF] Argence | Quantum cascade laser frequency stabilization at the sub-Hz level[END_REF][START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF] (see also related similar works, which however do not reach such high levels of spectral purity [START_REF] Insero | Measuring molecular frequencies in the 1-10 µm range at 11-digits accuracy[END_REF][START_REF] Sinhal | Frequency stabilisation and SI tracing of mid-infrared quantum-cascade lasers for precision molecular spectroscopy[END_REF] ). Efforts in combining metrology-grade spectral performance and wide continuous tunability are scarce. We have previously demonstrated respectively 10 GHz and 0.4 GHz continuous tunability for: (i) the 10-Hz narrow 10.6 µm QCL phase-locked to the mid-IR secondary frequency standard (∼ 10 -12 absolute frequency uncertainty) [START_REF] Sow | A widely tunable 10-µm quantum cascade laser phase-locked to a state-ofthe-art mid-infrared reference for precision molecular spectroscopy[END_REF] and (ii) the 0.1-Hz narrow 10.3 µm QCL ultimately calibrated to primary frequency standards (∼ 10 -14 absolute frequency uncertainty) [START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF] . Another interesting apparatus is the CO 2 -laser/microwave-sideband spectrometer demonstrated in [START_REF] Sun | Dual-mode CO 2 -laser/microwave-sideband spectrometer with broadband and saturation dip detection for CH 3 OH[END_REF] allowing 11.8 GHz wide windows around any CO 2 laser emission line to be covered in a single sweep. This spectrometer shows similarities with our metrology-grade equipment [START_REF] Sow | A widely tunable 10-µm quantum cascade laser phase-locked to a state-ofthe-art mid-infrared reference for precision molecular spectroscopy[END_REF][START_REF] Lemarchand | A revised uncertainty budget for measuring the Boltzmann constant using the doppler broadening technique on ammonia[END_REF] , but uses free-running CO 2 lasers and allows broadband scans at Doppler-limited resolutions only. Note that QCLs can also be phase-locked to a frequency comb controlled with a radiofrequency (RF) reference. This enables higher tuning ranges, but at the expense of a barely reduced frequency noise (resulting in ∼1 kHz line width at best) and a frequency uncertainty limited to around 10 -13 at best by the RF reference [START_REF] Mills | Coherent phase lock of a 9 µm quantum cascade laser to a 2 µm thulium optical frequency comb[END_REF][START_REF] Galli | Comb-assisted subkilohertz linewidth quantum cascade laser for high-precision mid-infrared spectroscopy[END_REF][START_REF] Bartalini | Frequency-comb-referenced quantum-cascade laser at 4.4 µm[END_REF][START_REF] Gambetta | Frequency metrology in the near-infrared spectrum of H 17 2 O and H 18 2 O molecules: testing a new inversion method for retrieval of energy levels[END_REF][START_REF] Borri | Direct link of a mid-infrared QCL to a frequency comb by optical injection[END_REF][START_REF] Gatti | Frequency-comb-calibrated doppler broadening thermometry[END_REF][START_REF] Gambetta | Direct phase-locking of a 8.6-µm quantum cascade laser to a mid-IR optical frequency comb: application to precision spectroscopy of N 2 O[END_REF][START_REF] Hansen | Quantum cascade laser-based mid-IR frequency metrology system with ultra-narrow linewidth and 1 × 10 -13 -level frequency instability[END_REF][START_REF] Gambetta | Absolute frequency measurements of CHF 3 doppler-free ro-vibrational transitions at 8.6 µm[END_REF][START_REF] Lamperti | Absolute spectroscopy near 7.8 µm with a comb-locked extended-cavity quantum-cascade-laser[END_REF] . Mid-IR frequency combs, based [START_REF] Chomet | Highly coherent phase-lock of an 8.1 µm quantum cascade laser to a turn-key mid-IR frequency comb[END_REF][START_REF] Lepère | A mid-infrared dual-comb spectrometer in step-sweep mode for high-resolution molecular spectroscopy[END_REF][START_REF] Agner | High-resolution spectroscopic measurements of cold samples in supersonic beams using a QCL dual-comb spectrometer[END_REF][START_REF] Komagata | Absolute frequency referencing for swept dual-comb spectroscopy with midinfrared quantum cascade lasers[END_REF] or not [START_REF] Yang | Self-broadening and self-shift in the 3ν 2 band of ammonia from mid-infrared-frequency-comb spectroscopy[END_REF][START_REF] Laumer | Sub-Hz relative linewidths from an interferometrically stabilized mid-infrared frequency comb[END_REF] on QCL technologies, are emerging as flexible sources of broadband coherent radiation. They can address a very wide spectral range, potentially via the dual-comb spectroscopy methods [START_REF] Picqué | Frequency comb spectroscopy[END_REF] , but metrology-grade resolutions and frequency uncertainties have yet to be demonstrated.

In the following, we describe our high-resolution molecular spectrometer. It exploits a frequency comb to transfer the spectral purity of a 1.54 µm remote SI-traceable optical frequency reference signal to a 10.3 µm QCL. The near-IR reference is operated at the French time/frequency metrology institute (LNE-SYRTE), calibrated there to primary frequency standards, and transferred to Laboratoire de Physique des Lasers (LPL) via a long-haul optical fiber link [START_REF] Argence | Quantum cascade laser frequency stabilization at the sub-Hz level[END_REF] . Compared to our previous work [START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF] , the QCL's tuneability has been extended by a factor of more than three. We use our SI-traceable QCL of record spectral purity to conduct sub-Doppler saturation spectroscopy of methanol in a multi-pass cell over an unprecedented continuous frequency span of 1.4 GHz. Methanol is found in a wide variety of astronomical sources [START_REF] Batrla | Detection of strong methanol masers towards galactic H II regions[END_REF] and is the second most abundant organic molecule in the Earth's atmosphere [START_REF] Singh | Evidence from the pacific troposphere for large global sources of oxygenated organic compounds[END_REF] . It is an excellent probe of the physical conditions, the chemistry and the history of these environments. Reliable laboratory spectroscopic data of methanol are thus much needed for interpreting astrophysical and planetary spectra, for air quality monitoring and atmospheric concentration retrieval. Although the simplest of the organic alcohols, it is a slightly asymmetric-top with a hindered internal rotor (torsion) leading to a triple-well internal tunnel dynamics and therefore to a rich rotation-torsion-vibration energy structure. Methanol is thus also an important molecule for fundamental spectroscopy [START_REF] Lankhaar | Hyperfine interactions and internal rotation in methanol[END_REF] , metrological applications and frequency calibration [START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF][START_REF] Sun | Sub-Doppler spectroscopy of the C-O stretching fundamental band of methanol by use of microwave sidebands of CO 2 laser lines[END_REF] , the realization of optically pumped far-IR gas lasers [START_REF] Jiu | Pulsed CH 3 OH terahertz laser emission pumped by a TEA CO 2 laser[END_REF] , or for probing the limits of the standard model, its internal tunnel dynamics making it one of the most sensitive molecules for a search of a varying proton-to-electron mass ratio [START_REF] Muller | A study of submillimeter methanol absorption toward PKS1830 -211: Excitation, invariance of the proton-electron mass ratio, and systematics[END_REF] . Here, we measure the resonance frequencies of fourteen rovibrational transitions, including very weak lines, some observed for the first time. We demonstrate record frequency uncertainties two to four orders of magnitude improved over previous measurements, at the few kilohertz level for the most intense lines and bring to light inaccuracies and gaps in the HITRAN database. We resolve subtle weak intensity doublets never observed before induced by a combination of the asymmetry and the tunneling dynamics in methanol, demonstrating both the high detection-sensitivity and high resolution of our widely tunable spectrometer and its potential for unraveling subtle internal molecular processes.

