Ghania Khraimech
email: gkhraimech@usthb.dz

Fatiha Merazka
email: fmerazka@usthb.dz

A Cooperative NOMA User Pairing in UAV-Based Wireless Networks

Keywords: NOMA, UAV, User pairing, Cooperative communication, Amplify-and-forward

NOMA (Non-Orthogonal Multiple Access) is deemed a promising technology for enhancing spectrum efficiency in future cellular networks. The use of unmanned aerial vehicle (UAV) Amplify-and-Forward (AF) relaying as a moving access point or BS, on the other hand, has emerged as a potential solution to high traffic demands in wireless networks. In this paper, we study joint user pairing and resource allocationbased distance to optimize fair throughput in a downlink scenario, with a focus on UAV-aided communication from different wireless powered nodes. Several transmission methods are adopted, including NOMA as well as UAV cooperative relaying along with two representative nodepairing strategies. The simulation results illustrate that the proposed user pairing strategies for cooperative NOMA and UAV-based cellular networks enhance downlink transmission performance and ensure optimum use of power and bandwidth resources.

Introduction

Due to considerable technological developments across several drone-related domains ranging from embedded systems to autonomy, control, security, and communications, the domain of unmanned aerial vehicles (UAVs), sometimes known as drones, has undergone a significant revolution in recent years. UAV communications are also a useful strategy to provide communication links throughout temporary events and in the aftermath of catastrophes in distant regions missing cellular infrastructure [START_REF] Zhang | Spectrum sharing for drone networks[END_REF]- [START_REF] Zeng | Energy minimization for wireless communication with rotary-wing UAV[END_REF]. One of the encouraging use case scenarios of UAVs is designing a flexible, adaptable, and wireless multiple antennas in the sky [START_REF] Mozaffari | Unmanned aerialvehicle with underlaid device-to-device communications: Performance and tradeoffs[END_REF]. Because of the high elevation of UAVs, line-of-sight (LoS) in UAV communications can provide better small-scale fading between UAVs and ground customers than in traditional ground links, which presents both opportunities as well as challenges in the development of UAV cellular networks [START_REF] Liu | Analysis of frequency-dependent line-of-sight probability in 3-D environment[END_REF]. Due to the obvious low -power consumption on UAV, achieving enhanced spectrum and energy efficiency is essential to the ability to get the most of UAV-based network technologies [START_REF] Mozaffari | Mobile unmanned aerial vehicles (UAVs) for energy-efficient Internet of things communications[END_REF]- [START_REF] Zeng | Energy-efficient UAV communication with trajectory optimization[END_REF].

NOMA seems to be a useful product for achieving both spectrum efficiency and energy efficiency in next-generation wireless technologies and beyond, especially in UAV communication networks [START_REF] Farajzadeh | UAV data collection over NOMA backscatter networks: UAV altitude and trajectory optimization[END_REF]. The potential advantage of NOMA over OMA in a cellular communication system where a base station (BS) is equipped with enormous antenna arrays was successfully investigated in [START_REF] Wei | On the performance gain of NOMA over OMA in uplink communication systems[END_REF]. In compared to conventional OMA techniques, NOMA is capable of more efficient manner utilizing available resources by arbitrarily maximizing on users' specific fading channel on both single-cell and cellular systems [START_REF] Ding | Impact of user pairing on 5G nonorthogonal multipleaccess downlink transmissions[END_REF], [START_REF] Shirvanimoghaddam | Massive non-orthogonal multiple access for cellular IoT: Potentials and limitations[END_REF], and it is ready to accommodate multiple members with specific quality-of-service (QoS) criteria in the same users access required for the Internet of Things (IoT) [START_REF] Liu | Nonorthogonal multiple access for 5G and beyond[END_REF]- [START_REF] Islam | Powerdomain non-orthogonal multiple access (NOMA) in 5G systems: Potentials and challenges[END_REF]. Theoretically, spectral efficiency combining within a frequency, time, and code block delivers the composite signal to multiple customers concurrently in NOMA. NOMA is based on the use of superposition coding (SC) at the transmitter and successive interference cancellation (SIC) techniques at the receiver [START_REF] Wang | Adaptive deployment for UAV aided communication networks[END_REF], [START_REF] Wei | On the performance gain of NOMA over OMA in uplink communication systems[END_REF]. Multiple access users in the power domain can be implemented in this scenario by using various energy degrees for consumers in the same resource block. As a result, by employing NOMA techniques to increase the available spectrum and energy efficiency, UAV networks may serve several users at the same time [START_REF] Ni | Resource Allocation for Multi-Cell IRS-Aided NOMA Networks[END_REF].

The downlink of a UAV-enabled wireless communication is investigated in this article using power-domain NOMA and cooperative relaying, with the ground nodes being wireless powered equipment. Air-to-ground (A2G) communication channels [START_REF] Iswarya | A Survey on Successive Interference Cancellation Schemes in Non-Orthogonal Multiple Access for Future Radio Access[END_REF] are used by these devices, and they are characterized by an altitudedependent route loss exponent and fading. We focus on a user pairing system for NOMA with cooperative relaying cellular networks, in which access devices are separated into two groups and a unit pair is produced from each group. After that, the channel capacity is divided by the number of pairs, with each pair sharing the same sub-channel to deliver their data. The primary issues that we address in this work are related to : how to pair NOMA users within each available orthogonal resource, user-pairing strategy for NOMA as well as cooperative relaying in our UAV model, which has a better performance given a specific user pairing strategy.

The remainder of this paper is organized as follows. In Section 2, we present the system and channel models considered throughout this article. Section 3 details user pairing in cooperative NOMA. Performance analysis of the cooperative NOMA based transmission and pairing strategies evaluated in this work are described in Section 4. The simulation results are provided and discussed in Section 5. Finally, Section 6 presents general conclusions regarding the obtained results in this paper as well as prospects for our future work.

System Model

As shown in Fig. 1, we consider a system composed of a transmitting BS, 4 users and a UAV playing the role of a relay node, separated by a distance R. We assume that direct communication between the transmitter and the receivers is possible, where there is one source node BS, half-duplex AF UAV relay node and 4 users U 1 , U 2 , U 3 , and U 4 , respectively. We use UAV, each equipped with a single antenna, to enable this link and maximize its capacity. Each UAV acts as an AF relay where it simply receives the signal, amplifies, and re-transmits it in a synchronized manner.

[H] 1 ∈ C N T ×N R denotes the communication between the N T transmitter and the N U UAVs, whereas, [H] 2 ∈ C N R ×N U denotes the channel between the UAVs and the N U receiver. We assume that both channels are strong LoS channels with each element defined as [H] i,j = λ 4πdi,j e j2πd i,j λ

, where λ is the signal frequency band and d i,j is the distance between antenna elements i and j at the transmitter, the relay or the receiver. We know that NOMA exploits successive interference cancellation (SIC) [START_REF] Won | A survey on UAV placement and trajectory optimization in communication networks[END_REF], in which one user detects the other user's message from the superposition coded incoming signal before decoding his own. Thus, while conducting SIC, the near user decodes the information of the far user. This is a step that cannot be ignored. In every case, the data of the far user must be decoded by the near user. Now that the near user has access to the far user's data, they can help them by relaying it. Because the distant user's channel with the broadcasting BS is poor, the close user's re-transmission of his data will supply him with diversity.