II. EXPERIMENTAL SETUP

FIG. 1 illustrates our widely tunable SI-traceable frequency-comb-stabilized-QCL-based high-resolution mid-IR spectrometer. It combines a widely tunable optical local oscillator (OLO) locked to a remote ultrastable frequency reference of LNE-SYRTE, a 10.3 µm cw DFB QCL stabilized to the OLO using an optical frequency comb (OFC) and sum-frequency generation, and a multi-pass absorption cell for carrying out saturation spectroscopy. The reader is referred to Ref. [START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF] for a detailed description of most of this setup.

A. Widely tunable, SI-traceable and ultrastable optical local oscillator

The widely tunable OLO assembly is shown in FIG. 1(b). The frequency reference ν ref located at LNE-SYRTE is produced by a ∼1.54 µm fiber laser locked to an ultrastable cavity which exhibits a relative frequency stability lower than 10 -15 between 0.1 s and 10 s integration time [START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF][START_REF] Argence | Prototype of an ultra-stable optical cavity for space applications[END_REF] . This reference is calibrated against a combination of a liquid-helium cooled cryogenic sapphire oscillator (CSO) and a hydrogen maser (H-maser), themselves monitored on the atomic fountains of LNE-SYRTE, and its absolute frequency is thus SI-traceable to the primary standards with an uncertainty at the few 10 -14 level. This reference signal is transferred to LPL [START_REF] Cantin | An accurate and robust metrological network for coherent optical frequency dissemination[END_REF][START_REF] Xu | Reciprocity of propagation in optical fiber links demonstrated to 10 -21[END_REF] through a 43-km long fiber link of the REFIMEVE infrastructure [START_REF] Cantin | An accurate and robust metrological network for coherent optical frequency dissemination[END_REF] without any degradation of its stability and absolute uncertainty thanks to an active compensation of the propagationinduced phase noise [START_REF] Newbury | Coherent transfer of an optical carrier over 251 km[END_REF] . A 200 MHz ultrastable RF reference f ref derived from the CSO/H-maser combination (∼ 10 -15 stability at 1 s, a few 10 -14 absolute frequency uncertainty [START_REF] Guena | Progress in atomic fountains at LNE-SYRTE[END_REF] ) is also transferred to LPL through the same 43-km long fiber link using an amplitude modulated auxiliary 1.5 µm laser diode.

At LPL, a laser diode of frequency ν OLO ∼1.54 µm is used as an OLO. Two sidebands are generated in the OLO signal by an electro-optic modulator (EOM in FIG. 1(b)). The EOM frequency f EOM is precisely controlled using a homemade phase-jump-free microwave synthesizer based on a cur- rent driven Yttrium Iron Garnet (YIG) oscillator with a continuous tuning range of 9 GHz (7.5-16.5 GHz) [START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF] . FIG. 1(a) shows a simplified scheme of the synthesizer. A 1 GHz signal is generated by a RF tracking oscillator (TO) referenced to a 10 MHz signal (stability at 1 s and relative frequency uncertainty better than 10 -12 and 10 -13 respectively 59 ) synthesized from the remote LNE-SYRTE 200 MHz reference f ref . A 0-400 MHz direct digital synthesizer (DDS) uses the 1 GHz signal as a reference. After division by 32, the YIG frequency is mixed with the doubled DDS frequency. The resulting phase error signal is converted into correction signals (using phaselock loop PLL 1 in FIG. 1(a)) applied to the YIG oscillator's current, the frequency of which is thus locked to the DDS with a frequency ratio of 64. This allows not only the spectral performance of the local 10 MHz reference to be transferred to the resulting microwave signal, but also the synthesizer frequency to be tuned over 9 GHz without any phase jumps by adjusting the DDS frequency. As illustrated in FIG. 1 and detailed below, the QCL is stabilized to the OLO via a series of three cascaded phase-lock loops. The wide tunability and, crucially, the phase-jump-free nature of the YIG-based syn-thesizer are key to allow broad but also continuous tunability of the stabilized QCL, and therefore demonstrate both broadband and ultra-high resolution spectroscopy.

The upper OLO sideband of frequency ν OLO + f EOM is phase-locked to the reference frequency ν ref using PLL 2 (500 kHz locking bandwidth) in FIG. 1(b) with an offset frequency ∆ 1 . Adjusting f EOM then allows the OLO carrier frequency ν OLO to be tuned over 9 GHz at the precision of the frequency reference ν ref [START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF] . In addition to copying the 10 -15 stability (for 0.1 s to 10 s averaging times) of the LNE-SYRTE optical reference, the OLO carrier frequency can be used as a tunable SI-traceable local oscillator of frequency:

ν OLO = ν ref -f EOM -∆ 1 .
(1)

B. Widely tunable, SI-traceable and ultrastable frequency comb-assisted quantum cascade laser

A f rep ∼250 MHz repetition rate Erbium-doped fiber optical frequency comb (Menlo Systems, F1500) is used to trans-fer the spectral performances of the remote frequency reference and the wide-tunability of the OLO to the QCL.