In other words, they will get two copies of the same message. One comes from the BS, and the other comes from a nearby user who is serving as a relay. As a result, we can assume the far user's outage probability to reduce.

Cooperative communication/cooperative relaying [START_REF] Ajmal | A Novel Hybrid AF/DF Cooperative Communication Scheme for Power Domain NOMA[END_REF] is the name for this concept. We can see that NOMA naturally encourages cooperative communication because the near user has access to the data of the far user, which they must decode anyway. The benefit of cooperative communication is the link redundancy, which allows a message to be transmitted twice. In other words, if one link is down, the other is very certainly up and running. The scenario in which both links fall down at the same time is less likely than one link going down at a time. We have a lower outage probability and, as a result, more diversity benefit without the need for extra antennas in this situation (i.e., MIMO).Another benefit of relaying is that it might essentially extend the BS's coverage area. Let us create a cooperative NOMA network now that we have seen what cooperative communication is and how it might benefit our network. Under a line-of-sight channel, we will consider a downlink transmission with a BS and two NOMA users relayed by a UAV acting as an AF relay [START_REF] Nayak | Enhanced Physical Layer Security for Cooperative NOMA Network with Hybrid-Decode-Amplify-Forward Relaying via Power Allocation Assisted Control Jamming[END_REF]. With the UAV, we have a near user with a stronger channel and a distance user with weak channel circumstances. The transmission is split into two parts. Let us refer to the first time slot as "direct transmission" and to the second as "relaying".

Direct transmission slot

In the direct transmission slot, the BS uses NOMA to transmit data intended for the near user x n and the far user x f . The near user uses SIC to decode the far user's data first, and then proceeds to decode its own data. The far user will only perform direct decoding.

At the end of the direct transmission slot, the achievable data rates at the near user and far user are,

R n = 1 2 log 2 (1 + α n ρh 2 n ). ( 1
)
for detection SIC, the desired signal and interference signalare given as,

R f,1 = 1 2 log 2 (1 + α f ρh 2 f α n ρh 2 f + 1
).

(

Notation α n : power allocation coefficient for the near user.

α f : power allocation coefficient for the far user.

h n : channel between BS and the near user.

h f : channel between BS and the far user.

ρ: transmit SN R = P σ 2 , where P is the transmit power and σ 2 is the noise variance -As usual, α f > α n , and, α n +α f =1

We have this factor of 1 2 in front of the achievable rates because we have two time slots of equal duration and R n , R f are the achievable rates during the first time slot alone.

Relaying slot The next half of the time slot is called the relaying slot.As we have seen, the near user already has the far user's data because he decoded it in the previous time slot. In the relaying time slot, the near user just transmits this data to the far user. The achievable rate of the far user at the end of the relaying slot is,

R f,2 = 1 2 log 2 (1 + ρh 2 nf ) (3)
here, h nf is the channel between the near user and the far user. We can already see that R f,2 > R f,1 because of two reasons:

-There is no interference from other transmissions -There is no fractional power allocation. The whole transmit power is given to the far user Diversity combining Now, at the end of the two time slots, the far user has two copies of the same information received through two different channels. The far user can now use a diversity combining technique. For example, he can use selection combining to choose the copy which was received with high SNR. After selection combining, the achievable rate of the far user would be,

R f = 1 2 log 2 (1 + max( α f ρh 2 f α n ρh 2 f + 1 , ρh 2 f )) (4)
If we do not use cooperative relaying, the achievable rate of the far user would be,

R f,noncoop = log 2 (1 + α f ρh 2 f α n ρh 2 f + 1 ) (5)
The factor of 1 2 does not exist here because the entire time slot will be used for transmission in non-cooperative communication. If we DID NOT use NOMA, for example, if we use Time-division multiple access (TDMA), we will allocate half of the time slot for transmission of the far user data. Hence, the achievable rate of the far user would be,

R f,OM A = 1 2 log 2 (1 + ρh 2 f ) (6)
Now that we have seen what cooperative communication is, and how it is useful to our network, let us design a cooperative NOMA network, in which we verify that the order of performances would be: cooperative NOMA > non-cooperative NOMA > Orthogonal Multiple Access (OMA). In fact, we can see from Fig. 2 that cooperative communication is clearly beneficial.

User Pairing in NOMA

We have previously shown that NOMA can support numerous consumers in the same band simultaneously. The next logical step will be to figure out how many users can be supported per frequency carrier. In practice, if the number of users is increased beyond a certain point, the network's total throughput begins to decline. As a result, the number of users per carrier cannot be increased forever.

A feasible option to support all users is to use hybrid NOMA. Hybrid NOMA is a technique that combines NOMA and any of the OMA approaches. Consider the combination of TDMA and NOMA, as indicated in Fig. 3. Assume we have a time slot with a duration of 4ms. Within this time frame, we must support four users. Now, TDMA will divide the 4 ms slot into four 1 ms slots, and each user will be assigned one slot. The four users will be given the entire 4 ms slot by NOMA. This, as we all know, will increase SIC complexity and processing time. In contrast, hybrid NOMA divides the 4 ms slot into two 2 ms slots and assigns two NOMA users to each slot. As can be shown, hybrid NOMA can provide service to all users while reducing complexity. As we progress through this article, we will talk more about hybrid NOMA methods. Now we must figure out how to match users within each orthogonal resourcethat is offered ( (1,2), [START_REF] Zhang | Cellular UAV-to-X communications: Design and optimization for multi-UAV networks[END_REF][START_REF] Wang | Adaptive deployment for UAV aided communication networks[END_REF] or [START_REF] Zhang | Spectrum sharing for drone networks[END_REF][START_REF] Zhang | Cellular UAV-to-X communications: Design and optimization for multi-UAV networks[END_REF], [START_REF] Ji | Joint cache placement, flight trajectory, and transmission power optimization for multi-UAV assisted wireless networks[END_REF][START_REF] Wang | Adaptive deployment for UAV aided communication networks[END_REF] or (1,4),(2,3)). users, respectively, from the UAV. U 1 is the nearest user and U 4 is the farthest user. Therefore, their channel conditions are ordered as

Performance Analysis

h 1 2 > h 2 2 > h 3 2 > h 4 2 .
We have two orthogonal resource blocks (time/ frequency/ subcarriers) to which two users must be allocated. We will pair users based on various distances. There are two basic ways to accomplish this:

-Near-far pairing (N-F) -Near-near, far-far pairing (N-N, F-F) Near-far pairing (N-F) In this method, the nearest user to UAV is paired with the farthest user from UAV. The next nearest user is paired with the next farthest user and so on. In our example, U 1 is the nearest user and U 4 is the farthest user. So, N-F pairing will pair U 1 with U 4 in one resource block. U 2 will be paired with U 3 in the next resource block.