A beat-note signal at

∆ 2 = ν OLO -p × f rep (2)
is obtained after beating the OLO carrier frequency with the p th nearest comb tooth at frequency p × f rep + f 0 (with p ∼ 780 000) and subsequently removing the comb offset frequency ( f 0 ). This signal is used to lock f rep . To this end, it is processed via PLL 3 to generate three correction signals respectively applied to a stepper motor (very slow correction), a piezo-electric actuator (PZT, slow-correction) and an intracavity EOM (fast-correction), all used to act on the comb cavity optical length (see panel (c) in FIG. 1, the stepper motor and the PZT are both mounted on a cavity mirror). Phaselocking f rep also involves transferring the tunability of the OLO carrier to every comb modes. Here, major improvements have been made to the previous setup [START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF] . The dynamic range of the EOM control voltage has been doubled, resulting in a larger locking bandwidth of ∼700 kHz. The most significant upgrade lies in the use of the stepper motor in order to enhance the tuning range of the cavity length and thus of f rep (limited in the previous setup by the course of the PZT actuator, corresponding to a 3 GHz span of a comb mode). To prevent the PZT from reaching the end of its course in our new setup, its voltage is maintained within a reduced range around its midpoint by acting on the stepper motor. When the PZT voltage falls outside the allowed range, a correction signal is sent to the stepper motor which performs a translation that brings the PZT back to its mid-point. This allows to lock f rep and tune it over a span corresponding to a 9 GHz tunability of the comb modes. This span is 3 times broader than allowed by acting on the PZT only, and now limited by the EOM tuning range, given by the YIG span. Sweeping f rep over such extended spans however also involves variations of f 0 of several tens of megahertz, thus pushing some beat notes (not only f 0 itself but also the beat signal between the OLO and the p th comb tooth) outside filters' bandwidths. The comb offset frequency is thus loosely locked (sub-10 Hz bandwidth) by acting on the comb pump lasers' currents.

We then lock the QCL (frequency ν QCL ) to a high harmonic of f rep (FIG. 1(d)). To this end, a fraction of the 1.54 µm frequency comb power is amplified in an Erbium-doped fiber amplifier (EDFA) and fed to a wavelength shifted fiber (WSF) to generate a new comb output centered at 1.85 µm with comb teeth frequencies q × f rep + f 0 (q ∼ 650 000). The QCL and 1.85 µm comb beams are overlapped in a non-linear crystal (AgGaSe 2 ) to perform sum frequency generation (SFG) which yields a shifted comb centered at 1.55 µm (SFG comb). The beat signal between the SFG and original combs at frequency:

∆ 3 = n × f rep -ν QCL (3)
with n = pq ∼ 120 000 is used to lock the QCL frequency.

It is obtained after overlapping the pulses in the time domain using a fiber delay line (DL in FIG. 1). It is processed by PLL 4 to generate a correction signal applied to the QCL's current.

The QCL frequency is thus directly traceable to the remote frequency reference in the following way:

ν QCL = n p (ν ref -f EOM -∆ 1 -∆ 2 ) -∆ 3 (4)
with n/p ∼ 0.15. Integers n and p are unequivocally determined by measuring the comb repetition rate using a RF counter, the QCL and OLO frequencies using an optical spectrum analyzer (Bristol Instruments, model 771A-MIR, 0.2 ppm accuracy), and by exploiting Equations ( 2) and (3) [START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF] .

As demonstrated in Ref. [START_REF] Argence | Quantum cascade laser frequency stabilization at the sub-Hz level[END_REF][START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF] , the QCL frequency (i) exhibits a stability at the level of the remote reference signal, below 0.03 Hz (10 -15 in relative value) for averaging times from 0.1 to 10 s, and a linewidth of ∼0.1 Hz at 100 ms timescales; (ii) is SI-traceable and known within a total uncertainty better than 4 × 10 -14 after 1 s averaging time.

Scanning the OLO frequency thus allows us to continuously tune f rep and in turn the QCL frequency at the precision of the remote reference. In a previous work [START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF] , we have demonstrated a QCL continuous tunability over a span of ∼400 MHz, limited by the course of the PZT actuator. With the additional capability provided by the stepper motor control, scanning the OLO carrier frequency over its entire 9 GHz span (limited by the YIG tunability) enables the QCL, which is both ultrastable and SI-traceable, to be continuously tuned over a record range of 1.4 GHz (given by the ∼6.7 mid-IR to near-IR wavelength ratio). Note that if the time delay between the SFG and original combs is not cancelled, tuning f rep comes with a deterioration of the pulses time overlap and in turn of beat-note signal ∆ 3 's signal-to-noise ratio, preventing us from carrying out long scans. Time delay cancellation is achieved by adding fiber length in order to balance the optical paths.

C. Saturated absorption spectroscopy setup

As illustrated in FIG. 1(e), the stabilized QCL has been used to carry out saturated absorption spectroscopic measurements in a multi-pass absorption cell (Aerodyne Research, model AMAC-36, 182 paths). With a 20-cm long distance between two astigmatic mirrors, the cell provides an effective absorption length of 36.4 m and allows us to perform spectroscopic measurements of weak lines at low pressures.

The QCL beam is split in two using an 80/20 beam splitter (BS 1 ). Around 8 mW is needed to phase-lock the laser to the comb while about 2 mW remains for the spectroscopy. Using BS 2 , the beam is further split into a pump and a probe beam, which are coupled into the multi-pass cell (incident powers of ∼1.3 mW and ∼0.7 mW, respectively) for conducting saturation spectroscopy. The mirrors' reflectivity leads to a ∼20% transmission after 182 passes which in turn results in pump and probe powers that vary inversely by almost an order of magnitude through the cell. This also goes with a ∼50% total power variation depending on the pass number and to an intracell averaged power of 0.81 mW. After exiting the cell, the probe beam is detected by a liquid-nitrogen-cooled mercurycadmium-telluride photodetector (MCT in FIG. 1(e)). Undesirable interference fringes as typically observed with multi-pass cells [START_REF] Mcmanus | Astigmatic mirror multipass absorption cells for long-path-length spectroscopy[END_REF] are averaged out by vibrating mirror M 2 held in a piezo-actuated mount and shaking the multi-pass cell assembly with a fan.

Frequency modulation (FM) of the QCL is used to improve the signal-to-noise ratio. This is done by modulating the frequency of the DDS used as a reference for PLL 4 (to lock the QCL to the comb) at a frequency of 20 kHz (well within PLL 4 's 250 kHz bandwidth). The signal detected by the MCT photodetector is fed to a lock-in amplifier for demodulation.

III. PRECISE SPECTROSCOPY OF METHANOL

Saturated absorption spectroscopy is carried out by tuning the QCL frequency in a series of discrete steps. To cancel out the frequency shifts induced by the limited detection bandwidth [START_REF] Rohart | Absorption-line-shape recovery beyond the detection-bandwidth limit: Application to the precision spectroscopic measurement of the Boltzmann constant[END_REF][START_REF] Rohart | Overcoming the detection bandwidth limit in precision spectroscopy: The analytical apparatus function for a stepped frequency scan[END_REF] we have carried out frequency scans in both directions with increasing and decreasing frequencies and only consider the averaged spectrum of pairs of up and down scans (see below). FIG. 2(a) shows a saturated absorption spectrum of methanol averaged over five such pairs spanning the full 1.4 GHz tuning range of our system (from ∼971.312 to ∼971.357 cm -1 ) after demodulation on the first harmonic. It has been recorded at a pressure of 1.5 Pa with a frequency step ∼15 kHz and a step duration and lock-in amplifier time constant both of 10 ms. The SI-traceability detailed in Section II B allows us to retrieve the absolute frequency scale using Equation (4). Although ν ref is measured and made available every second at LNE-SYRTE, its value varies by typically a few hertz over the duration of our scans. Therefore, for each pairs of scans, we fix ν ref in Equation (4) to its value measured halfway through the scans.

The spectrum exhibits fourteen rovibrational transitions of methanol which belong to the P branch of the ν 8 C-O stretch vibrational mode [START_REF] Xu | New assignments, line intensities, and HITRAN database for CH 3 OH at 10 µm[END_REF] . The red solid line is a fit to the data used as a guide-to-the-eye. Each line is fitted with the first derivative of the sum of a Lorentzian and a Gaussian profile to model the saturated absorption and Doppler (oscillations in the baseline) contribution, respectively. The transitions reported in the HITRAN database [START_REF] Gordon | The HITRAN2020 molecular spectroscopic database[END_REF] falling in the spectral window covered are shown as blue sticks. As an example, FIG. 2(b) shows a zoom on the saturation feature of the P(A, co,t = 0, K = 2 -, J = 33) methanol line around 971.319 cm -1 (in this work, we adopt the notations of Ref. [START_REF] Xu | New assignments, line intensities, and HITRAN database for CH 3 OH at 10 µm[END_REF] for the spectroscopic assignment of methanol transitions, see also Appendix A). It exhibits a signal-to-noise ratio of ∼370 and a ∼760 kHz full-width-at-half maximum, which is a combination of pressure, power and transit time broadening, as well as FM-induced distortion.

To achieve a reasonable signal-to-noise ratio, the FM amplitude has been set at 250 kHz, resulting in a line shape slightly distorted compared to the typical Lorentzian profile. Furthermore, the residual amplitude modulation associated with FM and the power variation over a scan can both contribute to an asymmetry of the line shape. We fit our data to a model described in Appendix B that takes into account these distortions.

To determine line-center frequencies of the fourteen re-solved transitions, we first select a spectral range of ∼6 MHz around each. We perform the following "pair by pair" analysis already established in [START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF] . We average each pair of up and down scans resulting in five averaged spectra for each of the fourteen transitions. To all data points of an averaged spectrum, we assign the same experimental error bar, calculated as the standard deviation of the residuals obtained after fitting a second-order polynomial to a small portion of the averaged spectrum far from resonance. To each of the five averaged spectra of a given transition is fitted the sum of the line profile described in Appendix B and of a second order polynomial to account for the baseline (see Appendix C). The absolute frequency and associated statistical uncertainty of each line are estimated by calculating the weighted mean and weighted standard error of the five fitted center frequencies, with the weights determined from the fits' error bars. Unlike in our previous work [START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF] , we have conducted measurements at a single pressure (1.5 Pa) and power (0.81 mW intra-cell averaged power) preventing us from deducing zero-power and collision-free transition frequencies. However, we estimate the resulting overall pressure-and power-shift to be of the order of 10 kHz. The reader is referred to Refs. [START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF][START_REF] Tran | Widely tunable and SI-traceable frequency-comb-stabilised mid-infrared quantum cascade laser: application to high precision spectroscopic measurements of polyatomic molecules[END_REF] and to the summary given in Appendix D for a detailed description of the line positions uncertainty budget which result in a systematic uncertainty of 5.4 kHz on the frequency of all the resonances studied here. TABLE I lists the line center frequencies of the fourteen transitions recorded at a pressure of 1.5 Pa and an intra-cell averaged power of 0.81 mW. 1-σ total uncertainties are quoted into parentheses. The absolute frequencies of three unambiguously assigned high intensity transitions have been determined with a sub-10-kHz total uncertainty, an improvement of ∼2000 over previous measurements based on the Fourier-transform IR (FTIR) spectroscopic technique [START_REF] Xu | New assignments, line intensities, and HITRAN database for CH 3 OH at 10 µm[END_REF][START_REF] Gordon | The HITRAN2020 molecular spectroscopic database[END_REF] .

All other absolute frequencies are determined with a total uncertainty ranging from ∼10 kHz to ∼350 kHz, to be compared to the typical 15 MHz FTIR uncertainty [START_REF] Xu | New assignments, line intensities, and HITRAN database for CH 3 OH at 10 µm[END_REF] . The data shown in FIG. 2 demonstrates both the high detection-sensitivity and high resolution of our spectrometer. An illustration of this double asset is the resolution of two weak doublets (see insets of FIG. 2(a)), which exhibit a splitting of 2.66 ± 0.07 and 1.70 ± 0.16 MHz, respectively. The use of a multipass cell allows relatively high-J (J > 30) rovibrational lines to be probed at the low pressures required for ultra-high resolution measurements, including weak lines belonging to hot-bands (torsional excitation t 1). The intensities of the strongest transitions shown here are two to three orders of magnitude weaker than the most intense lines of this vibrational mode.