In the first pair of users, U 1 is the near user and U 4 is the far user. Therefore, we have to choose the power allocation coefficients as α 1 < α 4 . So, U 1 should perform SIC, while U 4 will perform direct decoding. Similarly, in the second pair of users, U 2 is the near user and U 3 is the far user. Therefore, we have to choose α 2 < α 3 . Here, U 2 should perform SIC while U 3 will perform direct decoding.

The achievable rates for the users in the first pair are,

R 1,nf = 1 2 log 2 (1 + P α 1 h 2 1 σ 2 )(af ter SIC). (7)
R 4,nf = 1 2 log 2 (1 + P α 4 h 2 4 P α 1 h 2 4 + σ 2 ). (8)
Similarly, for the second pair,

R 2,nf = 1 2 log 2 (1 + P α 2 h 2 2 σ 2 )(af ter SIC). (9)
R 3,nf = 1 2 log 2 (1 + P α 3 h 2 3 P α 2 h 2 3 + σ 2 ). ( 10
)
The sum rate of the (N-F) scheme will be:

R nf = R 1,nf + R 2,nf + R 3,nf + R 4,nf . (11)
Near-near, far-far pairing (N-N, F-F) Another way to perform user pairing is to group the nearest user with the next nearest user. The farthest user is grouped with the next farthest user. If we follow this strategy, in our example, U 1 will be paired with U 2 in one resource block. U 3 will be paired with U 4 in the next resource block. Now, in the first pair of users, U 1 is nearest to the BS when compared to U 2 . Therefore, we have to choose α 1 < α 2 . U 1 should perform SIC, U 2 will perform direct decoding. Similarly, U 3 is closer to the BS than U 4 . So, we have to choose α 3 < α 4 . U 3 should perform SIC, while U 4 will perform direct decoding.

The achievable rates for the users in the first pair are,

R 1,nn = 1 2 log 2 (1 + P α 1 h 2 1 σ 2 )(af ter SIC). (12)
R 2,nn = 1 2 log 2 (1 + P α 2 h 2 2 P α 1 h 2 2 + σ 2 ). (13)
Similarly, for the second pair,

R 3,nn = 1 2 log 2 (1 + P α 3 h 2 3 σ 2 )(af ter SIC). (14)
R 4,nn = 1 2 log 2 (1 + P α 4 h 2 4 P α 3 h 2 4 + σ 2 ). (15)
The sum rate of the (N-F) scheme will be

R nn = R 1,nn + R 2,nn + R 3,nn + R 4,nn . (16)

Simulation Results

We have observed two different AF relaying UAV cooperative NOMA approaches.

The quest now is to find out which is better. A possible set of experiments would possibly start off by just multiplexing all the users on the same carrier (SC-NOMA), with no user pairing. Given the additional work required for user pairing, one would even question the very need of NOMA and how can using its TDMA counterpart instead would influence the overall performance. Let try to seek answers by performing a set of simulations in MATLAB. In this simulation, N users with N = 4 are considered. When we utilize each of the user pairing strategies with the UAV that we looked at, we will plot the network's sum rate. We will also examine the network's sum-rate performance with simply SC-NOMA and TDMA with UAV.

When the near user is coupled with the far user, as seen in Fig. 4, a higher sum rate is produced. This supports the well-known notion that cooperative NOMA operates better when the channel conditions, or the Los probability model in UAV communication, are different between the two users. When nearnear, far-far pairing is employed, NOMA-based UAV still outperforms TDMAbased UAV in terms of sum rate, but the difference is not significant. When compared to TDMA, the performance of SC-NOMA-based UAVs is poor since all users are crammed onto the same carrier, generating interference. This also validates our assumption that increasing the number of customers sharing the same carrier without paying for a premium subscription. Another benefit of the N-F pairing UAV relay over the N-N-F-F pairing UAV relay is the careful selection of users who may be impacted by interference. In contrast to the N-N-F-F system, which gives interference-free transmission to one near the user and one far user (assuming perfect SIC), the N-F strategy provides interferencefree reception to both near users (assuming perfect SIC). As a result, N-F has a higher sum rate. When noise dominates the interference power, however, the choice of users who experience interference has little effect. Both systems are projected to perform similarly.

Conclusion and Future Work

In this paper, a cooperative NOMA user pairing under a LoS channel is proposed, which is provided by a UAV serving as an AF relay. We investigated two user pairing strategies in order to improve downlink transmission performance. According to our results, implementing a user pairing strategy offers the most appropriate use of power and bandwidth resources. If we couple users at random, we may not be able to maximize the network's potential. The conversion of our single-antenna UAV relay model to a multiple-antenna UAV relay should be interesting. These issues will be addressed in the future.

Fig. 1 .

 1 Fig. 1. Illustration of the system model.

Fig. 2 .

 2 Fig. 2. Outage probabilities of the considered transmission schemes : cooperative NOMA, non-cooperative NOMA, Orthogonal Multiple Access (OMA).

  Let us consider a downlink cooperative communication based UAV scenario with four users U 1 , U 2 , U 3 and U 4 . Let d 1 , d 2 , d 3 and d 4 denote the distances of those

Fig. 3 .

 3 Fig. 3. Hybrid NOMA.

Fig. 4 .

 4 Fig. 4. Sum-rate performance of the network with SC-NOMA, TDMA, hybrid NOMA N-F pairing and hybrid NOMA NN-FF pairing with/without UAV.


Jörg Hermann
email: jorg.hermann@cnrs.fr

Christoph Gerhard

Miloš Burger

Valentin Craciun

Frédéric Pelascini

Progress in calibration-free laser-induced breakdown spectroscopy

Keywords: LIBS, calibration-free, elemental analysis, measurement uncertainty, analytical performance, self-absorption, trace elements

Direct elemental analysis of materials via modeling of the laser-produced plasma emission attracted great interest since the introduction of calibration-free laser-induced breakdown spectroscopy in 1999. However, the large number of different approaches and their validation in diverse experimental conditions make it difficult to evaluate the achievable performance with this revolutionary type of analytical measurements. The purpose of the present paper is to provide a critical review of the development of calibration-free LIBS, to highlight the progress being achieved until now, and to estimate its available analytical performance. Based on the identification of the principal sources of measurement uncertainty, solutions are proposed to further improve the accuracy in order to bring calibration-free LIBS to the standard of fully recognized analytical methods.