As shown in FIG. 2(a), our spectrum is a very rich source of information. First of all, we resolve fourteen lines where the HITRAN database lists only nine (of which two pairs reported as degenerate). Several transitions have to our knowledge never been reported elsewhere. Based on HITRAN, we are in fact able to unequivocally assign only four of the lines (see TABLE I). Five weak transitions that could not be assigned are indicated with blue arrows and most probably belong to hot-bands with torsional excitation t 2. As discussed in the following and summarized in TABLE I, the five remaining measured resonances are tentatively assigned to the five other HITRAN transitions. As shown in FIG. 2(c), around 971.328 cm -1 we resolve one weak and two intense lines labelled (1), ( 2) and (3). We tentatively assign these to transitions P(E, co, 1, 1, 33), P(E, co, 0, 5, 33) and P(A, co, 0, 6 + , 35), without being able to decide which one is which. Those three transitions are indeed listed in Ref. [START_REF] Lees | Rotationtorsion-vibration term-value mapping for CH 3 OH: Torsion-mediated doorways and corridors for intermode population transfer[END_REF] at the same degenerate frequency of 971.3288 cm -1 . In HITRAN however and as shown in FIG. 2, only P(E, co, 1, 1, 33) and P(E, co, 0, 5, 33) are reported as degenerate at 971.32894 cm -1 , while P(A, co, 0, 6 + , 35) is ∼150 MHz blue-shifted at 971.33377 cm -1 . It is however unlikely for this latter to correspond to one component of the doublet observed at ∼971.334 cm -1 (based on intensities, this doublet looks very much like a single structure, not like two fortuitously quasi-coincident lines). P(A, co, 1, K = 10 ± , 35) (red arrow in FIG. 2(a)) corresponds to a so-called closed-lying K-doublet that splits for A-symmetry lines due to the combination of the slight asymmetry and internal tunnel dynamics of the methanol molecule [START_REF] Lees | Rotationtorsion-vibration term-value mapping for CH 3 OH: Torsion-mediated doorways and corridors for intermode population transfer[END_REF] . It is reported as degenerate in the HITRAN database, but no matching resonance is observed around. Even at the high resolution demonstrated here, we do not expect to resolve such K-doublets for K 7 (here K = 10), but enhanced splittings are possible as a result of mixing of the considered transition upper states that belong to the C-O stretching vibrational mode with a closely lying state from another vibrational mode [START_REF] Lees | Rotationtorsion-vibration term-value mapping for CH 3 OH: Torsion-mediated doorways and corridors for intermode population transfer[END_REF] . P(A, co, 1, K = 10 ± , 35) could then well be assigned to either of the two observed resolved Our data is a source of information much richer than what is currently available in the literature. It has yet to be fully exploited to build a more accurate spectroscopic model of methanol. As seen, it brings to light some inaccuracies and gaps in the HITRAN database. In addition, our measured and unequivocally assigned resonance frequencies are red-shifted by 5 to 20 MHz with respect to previous FTIR measurements at the origin of the current HITRAN edition line list [START_REF] Gordon | The HITRAN2020 molecular spectroscopic database[END_REF] (see FIG. 2 andTABLE I). These shifts are about three orders of magnitude larger than our uncertainties and more than an order of magnitude larger than the HITRAN frequency accuracies reported to be between 30 kHz and 300 kHz. It is consistent with deviations between FTIR and saturation spectroscopy data previously observed in the C-O stretch of methanol in methanol and is attributed to FTIR spectrometers calibration imperfections [START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF][START_REF] Sun | Sub-Doppler spectroscopy of the C-O stretching fundamental band of methanol by use of microwave sidebands of CO 2 laser lines[END_REF][START_REF] Sun | Sub-Doppler measurements of ν CO = 1 ← 0, K=0, A/E lines of methanol with microwave sidebands of CO 2 laser lines[END_REF][START_REF] Sun | Sub-Doppler spectroscopy by use of microwave sidebands of CO 2 laser lines applied to the CO stretching fundamental band of methanol[END_REF] . Our data also offers new information on weak hot-band transitions for which molecular databases remain largely incomplete. Exploiting this in refined models of methanol is essential for atmospheric quantification. Moreover, mixing between near-degenerate levels of two different modes is known to lead to collision-induced population transfer from one vibrational mode to another. Doublets exhibiting enhanced splittings such as those resolved in our work, a signature of this type of mixing, may thus help understand how molecules transfer among different modes and give insight on thermal equilibration in gases [START_REF] Lees | Rotationtorsion-vibration term-value mapping for CH 3 OH: Torsion-mediated doorways and corridors for intermode population transfer[END_REF] .

Other saturation spectroscopy measurements of the C-O stretch of methanol exist. Almost 700 frequencies have been measured with an accuracy of ∼100 kHz using a CO 2laser spectrometer [START_REF] Sun | Dual-mode CO 2 -laser/microwave-sideband spectrometer with broadband and saturation dip detection for CH 3 OH[END_REF][START_REF] Sun | Sub-Doppler spectroscopy of the C-O stretching fundamental band of methanol by use of microwave sidebands of CO 2 laser lines[END_REF][START_REF] Sun | Sub-Doppler measurements of ν CO = 1 ← 0, K=0, A/E lines of methanol with microwave sidebands of CO 2 laser lines[END_REF][START_REF] Sun | Sub-Doppler spectroscopy by use of microwave sidebands of CO 2 laser lines applied to the CO stretching fundamental band of methanol[END_REF][START_REF] Sun | Saturation-dip measurements for the ν 8 CO stretching band of CH 3 OH with a CO 2 -laser-microwave-sideband spectrometer[END_REF] , however mostly in the Q and R branches, with a few low J lines in the P branch. To our knowledge, there are only two other methanol frequencies that have been measured with an uncertainty comparable to the present work: our previous measurement [START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF] , and a weak unassigned line around 947.7 cm -169 .

IV. CONCLUSIONS

We have developed a widely tunable SI-traceable frequency-comb-stabilized high-resolution spectrometer potentially covering the 8-12 µm spectral window. A modelocked frequency comb and a metrological fiber link are used to transfer the spectral purity and SI-traceability of a 1.54 µm frequency reference located at LNE-SYRTE to a 10.3 µm QCL located at LPL. The QCL exhibits a sub-10 -15 frequency stability from 0.1 to 10 s, a linewidth of ∼0.1 Hz at 100 ms and its frequency is SI-traceable with a total uncertainty better than 4 × 10 -14 after 1 s of averaging time. In addition, we have developed the instrumentation allowing comb modes to be continuously tuned over 9 GHz, resulting in a continuous tunability of 1.4 GHz for the QCL, a three-fold improvement compared to previous measurements at such levels of spectral purity. This tuning range can potentially be increased by transferring the full 9 GHz tunability of comb modes directly to the QCL, or by using a commercially available 40 GHz telecom EOM together with a home-made microwave synthesizer of wider tunability. We have carried out saturation spectroscopy of methanol in a multi-pass absorption cell. We report linecenter frequencies of fourteen transitions of methanol in the P branch of the ν 8 C-O stretch vibrational mode, including very weak transitions belonging to hot-bands, and some observed for the first time. We demonstrate record global uncertainties ranging from few kilohertz to few hundred kilohertz -two to four orders of magnitude improved over previous measurements -depending on the line intensity. We expose manifest discrepancies with the HITRAN database and resolve subtle weak intensity doublets never observed before induced by a combination of the asymmetry and the tunneling dynamics in methanol. All these result demonstrate the potential of our apparatus for providing information on hot-bands essential to atmospheric quantification, for probing subtle internal molecular processes, for building more accurate spectroscopic models of polyatomic molecules of atmospheric or astrophysical interest. This work is also an important forward for our on-going efforts towards using polyatomic molecules to perform spectroscopic tests of fundamental physics in the mid-IR, such as searching for a varying proton-to-electron mass ratio [START_REF] Barontini | Measuring the stability of fundamental constants with a network of clocks[END_REF][START_REF] Muller | A study of submillimeter methanol absorption toward PKS1830 -211: Excitation, invariance of the proton-electron mass ratio, and systematics[END_REF][START_REF] Segal | Studying fundamental physics using quantum enabled technologies with trapped molecular ions[END_REF] or measuring the tiny parity-violating energy difference between enantiomers of a chiral molecule [START_REF] Fiechter | Toward detection of the molecular parity violation in chiral Ru(acac) 3 and Os(acac) 3[END_REF][START_REF] Segal | Studying fundamental physics using quantum enabled technologies with trapped molecular ions[END_REF] .

the signal into a Fourier series gives:

s(x) = A ∞ ∑ n=0 s np (x) cos n2π f m t - ∞ ∑ n=0 s nq (x) sin n2π f m t (B2)
with x = (νν 0 )/γ the normalized frequency. The amplitude of the in-phase and quadrature signal (with respect to the intensity modulation) after demodulation on the n th harmonic are respectively given by 70 Our experimental procedure consists in choosing the phase that maximises the signal. At n th harmonic detection, a signal s nΦ n,max (x) of maximum amplitude is reached for the detection phase [START_REF] Schilt | Wavelength modulation spectroscopy: combined frequency and intensity laser modulation[END_REF] Φ n,max = nΨ + kπ, (B7)

where k is an integer. By introducing Equation B7 into Equation B6, we obtain the signal of maximum amplitude that can be detected

s nΦ n,max (x) = (γB 1 x + 1)s n (x) -γB 1 m 2 ε n 2 -ε n + 1 s n+1 (x)
-γB 2 m 2 cos Ψ 2

ε n-1 s n-1 (x) + (ε n -1)s n+1 (x) .

(B8)

The experimental line shape obtained after demodulation on the first harmonic, as used in this work is finally found to be: We use the three unambiguously assigned highest intensity lines shown in FIG. 2(a) to determine the systematic uncertainty resulting from the inaccuracy of our model for the line shape (point (ii) above). We fit averaged pairs of up and down scans with the sum of the spectral line shape described in Appendix C and a polynomial of order k with k ranging from 2 to 5. For all averaged spectra, we find the standard deviation of the four frequencies resulting from the corresponding fits to be less that 2 kHz, which we take as our conservative systematic uncertainty.

s 1Φ 1,max (

FIG. 1 .

 1 FIG. 1. Widely-tunable SI-traceable frequency-comb-stabilized-QCL-based high-resolution mid-IR spectrometer. (a) Yttrium Iron Garnet (YIG) oscillator-based microwave (MW) synthesizer disciplined to a 10 MHz RF reference. (b-d): QCL frequency stabilization to a remote 1.54 µm frequency reference using an optical frequency comb (OFC). The frequency reference operated at LNE-SYRTE is transferred to LPL via a 43-km long fiber link. In (b), a laser diode of frequency 1.54 µm is used as an optical local oscillator (OLO). One sideband generated by an electro-optic modulator (EOM) is locked to the remote reference using phase-lock loop PLL 2 , ensuring wide tunability of the OLO carrier. In (c), the comb repetition rate f rep is locked to the OLO using PLL 3 . In (d), a mid-IR quantum cascade laser (QCL) at 10.3 µm is stabilized to f rep using PLL 4 . (e): Simplified setup for saturation spectroscopy in a multi-pass absorption cell using frequency modulation. RF TO: radiofrequency tracking oscillator, DDS: direct digital synthesizer, MX i : mixers, PD i : photodiodes, WDM: wavelength division multiplexer, f -2 f : comb offset frequency detection, PBF: band pass filter, PZT: piezo-electric actuator, WSF: wavelength-shifting fiber, DL: delay line, SFG: sum frequency generation crystal, BS i : beam splitters, M i : mirrors, MCT: mercury-cadmium-telluride photodetector, DAQ: data acquisition card.

FIG. 2 .

 2 FIG. 2. (a) Saturated absorption spectrum of methanol spanning ∼1.4 GHz recorded using frequency modulation and first-harmonic detection (black curve). Blue sticks indicate the line frequencies and intensities reported in the HITRAN database (intensities of degenerate lines are summed). Experimental conditions: pressure, 1.5 Pa; modulation frequency, 20 kHz; modulation amplitude, 250 kHz; frequency step, ∼15 kHz; average of five pairs of up and down scans; total integration time per point, 100 ms; whole spectrum measurement time, 9000 s. (b) Zoom on the saturated absorption spectrum of the P(A, co, 0, 2 -, 33) rovibrational transition. (c) Zoom of the saturated absorption features observed around 971.328 cm -1 . Lines (1-3) are tentatively assigned to the transitions listed in the grey rectangle.

2 ε n 2 -γB 2 m 2 2 ε 2 ε n 2 -γB 2 m 2 2 ε 2 i n ε n ( 1 -

 22222221 s np (x) = (γB 1 x + 1) cos nΨs n (x) -γB 1 m ε n + 1 s n+1 (x)n-1 cos [(n -1)Ψ] s n-1 (x) + (ε n -1) cos [(n + 1)Ψ] s n+1 (x) (B3) s nq (x) = (γB 1 x + 1) sin nΨs n (x) -γB 1 m ε n + 1 s n+1 (x) n-1 sin [(n -1)Ψ] s n-1 (x) + (ε n -1) sin [(n + 1)Ψ] s n+1 (x) (B4) in which m = ∆ν m /γ, ε 0 = 1, ε n = 2 for n ≥ 1.s n (x) corresponds to the signal obtained after demodulation on the n th harmonic when only considering pure FM (Arndt's model, no intensity modulation) and is given by 71s n (x) = 1 ix) 2 + m 2 -(1ix) n m n (1ix) 2 + m 2 + c.c. (B5)where c.c. is the complex conjugate. The lock-in amplifier allows us not only to detect the in-phase and quadrature signals but also the signal s nΦ (x) at any detection phase Φ, with respect to the phase of the intensity modulation s nΦ (x) = [s np (x) cos Φ + s nq (x) sin Φ] . (B6)

  x) = [γB 1 x + 1]s 0 (x) -γB 2 m 2 cos Ψ [s 0 (x) + s 2 (x)] . (B9) in which s 0 (x), s 1 (x), and s 2 (x) are given by Equation B5.

FIG. 3 .

 3 FIG. 3. Saturated absorption spectrum of the P(A, co, 0, 2 -, 33) rovibrational line of methanol, with fit and residuals. The data (grey dots) are recorded using frequency modulation and first-harmonic detection. The red solid line is a fit to the data. Residuals are shown on the bottom graph. Experimental conditions: pressure, 1.5 Pa; modulation frequency, 20 kHz; frequency modulation amplitude, 250 kHz; frequency step, ∼15 kHz; average of one pair of up and down scans (corresponding to one of the five pairs contributing to FIG. 2(b)); total integration time per point, 20 ms; whole spectrum measurement time, 8.02 s.