Introduction

The basic concept behind calibration-free laserinduced breakdown spectroscopy has been developed far before the invention of the laser. Plasma technologies emerged at the beginning of the last century [START_REF] Shenstone | Low voltage arc spectra of copper[END_REF], with principal objectives devoted to the understanding of astrophysical plasmas and to the investigation of the atomic structure. Later, the concept of local thermodynamic equilibrium (LTE) was introduced to enable a simplified description of the elementary processes that occur in a plasma [START_REF] Griem | Plasma spectroscopy[END_REF]. Among them, we can distinguish collisional processes, such as excitation and ionisation by electron collisions and their invers processes, and radiative processes, such as spontaneous emission and absorption between bound and free electronic states. Out of equilibrium, complex collisonal-radiative modeling is required to describe the plasma, and the rates of all processes must be known [START_REF] Capitelli | Non-equilibrium plasma kinetics: a state-to-state approach[END_REF]. In thermodynamic equilibrium, the principle of microscopic reversibility applies and each process is counterbalanced by its inverse process. A simplified description of the plasma is thus possible via the statistical laws of equilibrium. So, the species velocities are described by the Maxwellian distribution, the excited-states population number densities are given by the Boltzmann law, the chemical and ionization equilibrium is described by Saha-equations, and the radiation equilibrium is governed by Planck's law [START_REF] Griem | Plasma spectroscopy[END_REF]. The size of laboratory plasmas being typically smaller than the characteristic length of absorption, the principle of microscopic reversibility does not apply to radiative processes. The plasma may, however, reach the state of equilibrium, if the collisonal processes dominate. In that case, the plasma is called in local thermodynamic equilibrium and described by all above mentioned equilibrium laws, except Planck's law [START_REF] Griem | Plasma spectroscopy[END_REF]. In strongly ionized plasmas, the collisonal processes are dominated by the electrons and the establishment of LTE requires thus a large enough electron density, above a minimum value that can be estimated using the criteria proposed by Griem [START_REF] Griem | Validity of local thermal equilibrium in plasma spectroscopy[END_REF], McWhirter [START_REF] Mcwhirter | Plasma diagnostic techniques[END_REF], and Drawin [START_REF] Drawin | Validity conditions for local thermodynamic equilibrium[END_REF]. In literature, most studies refer to the criterion of McWhirter due to the simplicity of its implementation.

To investigate the LTE state, appropriate plasma sources such as stabilized arcs [START_REF] Shenstone | Low voltage arc spectra of copper[END_REF], shock tubes [START_REF] Wiese | Measurements of temperatures and densities in shock-heated hydrogen and helium plasmas[END_REF] or spark discharges [START_REF] Bye | Saha-Boltzmann statistics for determination of electron-temperature and density in spark discharges using an echelle CCD system[END_REF] have been developed. Operated at atmospheric pressure, the time of thermalization -that is the time required for the establishment of equilibrium -equals the characteristic times of particle and heat diffusion [START_REF] Belmonte | Theoretical background of optical emission spectroscopy for analysis of atmospheric pressure plasmas[END_REF]. Plasmas in LTE are therefore spatially nonuniform and their diagnostics require space-resolved spectroscopic measurements and complex data analyses [START_REF] Glasser | Abel inversion applied to plasma spectroscopy -new interactive method[END_REF]. Laser-produced plasmas appeared later. Their early spectroscopic observations were mostly dedicated to the study of laser-matter interaction [START_REF] Meyerand | Optical-energy absorption and high-density plasma production[END_REF][START_REF] Dawson | Optical absorption and expansion of laser-produced plasmas[END_REF] and to analytical measurements [13,14]. Their use as a source for plasma spectroscopy was limited. Laser-produced plasmas have small sizes and are characterized by fast expansion dynamics. They had low reproducibility during the early experiments due to the poor stability of the available laser sources. Technological advances changed the situation. Reliable gain-switched Nd:YAG laser sources, delivering pulses of nanosecond duration with high energy and pointing stabilities, enabled the generation of highly reproducible plasmas [START_REF] Hahn | Laser-induced breakdown spectroscopy (LIBS), Part II: Review of instrumental and methodological approaches to material analysis and applications to different fields[END_REF]. Fast and sensitive intensified charge-coupled detectors rendered the time-and spaceresolved spectroscopic observation of the small-sized and rapidly expanding laser-induced plasmas accessible. The interest for LIBS increased, in particular during the last two decades, where an exponential increase of the number of publications related to the subject was observed [START_REF] Noll | Laser-induced breakdown spectroscopy: fundamentals and applications[END_REF]. The increase of interest was not only dedicated to analytical measurements but also to investigations of plasma fundamentals [START_REF] Mercadier | Plume segregation observed in hydrogen and deuterium containing plasmas produced by laser ablation of carbon fiber tiles from a fusion reactor[END_REF][START_REF] Cristoforetti | Local thermodynamic equilibrium in laser-induced breakdown spectroscopy: beyond the McWhirter criterion[END_REF][START_REF] De Giacomo | Laser-induced plasma emission: from atomic to molecular spectra[END_REF]. The laser-induced plume as a plasma source for measurements of spectroscopic data became popular [START_REF] Aragón | Stark width measurements of Fe II lines with wavelengths in the range 260-300 nm[END_REF][START_REF] Cirisan | Stark broadening measurement of Al II lines in a laser-induced plasma[END_REF][START_REF] Burger | Stark broadening measurements in plasmas produced by laser ablation of hydrogen containing compounds[END_REF][START_REF] Ivković | Stark width and shift for electron number density diagnostics of low temperature plasma: Application to silicon LIBS[END_REF]. The small size appeared now as an advantage as it limits the amount of self-absorption. Plumes produced by pulsed laser ablation are singular plasma sources.

Compared to other atmospheric pressure plasmas, they are characterized by a much higher initial density, followed by fast expansion until a pressure equilibrium with the surrounding gas is reached [START_REF] Zel'dovich | Physics of shock waves and high temperature phenomena[END_REF]. The high density makes thermalization faster, and renders diffusion processes slower. Thus, plasmas produced by laser ablation are characterized by a time of thermalization smaller than the times of diffusion. They may therefore combine two properties which are generally not found together: the local thermodynamic equilibrium and the spatial uniformity. Both properties were found together for the plasma produced by ultraviolet nanosecond laser ablation under argon atmosphere [START_REF] Hermann | Ideal radiation source for plasma spectroscopy generated by laser ablation[END_REF]. They were evidenced by the simultaneous observation of three features in the plasma emission spectrum: (i) the emission spectrum is accurately described by the spectral radiance of a uniform LTE plasma, computed by using an analytical solution of the radiation transfer equation. Thus, the most intense lines were found to saturate at the blackbody spectral radiance, and the blackbody temperature equaled the atomic excitation temperature deduced from the Boltzmann plot of optically thin lines [START_REF] Hermann | Ideal radiation source for plasma spectroscopy generated by laser ablation[END_REF][START_REF] Hermann | Local thermodynamic equilibrium in a laser-induced plasma evidenced by blackbody radiation[END_REF]. (ii) The observation of symmetric profiles of Starkshifted spectral lines. Such lines appear asymmetric when emitted from a nonuniform plasma, like the vapor plume produced by laser ablation of a solid sample in air [START_REF] Hermann | Ideal radiation source for plasma spectroscopy generated by laser ablation[END_REF][START_REF] Hermann | Comparative investigation of laser ablation plumes in air and argon by analysis of spectral line shapes: insights on calibration-free laserinduced breakdown spectroscopy[END_REF]. (iii) The absence of absorption dips in the profiles of strongly self-absorbed resonance lines [START_REF] Hermann | Ideal radiation source for plasma spectroscopy generated by laser ablation[END_REF][START_REF] Hermann | Comparative investigation of laser ablation plumes in air and argon by analysis of spectral line shapes: insights on calibration-free laserinduced breakdown spectroscopy[END_REF]. The combination of both properties in the laser-induced plasma constitutes a breakthrough in the development of calibration-free LIBS as it enables, via robust and accurate modeling of the plasma emission spectrum, elemental analysis of materials with an accuracy not attainable before [START_REF] Hermann | Analysis of multi-elemental thin films via calibration-free laser-induced breakdown spectroscopy[END_REF][START_REF] Gerhard | Quantification of surface contamination on optical glass via sensitivity-improved calibration-free laser-induced breakdown spectroscopy[END_REF]. Moreover, straightforward measurements of spectroscopic data, such as Einstein coefficients of spontaneous emission and Stark broadening parameters, become accessible without the need of space-resolved spectroscopic observations, simplifying thus this kind of measurements [START_REF] Burger | Stark broadening measurements in plasmas produced by laser ablation of hydrogen containing compounds[END_REF][START_REF] Hermann | Simulation of emission spectra from nonuniform reactive laser-induced plasmas[END_REF]. This opens multiple perspectives for all applications requiring spectroscopic data, as the number of reliable data and their quality are expected to grow rapidly within the next years. In the following, we briefly recall basic fundamentals of calibration-free LIBS, before presenting a short review of its development in Section 3. Then, the analytical performance will be critically analyzed in Section 4, and upcoming still required improvements will be discussed in Section 5.

Fundamentals

The simplest way to calculate the emission spectrum of a spatially uniform plasma is to approximate its shape by a layer of thickness L as shown in Fig. 1, in order to describe the radiation transfer as a one-dimensional problem. The intensity variation of the radiation flux propagating along the z -axis towards the observer [see Fig. 1(c)] is given by

n(z ) d dz I λ (z ) n 2 (z ) = ε λ (z ) -α(z ) I λ (z ) , (1)
where ε λ , α and n are the emission coefficient, the absorption coefficient and the refractive index of the medium, respectively. For a plasma, we have n 1. Assuming a spatially uniform medium, integration of Eq. (1) along the line of sight leads to

I λ (z) = ε λ α 1 -e -αz . (2)
In local thermodynamic equilibrium, the coefficients of emission and absorption are related via Kirchhoff's law of thermal radiation to the blackbody spectral radiance

[31] B 0 λ (λ) = ε λ (λ) α(λ) , (3)
and the spectral radiance of the uniform plasma in LTE is given by

B λ (λ) = B 0 λ (λ) 1 -e -τ(λ) . (4)
Here, τ(λ) = α(λ, z)dz = α(λ)L is the optical thickness that accounts for the amount of self-absorption.

In the case of negligible self-absorption (τ 1), Eq. ( 4) becomes

B λ (λ) = ε λ (λ)L. ( 5
)
The spectral radiance of the plasma, and thus the measured intensity, are proportional to the emission coefficient. In the opposite case of strong self-absorption (τ 1), Eq. ( 4) simplifies to

B λ (λ) = B 0 λ (λ). ( 6
)
Microscopic reversibility between the processes of emission and absorption applies in a given spectral range, a situation that is generally limited to strong resonance transitions and a narrow wavelength interval close to the line center wavelength. Figure 2: Space-integrated spectra recorded during laser ablation of steel in argon at different times. The strongest transitions are shown to saturate at the blackbody spectral radiance. The excellent agreement with the spectral radiance computed using Eq. ( 4) shows that the emission originates from a uniform plasma in LTE. (Adapted from Ref. [START_REF] Hermann | Ideal radiation source for plasma spectroscopy generated by laser ablation[END_REF])

Evidence of blackbody radiance

The saturation of most intense lines at the blackbody radiance is illustrated in Fig. 2, where LIBS spectra of steel, recorded for two different delays between the laser pulse and the detector gate, are presented. Numerous lines are shown to reach the blackbody spectral radiance that was calculated for the temperature deduced from the intensity distribution of atomic and ionic lines. For three spectral ranges, the measured spectrum is displayed together with the spectral radiance computed according to Eq. ( 4). The excellent agreement between measured and computed spectra supports the hypothesis of a uniform plasma in LTE.

Symmetric profile of Stark-shifted line

In strongly ionized laser-produced plasmas, the shape of the spectral lines is mostly determined by the Stark effect [START_REF] Zhao | Spatial distributions of electron density in microplasmas produced by laser ablation of solids[END_REF]. The lines are therefore broadened and shifted, and the Stark widths and shifts increase linearly with the electron density in the case of nonhydrogenated transitions [START_REF] Griem | Spectral line broadening of plasmas[END_REF]. For a given transition, the Stark shift is typically smaller than the Stark width, and ranges from 0 to 30% for most lines [START_REF] Konjević | A critical review of the Stark widths and shifts of spectral lines from non-hydrogenic atoms[END_REF]. The space-integrated observation of a strongly Stark-shifted line delivers information about the spatial distribution of the electron density n e within the plasma. In the case of a nonuniform plasma, typically characterized by a hot core and a cold border, the line profiles emitted from the plasma volumes of different n e -values have unequal widths and shifts as illustrated in Fig. 3. The resulting line shape recorded by the spectroscopic apparatus is therefore asymmetric. In opposition, the observation of a symmetric shape of a strongly Stark-shifted line proves that the plasma has a uniform electron density distribution.

Measured profiles of a strongly Stark-shifted line are displayed in Fig. 4. They were recorded during ultraviolet laser ablation of optical glass in ambient air and argon background gas. The comparison with computed symmetric line shapes shows that the line profile observed for ablation in air is asymmetric, indicating thus a nonuniform n e -distribution. In opposition, the line profile observed for ablation in argon appears symmetric, showing thus that the plume generated in the inert gas has a uniform electron density distribution. The observed difference illustrates the influence of the background gas on the properties of the ablated vapor plume. For a molecular gas such as air, the interaction at the vapor-gas contact front is dominated by inelastic collisions, due to the numerous rovibrational levels of the gas molecules. This leads to efficient cooling of the peripheral volume of the vapor plume. In an inert gas such as argon, most collisions are elastic due to the high energy required for excitation (11.55 eV for Ar). The electrons conserve most of their energy, and the surrounding gas acts like an isolating layer.

Absorption dip of resonance line

In a nonuniform plasma, the values of temperature and electron density are typically lower in the peripheral volume compared to their values in the plasma core. The population number densities of ground state species are therefore large, and the resonance lines undergo strong absorption in the border volume. According to the reduced electron density, the strongly broadened line emitted from the core volume, is reabsorbed in a narrow spectral range, close to the resonance wavelength of the transition (see Fig. 5). The resulting spectral line shape is thus characterized by an absorption dip that illustrates the occurence of nonuniform distributions of electron density and temperature [START_REF] Cowan | Self-absorption of spectrum lines[END_REF][START_REF] Hermann | Diagnostics of the early phase of an ultraviolet laser induced plasma by spectral line analysis considering self-absorption[END_REF]. Measured profiles of a resonance line are displayed in Fig. 6. They were recorded during ultraviolet laser ablation of pure indium in ambient air and argon background gas. The line observed in air exhibits an absorption dip that confirms the nonuniform plasma distribution deduced from the observation of a strongly Stark-shifted transition (see Section 2.2). In opposition, the absorption dip is not observed for line emission in argon. This confirms the results presented in Figs. 2 and4, showing that the vapor plume produced by ultraviolet laser ablation in argon has almost uniform distributions of temperature and electron density. The uniform spatial distributions observed in argon are however restricted to the vapor plume, and gradients of temperature and electron density are naturally expected in the surrounding background gas, to provide the transition from the heated gas layer at the vapor-gas contact front towards the cold environment.