TABLE I .

 I Line-center frequencies of rovibrational transitions of methanol shown in FIG.2, (tentative) assignments and corresponding values reported in the HITRAN database[START_REF] Gordon | The HITRAN2020 molecular spectroscopic database[END_REF] and in Ref.[START_REF] Lees | Rotationtorsion-vibration term-value mapping for CH 3 OH: Torsion-mediated doorways and corridors for intermode population transfer[END_REF] when a match is possible. Tentative assignments of the three measured and resolved lines without being able to decide which one is which.

	This work	Assignment	HITRAN		Ref. 65	ν TW -ν HITRAN	ν TW -ν Ref. 65
	(ν TW , kHz)		(cm -1 ) Wavenumber	(cm -1 /mol.cm -2 ) Intensity Wavenumber (cm -1 )	(kHz)	(kHz)
	29 119 218 300.4 (114.6)	-	-	-	-
	29 119 429 316.2 (8.5)	P(A, co, 0, 2 -, 33)	971.31977	5.950×10 -22		971.31966	-4 819.0	-1 521.3
	29 119 492 935.1 (121.7)	-	-	-	-
	29 119 689 067.7 (8.8) 29 119 690 623.2 (343.1)	P(E, co, 1, 1, 33) (a) P(E, co, 0, 5, 33) (a)	971.32894	4.712×10 -22 3.083×10 -22	  	971.32880
	29 119 713 238.5 (10.0)	P(A, co, 0, 6 + , 35) (a) 971.33377	2.076×10 -22	 
	29 119 850 408.4 (117.4) (b) 29 119 853 004.6 (82.2) (b)	P(A, co, 1, 10 ± , 35) (c) 971.34300 971.34300	3.215×10 -23 3.215×10 -23
	29 120 094 740.7 (6.1)	P(A, co, 0, 1 -, 33)	971.34199	6.251×10 -22		971.34176	-5 533.4	+1 361.9
	29 120 216 333.6 (7.3)	P(E, co, 0, 1, 33)	971.34604	6.263×10 -22		971.34592	-5 356.4	-1 758.9
	29 120 231 092.8 (259.2) (b) -	-	-	-
	29 120 232 719.0 (274.9) (b) -	-	-	-
	29 120 472 397.6 (84.7)	-	-	-	-
	29 120 485 592.6 (25.0)	P(E, co, 1, -7, 33)	971.35548	1.181×10 -22		971.35536	-19 101.5	+15 504.0
	(a) (b) Measured resolved doublets.
	(c) Tentative assignments.
	doublets. Based on intensities, we tentatively assign it to the
	strongest doublet around 971.334 cm -1 . All certain and ten-
	tative assignments are finally summarized in TABLE I.

ACKNOWLEDGMENTS This project has received funding from the EMPIR programme co-financed by the Participating States and from the European Union's Horizon 2020 research and innovation programme, through EMPIR project 15SIB05 "OFTEN". This work was supported by the Region Île-de-France in the framework of DIM Nano-K and DIM SIRTEQ, the Agence Nationale de la Recherche project PVCM (Grant No. ANR-15-CE30-0005-01), the LabEx Cluster of Excellence FIRST-TF (ANR-10-LABX-48-01), the EquipEx Cluster of Excellence REFIMEVE+ (ANR-11-EQPX-0039), CNRS and Université Sorbonne Paris Nord. D.B.A. Tran was supported by the Ministry of Education and Training, Vietnam (Program 911).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Appendix A: Methanol spectroscopic notations

In this work, we adopt the notations of Ref. [START_REF] Xu | New assignments, line intensities, and HITRAN database for CH 3 OH at 10 µm[END_REF] for the spectroscopic assignment of methanol transitions. The P(σ , co,t, K, J) resonances studied here are all ∆J = -1, ∆K = 0 and ∆t = 0 transitions between the ground vibrational state and the first excited state of the C-O stretch vibrational mode ν 8 , with σ , t, K, and J, the torsional symmetry (A or E), the torsional mode (ν 12 ) quantum number, and rotational quantum numbers of the lower sate (J is the total orbital angular momentum quantum number, K is the quantum number for the projection of the total orbital angular momentum onto the molecular symmetry axis). K-doublets of A symmetry have ± superscript on K to distinguish the A + or A -component of the doublet. For E symmetry levels, K is a signed quantum number with positive (respectively negative) values corresponding to levels often denoted as E1 (respectively E2).

Appendix B: Model used for line fitting

The QCL frequency is modulated at a frequency f m with a FM amplitude ∆ν m . The instantaneous QCL frequency can be written as ν QCL (t) = ν c + ∆ν m cos(2π f m t) with ν c the optical carrier frequency. To achieve a reasonable signal-tonoise ratio, the FM amplitude has been set at ∆ν m = 250 kHz, comparable to the measured resonances half-width-at-halfmaximum of ∼ 380 kHz resulting in a line shape slightly distorted compared to the typical Lorentzian profile. Furthermore, the residual amplitude modulation associated with FM and the power variation over a scan can both contribute to an asymmetry of the line shape. The power frequencydependency can result from (i) the QCL gain curve, (ii) the underlying Doppler envelop (potentially non-trivial in case of neighboring lines not resolved in the Doppler regime, see FIG. 2) or (iii) multi-pass-cell-induced residual interference fringes not fully averaged out (see text).

We thus fit our data to a model that takes into account both these intensity-modulation-induced asymmetries and the FMinduced distortions. It is based on a profile introduced by Schilt et al [START_REF] Schilt | Wavelength modulation spectroscopy: combined frequency and intensity laser modulation[END_REF] used to fit direct absorption spectra (itself derived from Arndt's model [START_REF] Arndt | Analytical line shapes for lorentzian signals broadened by modulation[END_REF] which considers only pure FM) that we have straightforwardly adapted to study saturation spectra (by not considering the Beer-Lambert law contribution) [START_REF] Tran | Widely tunable and SI-traceable frequency-comb-stabilised mid-infrared quantum cascade laser: application to high precision spectroscopic measurements of polyatomic molecules[END_REF] . The signal detected on the MCT photodetector is a combination of a Gaussian baseline and a saturated absorption signal s(ν). The latter can be expressed in the following form:

which corresponds to a frequency modulated Lorentzian profile multiplied by an amplitude that accounts for the intensity modulation at f m and the power variation over the scan. A is the line intensity factor, ν 0 is the center frequency of the transition, γ is the half-width-at-half-maximum line width of the underlying Lorentzian profile, B 1 and B 2 are the asymmetry factors related to the quasi-static intensity variations over a scan and to the FM-induced intensity modulation, respectively, and Ψ is the phase shift between intensity modulation and FM. Following the derivation in Ref. [START_REF] Schilt | Wavelength modulation spectroscopy: combined frequency and intensity laser modulation[END_REF] , the expansion of