The presence of temperature and electron density gradients in the argon background gas is illustrated by the absorption dip in the profile of the argon line displayed in Fig. 7. The Ar I 763.51 nm transition decays towards a metastable state. According to the large energy gap of 11.55 eV towards the ground state, the collisional desexcitation is slow, and the metastable state is expected to reach a large population number density. The spectral line behaves therefore like a resonance transition, characterized by strong self-absorption that enables here the visualization of gradients in the background gas.

Development of calibration-free LIBS

Succesful introduction of first method

The development of calibration-free laser-induced breakdown spectroscopy started in 1999 with the approach of Ciucci et al. [START_REF] Ciucci | New procedure for quantitative elemental analysis by laser-induced plasma spectroscopy[END_REF]. The method was based on the hypotheses that (i) the mass transfer from the solid into the plasma is congruent, (ii) the plasma is in local thermodynamic equilibrium, (iii) the plasma has spatially uniform distributions of temperature and densities, and (iv) self-absorption of spectral lines is negligible. In that case, the measured line intensity is proportional to the emission coefficient [see Eq. ( 5)]. Assuming furthermore moderate ionisation, the elemental fractions can be directly deduced from the multielemental Boltzmann plot. This method attracted great interest due to the simplicity of its implementation [START_REF] Tognoni | Calibration-free laser-induced breakdown spectroscopy: state of the art[END_REF]. Applied by many research groups on all kind of materials, the analytical performance was however found to be limited [START_REF] Tognoni | A numerical study of expected accuracy and precision in calibration-free laserinduced breakdown spectroscopy in the assumption of ideal analytical plasma[END_REF]. The low performance was due to the fact that the hypotheses of model validity were generally not all satisfied. Although it is now well accepted that experimental conditions can be easily found to satisfy the hypotheses of congruent ablation [START_REF] Mao | Preferential vaporization during laser ablation inductively coupled plasma atomic emission spectroscopy[END_REF][START_REF] Hermann | On the stoichiometry of mass transfer from solid to plasma during pulsed laser ablation of brass[END_REF] and LTE [START_REF] Cristoforetti | Thermodynamic equi-librium states in laser-induced plasmas: From the general case to laser-induced breakdown spectroscopy plasmas[END_REF][START_REF] Gerhard | Quantitative analyses of glass via laser-induced breakdown spectroscopy in argon[END_REF], and that spatial uniformity can be reached in appropriate experimental conditions (see Section 2), the condition of optically thin emission is hardly achieved for dense laser-induced plasmas [START_REF] Hermann | Diagnostics of the early phase of an ultraviolet laser induced plasma by spectral line analysis considering self-absorption[END_REF][START_REF] Takahashi | Quantitative methods for compensation of matrix effects and self-absorption in laser induced breakdown spectroscopy signals of solids[END_REF][START_REF] Touchet | Laser-induced breakdown self-reversal isotopic spectrometry for isotopic analysis of lithium[END_REF].

Amended methods

Amendments have thus been proposed, most of them dedicated to the compensation of self-absorption [START_REF] Lazic | Selfabsorption model in quantitative laser induced breakdown spectroscopy measurements on soils and sediments[END_REF][START_REF] Bulajic | A procedure for correcting selfabsorption in calibration-free laser-induced breakdown spectroscopy[END_REF][START_REF] Sherbini | Evaluation of self-absorption coefficients of aluminum emission lines in laser-induced breakdown spectroscopy measurements[END_REF]. Later, approaches based on the spectra simulation have been developed [START_REF] D'angelo | Laser induced breakdown spectroscopy on metallic alloys: Solving inhomogeneous optically thick plasmas[END_REF][START_REF] Hermann | System and method for quantitative analysis of the elemental composition of a material by laser-induced breakdown spectroscopy (LIBS)[END_REF][START_REF] Wester | Heuristic modeling of spectral plasma emission for laser-induced breakdown spectroscopy[END_REF]. These have the advantage that they intrinsically account for self-absorption. However, compared to the initial calibration-free LIBS method proposed by Ciucci et al. [START_REF] Ciucci | New procedure for quantitative elemental analysis by laser-induced plasma spectroscopy[END_REF], the amended approaches were associated to a significant increase of complexity in their implementation. Their use was therefore restricted to a limited number of research groupes, and numerous authors reported calibrationfree LIBS analysis by carefully selecting optically thin spectral lines [START_REF] Pedarnig | Element analysis of complex materials by calibration-free laser-induced breakdown spectroscopy[END_REF][START_REF] Dell'aglio | Stand-off laser induced breakdown spectroscopy on meteorites: calibration-free approach[END_REF]. As absorption and emission are correlated in LTE plasmas according to Kirchhoff's law of thermal radiation [see Eq. ( 3)], the selection of optically thin lines implies the restriction to transitions of weak intensity, lowering thus the analytical performance via the reduced signal-to-noise ratio. More recently, methods including a single calibration step have been introduced to improve the accuracy of the analytical measurement [START_REF] Cavalcanti | One-point calibration for calibrationfree laser-induced breakdown spectroscopy quantitative analysis[END_REF][START_REF] Gaudiuso | Laser-induced breakdown spectroscopy of archaeological findings with calibration-free inverse method: comparison with classical laser-induced breakdown spectroscopy and conventional techniques[END_REF][START_REF] Aragón | Direct analysis of aluminum alloys by CSigma laser-induced breakdown spectroscopy[END_REF]. Here, the spectrum measured for a single standard sample is used to compensate errors due to the apparatus response function and spectroscopic data. This improvement is however limited, as the spectroscopic data -that are transition probabilities and Stark broadening parameters -impact the amount of self-absorption [START_REF] Taleb | Measurement error due to self-absorption in calibration-free laser-induced breakdown spectroscopy[END_REF], whereas the apparatus response has a linear influence on the measured line intensity [START_REF] Taleb | Echelle spectrometer calibration by means of laser plasma[END_REF]. Consequently, the accuracy improvement due to the single calibration step works properly only when optically thin lines are used, or when the elemental fractions to be measured are close to those of the standard sample.

General remarks

The rich literature dedicated to the development of calibration-free LIBS analysis and its validation in a large variety of different experimental conditions makes it difficult to chose the appropriate method for a particular compositional measurement and to predict the achievable analytical performance. Thus, before discussing accuracy and current limitations, we would like to make several general comments on both the development of calibration-free LIBS and its validation: (i) Most of the proposed calibration-free LIBS approaches are based on the same physical model: the spatially uniform LTE plasma. They are therefore expected to have similar analytical performance. (ii) Methods based on spectra simulation and methods based on the multielemental Boltzmann plot are often considered as fundamentally different approaches, with distinguished pathways for obtaining the fundamental parameters that describe the plasma. Furthermore, spectra simulation is often associated to time-consuming calculations. The reason for this is that some authors combined spectra simulation with Monte-Carlo calculations to find the plasma parameters [START_REF] Gornushkin | Experimental verification of a radiative model of laser-induced plasma expanding into vacuum, Spectrochim[END_REF][START_REF] Hansen | Modeling of time-resolved LIBS spectra obtained in Martian atmospheric conditions with a stationary plasma approach[END_REF]. This is however not required for the simple case of the uniform LTE plasma for which the parameters can be found straightforwardly [START_REF] Gerhard | Quantitative analyses of glass via laser-induced breakdown spectroscopy in argon[END_REF]. Moreover, the number of parameters is independent of the kind of approach and depends uniquely on the phyiscal model. When the spectrum is computed via the analytical solution of the radiation transfer equation given by Eq. ( 4), the calculation is fast and no significant difference is expected in the computational duration of spectra-simulation-based calibration-free LIBS compared to the analysis through the multielemental Boltzmann plot. The fundamental difference relies in the data processing methology only: in the case of the multielemental Boltzmann plot, the analysis starts with the experimental spectrum via measurements of line intensities and widths. Contrarily, in spectra simulation, the starting point is the calculation of the plasma emission spectrum that is then compared to the measured one. Methods based on spectra simulation facilitate automation of calibration-free LIBS analysis as the treatment of line interferences is much easier to handle. (iii) The validation of the calibration-free LIBS methods is exclusively done by comparing the final result of the elemental analysis with the reference composition [START_REF] Aguilera | Application of calibration-free laser-induced breakdown spectroscopy to radially resolved spectra from a copper-based alloy laserinduced plasma[END_REF][START_REF] Herrera | Semi-quantitative analysis of metal alloys, brass and soil samples by calibration-free laser-induced breakdown spectroscopy: recent results and considerations[END_REF][START_REF] Praher | Calibration free laser-induced breakdown spectroscopy of oxide materials[END_REF]. This makes it difficult to identify individual error sources, and thus to predict the analytical performance for a given type of material.

Analytical performance

. Influence of closure condition on accuracy

Calibration-free LIBS is generally considered to have low accuracy in minor and trace element quantification [START_REF] Tognoni | Calibration-free laser-induced breakdown spectroscopy: state of the art[END_REF]. This is often atributed to the closure condition according to which the sum over all elemental fractions equals unity ( A C A = 1) [START_REF] Tognoni | A numerical study of expected accuracy and precision in calibration-free laserinduced breakdown spectroscopy in the assumption of ideal analytical plasma[END_REF][START_REF] Gornushkin | Extension and investigation by numerical simulations of algorithm for calibration-free laser induced breakdown spectroscopy, Spectrochim[END_REF]. It is thus expected that even small errors of major element fractions induce large errors to minor and trace element fractions. The mass fraction of an element A is given by C A = n A m A /ρ tot , where n A and m A are the atomic number density and the atomic mass of element A, respectively, and ρ tot = A n A m A the specific mass of the plasma. Applying the partial derivations with respect to the atomic number densities of all sample-composing elements, we obtain the uncertainty accociated to the fraction measurement

∆C A C A = (1 -C A ) 2 ∆n A n A 2 + N j A C 2 j ∆n j n j 2 . (7)
The factor (1 -C A ) 2 in the first term of the quadratic sum illustrates that elements of large abundance have reduced fraction measurement uncertainties. Contrarily, elements of small abundance are characterized by a measurement uncertainty that has significant contributions from the major elements as indicated by the sum of the quadratic errors weighted by the C 2 j -values. For clear illustration, we consider a sample composed of only two elements, a major and a trace element. Negelecting self-absorption and ignoring all error sources except the intensity measurement error, that is supposed to be equal for both elements, Eq. ( 7) simplifies to

∆C A C A = (1 -C A ) 2 ∆I I 2 + C 2 B ∆I I 2 . ( 8
)
For a major element (C A → 1) and a trace element (C B → 0), the uncertainties become ∆C A /C A → 0 and ∆C B /C B → √ 2∆I/I, respectively. The closure condition leads thus to a reduction of the fraction measurement uncertainty of major elements and a moderate uncertainty increase for minor and trace elements.

Influence of LTE condition on accuracy

Calibration-free LIBS requires emission spectra recorded in experimental conditions for which the plasma is in local thermodynamic equilibrium. The equilibrium state is established when the electron density is large enough to satisfy the LTE criteria (see Section 1). The large number density of charged particles is associated to intense continuum emission due to bremsstrahlung and radiative recombination.

As n e decreases in time, spectra used for calibrationfree LIBS measurements have to be recorded with a short enough detector gate delay, as illustrated in Fig. 8. In calibrated LIBS, there is no need of LTE and a larger gate delay is generally applied to achieve best signal-to-noise ratio. The need of LTE leads thus to lowering of the signal-to-noise ratio in calibration-free LIBS measurements. Consequently, the measurement accuracy is reduced for analytical lines of low emission intensity that generally belong to minor or trace elements.

The lowering of accuracy is particularly severe if the sample composition contains major elements with large energy gaps in their atomic structure, such as C, H, N or O [START_REF] Gerhard | Quantitative analyses of glass via laser-induced breakdown spectroscopy in argon[END_REF]. In that case, a larger electron density is required to establish LTE, and spectra recording at shorter delay leads to further reduction of the signal-to-noise ratio. The loss of sensitivity is thus amplified in calibrationfree LIBS analyses of glass [START_REF] Gerhard | Quantification of surface contamination on optical glass via sensitivity-improved calibration-free laser-induced breakdown spectroscopy[END_REF] or organic materials such as food [START_REF] Chen | Two-step procedure for trace element analysis in food via calibration-free laser-induced breakdown spectroscopy[END_REF].

To improve the sensitivity of calibration-free LIBS analysis of such materials, a two-step method was proposed [START_REF] Chen | Two-step procedure for trace element analysis in food via calibration-free laser-induced breakdown spectroscopy[END_REF]. It consists of recording two spectra with different delays between the laser pulse and the detector gate as illustrated in Fig. 9. The early measurement is performed with a delay sufficiently short, so that the electron density is large enough to ensure full LTE conditions. Here, the signal-to-noise ratio is high enough to enable the quantification of elements having large abundance, but too small for fraction measurements of trace elements. The late measurement is therefore performed in conditions of partial LTE, for which matrix elements with large energy gaps in their atomic structure are out of equilibrium, whereas atoms and ions of all other elements have Boltzmann equilibrium distributions. According to the reduced electron density, the signalto-noise ratio is increased, and trace elements can be quantified. The combination of both measurements enables thus the analysis of the entire sample composition, as demonstrated for glass [START_REF] Gerhard | Quantitative analyses of glass via laser-induced breakdown spectroscopy in argon[END_REF] and seafood [START_REF] Chen | Two-step procedure for trace element analysis in food via calibration-free laser-induced breakdown spectroscopy[END_REF].