Appendix C: Line fitting details

We fit each averaged pairs of up and down scans of a given transition to the sum of the line profile described in Appendix B and of a second order polynomial to account for the baseline (dominated by the underlying Gaussian contribution on the narrow frequency span used for fitting). All line shape parameters are left free in the fit except the FM amplitude fixed to 250 kHz. The doublets are fitted together using a single second order baseline and, apart from the intensity factors, line shape parameters set to be equal for the two components. For instance, Figure 3 shows the average of one pair of up and down scans of the P(A, co, 0, 2 -, 33) ro-vibrational transition of methanol (grey points), the fit of the line shape model to the data (red solid curve) and the corresponding residuals.

Appendix D: Frequency measurement uncertainty budget

The reader is referred to Refs. [START_REF] Santagata | High-precision methanol spectroscopy with a widely tunable SI-traceable frequency-comb-based mid-infrared QCL[END_REF][START_REF] Tran | Widely tunable and SI-traceable frequency-comb-stabilised mid-infrared quantum cascade laser: application to high precision spectroscopic measurements of polyatomic molecules[END_REF] for a detailed description of the line positions uncertainty budget. We only briefly list here the systematic effects and associated uncertainties: (i) we estimate a conservative upper bound on the frequency shift resulting from gas lens effects, wavefront-curvature-induced residual Doppler shifts, the second-order Doppler shift, Zeeman effects, black-body radiation shifts and the photon recoil doublet to be 5 kHz and take this as the corresponding uncertainty; (ii) we assign a conservative 2 kHz systematic uncertainty to the inaccuracy of our model for the line shape (see below); (iii) our 0.02 Pa pressure measurement accuracy and the 5% specified accuracy of the power meter result in pressureand power-shift-induced systematic uncertainties of ∼0.1 kHz and ∼0.5 kHz respectively (pressure and power fluctuations respectively smaller than 0.02 Pa and 10% during a scan and from one to another result in frequency fluctuations of <0.1 kHz and ∼1 kHz respectively which contribute to our statisti-


MCF Galassi Francesca
email: francesca.galassi@irisa.fr

Deep Learning in Medical Imaging: What's Needed for Training Data?

Data quality, volume and variety matter Ø Error analysis guides model improvements and data collection Ø A data readiness framework can help assessing data suitability for a DL process Ø Dataset shifts arise from factors like population, acquisition, and annotation variations Ø Causal diagram can help describing the shift in the specific project Ø Consider strategies for addressing these challenges, i.e., label noise reduction and domain adaptation

2

Applications in Empenn Deep Learning in Medical Imaging

New MS lesions Baseline Follow-up, 6 months later Follow-up, annotated Masson, A., Le Bon, B., Kerbrat, A., Edan, G., Galassi, F., & Combes, B. (2021). A nnUnet implementation of new lesions segmentation from serial FLAIR images of MS patients. MSSEG-2 challenge proceedings: Multiple sclerosis new lesions segmentation challenge using a data management and processing infrastructure, 5. MICCAI MSSEG-2 Challenge. Hussein, B. R., Meurée, C., Gaubert, M., Masson, A., Kerbrat, A., Combès, B., & Galassi, F. (2023, April). A study on loss functions and decision thresholds for the segmentation of multiple sclerosis lesions on spinal cord MRI. In 2023 IEEE 20th International Symposium on Biomedical Imaging (ISBI) (pp. 1-5). IEEE.

MS lesions Axial view MRI volume

Stroke lesion

Good vs. bad "Data, data everywhere and not a set to process." Ø Optimization problem: min γ,f,g ∑ i L s ( y s i , f(g(x s i ))) + ∑ i,j γ i,j (α || g(x s i )-g(x t j )|| 2 + β L t ( y s i , f(g(x t j ))))

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  Considerations for robustness and quality control Lavdas, I., Glocker, B., Rueckert, D., Taylor, S. A., Aboagye, E. O., & Rockall, A. G. (2019). Machine learning in whole-body MRI: experiences and challenges from an applied study using multicentre data. Clinical radiology, 74(5), 346-356. Training on homogenous data may not generalize well to heterogeneous test data. Ø Consider adopting a versatile or narrowly focused approach. MNI atlas Helminth Infections and their Impact on Global Public Health (pp.180-207). Edition: 1, Chapter: 7. Publisher: Life Sciences Springer Wien. Editors: Fabrizio Bruschi Data Volume Data Volume How much data do you need? -Clinician How much data do you have ? -DL Researcher Ø Traditional statistical rules don't always fit the machine learning paradigm Ø DL has mechanisms to determine whether more data could help Data Volume Data Volume Source of inspiration: Andrew NG machine learning course , N. D. (2017). Data readiness levels. arXiv preprint arXiv:1705.02245. Are data ready for a DL process? Ø Consider using a framework to evaluate readiness C B Are data ready for a DL process? Ø Consider using a framework to evaluate readiness Dataset shift is any situation in which the training and test data distributions disagree due to exogenous factors, e.g. dissimilar cohorts or inconsistent acquisition processes." Given an image X , train a model to predict some annotation Y P( Y | X ) Assumptions: Availability of sufficient training data (X, Y) -data volume v Consistency in training and testing data distribution -data variety Challenges in DL for medical imaging: Data Scarcity & Data Mismatch Image and Annotation -a causal perspective Example: MS Lesion segmentation X -structural brain MRI scan Y -manual segmentation masks What is the relationship between image X and annotation Y ? . C., Walker, I., & Glocker, B. (2020). Causality matters in medical imaging. Nature Communications, 11(1), 3673. Training and test data distributions may disagree due to ... Castro, D. C., Walker, I., & Glocker, B. (2020). Causality matters in medical imaging. Nature Communications, 11(1), 3673.

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  Unsupervised DA -unlabeled target data accessible at training time the coupling matrix , et al. Expert Variability and Deep Learning Performance in Spinal Cord Lesion Segmentation for Multiple Sclerosis Patients. IEEE CBMS 2023 Ø Considerable inter-and intra-rater variability observed Ø High precision and low sensitivity of individual experts

Data Readiness

Harvey , H., & Glocker, B. (2019). A standardised approach for preparing imaging data for machine learning tasks in radiology. Artificial intelligence in medical imaging: opportunities, applications and risks, 61-72.