Influence of probe volume on accuracy

Another error source of calibration-free LIBS measurements can be attributed to the probe volume, that generally differs from that of the reference method. As the fractions of trace elements on the sample surface often differ from their values in the bulk material, the measured value is probe volume-dependent.

To account for possible nonuniform elemental distributions, in-depth measurements can be performed by recording spectra for laser pulses applied successively to the same irradiations sites [START_REF] Mercadier | In-depth analysis of ITERlike samples composition using laser-induced breakdown spectroscopy[END_REF]. Such measurements are of particular interest in combination with calibration-free spectra analysis, as possible changes of electron density and temperature within the plasma produced by successive laser pulses are taken into account [START_REF] Hermann | On the stoichiometry of mass transfer from solid to plasma during pulsed laser ablation of brass[END_REF]. The fractions of trace elements as functions of depth measured via depth-resolved calibration-free LIBS are displayed in Fig. 10 for analysis of optical glass. Three types of trace elements can be distinguished: elements that belong to the glass matrix having a depth-independent mass fraction (a); elements originating from the polishing agent, characterized by an exponential decrease over a few micrometers towards zero (b); and elements with contributions from both the polishing solution and the glass matrix, characterized thus by an exponential decrease towards a constant value (c). Comparing the mass fractions deduced for the bulk to reference values measured via inductively-coupled plasma atomic emission spectrometry, a good agreement was found for the entire glass composition including major, minor and trace elements [START_REF] Gerhard | Quantification of surface contamination on optical glass via sensitivity-improved calibration-free laser-induced breakdown spectroscopy[END_REF]. In summary, compared to the analytical performance of calibrated LIBS, the measurement accuracy of calibration-free LIBS analysis is reduced for minor and trace elements. However, the accuracy lowering is moderate if the measurements were performed in appropriate experimental conditions (see Sections 4.1.2 and 4.1.3).

Accuracy lowering due to self-absorption

Numerous approaches have been proposed to account for self-absorption in calibration-free LIBS [START_REF] Lazic | Selfabsorption model in quantitative laser induced breakdown spectroscopy measurements on soils and sediments[END_REF][START_REF] Bulajic | A procedure for correcting selfabsorption in calibration-free laser-induced breakdown spectroscopy[END_REF][START_REF] Sherbini | Evaluation of self-absorption coefficients of aluminum emission lines in laser-induced breakdown spectroscopy measurements[END_REF]. Some authors reported methods that enable the use of even strongly self-absorbed transitions, having an optical thickness up to 10 2 [START_REF] Gornushkin | Extension and investigation by numerical simulations of algorithm for calibration-free laser induced breakdown spectroscopy, Spectrochim[END_REF][START_REF] Maali | Error analysis in optimization problems relevant for calibration-free laser-induced breakdown spectroscopy[END_REF]. Recently, a method was proposed to compensate self-absorption via the exploitation of blackbody radiation [START_REF] Li | Correction of self-absorption effect in calibration-free laser-induced breakdown spectroscopy (CF-LIBS) with blackbody radiation reference[END_REF]. All these methods have been proposed without the evaluation of the analytical measurement uncertainty that is expected to grow according to the decrease of the slope of the curve of growth with increasing optical thickness (see Fig. 11). The quantification of the measurement uncertainty due to self-absorption is however essential for choosing the most appropriate spectral lines, and consequently for the optimizing the analytical performance. To quantify the increase of analytical measurement uncertainty due to self-absorption, the uncertainty of the atomic number density ∆n A in Eq. ( 7) must be evaluated. Neglecting the uncertainties of temperature and electron density, we obtain in case of optically thin (τ 1) emission [START_REF] Taleb | Measurement error due to self-absorption in calibration-free laser-induced breakdown spectroscopy[END_REF] Optical thickness computed computed (without absorption)

∆n A n A = ∆I I 2 + ∆A ul A ul 2 , (9)

Elemental fraction Spectral line intensity

Figure 11: Curve of growth of a self-absorbed transition. The spectral line intensity measured for various elemental fractions is compared to the intensity computed according to Eq. ( 4). (Adapted from Ref. [START_REF] Taleb | Measurement error due to self-absorption in calibration-free laser-induced breakdown spectroscopy[END_REF])

where ∆I and ∆A ul are the uncertainties of the measured line intensity and the transition probability, respectively. The uncertainty ∆I accounts for line interferences, signal-to-noise ratio, and the uncertainty of the apparatus response function.

In the general case, when self-absorption reduces the emission intensity of the spectral line, the analytical measurement uncertainty is given by [START_REF] Taleb | Measurement error due to self-absorption in calibration-free laser-induced breakdown spectroscopy[END_REF] ∆n

A n A = ∆τ 0 τ 0 2 + ∆A ul A ul 2 + (1 -e -τ 0 )        ∆w sd w sd 2 + ∆L L 2        .( 10
)
Here, ∆τ 0 is the uncertainty of the line center optical thickness that accounts for the increase of analytical measurement uncertainty due to the reduction of the slope of the curve of growth (see Fig. 11). The spectral line width due to Stark and Doppler broadening w sd and the plasma size along the line of sight L determine the amount of optical thickness, and the associated uncertainties ∆w sd and ∆L determine thus the uncertainty of horizontal scaling of the curve of growth.

Accuracy lowering due to the reduction of the slope of the curve of growth

The uncertainty ∆τ 0 can be expressed as a function of the intensity measurement uncertainty via

∆τ 0 τ 0 = g(τ 0 ) ∆I I , (11)
where the error growth factor g(τ 0 ) is obtained from the I = f (τ 0 ) dependence that can be calculated using Eq. ( 4). For center of line intensity measurements, the error growth factor in case of negligible apparatus spectral broadening has an analytical expression given by [START_REF] Taleb | Measurement error due to self-absorption in calibration-free laser-induced breakdown spectroscopy[END_REF]

g 0 = 1 -e -τ 0 τ 0 e -τ 0 . ( 12
)
It describes the exponential error growth that is expected from the saturation of the center of line intensity at the blackbody radiance level [see Eq. ( 6)].

For measurements of the line-integrated intensity, no analytical expression I line = f (τ 0 ) exists for common line shapes represented by the Voigt profile, and g(τ 0 ) must be computed numerically. For most common transitions having a significant contribution of Stark broadening, a moderate error growth g ≤ 2 is expected according to the square root dependence of the curve of growth for τ 0 1 [START_REF] Gornushkin | Curve of growth methodology applied to laser-induced plasma emission spectroscopy, Spectrochim[END_REF].

Accuracy lowering due to line width uncertainty

The line width due to Stark and Doppler broadening determines the amount of self-absorption, and its uncertainty has an influence on the analytical measurement uncertainty that increases with the optical thickness according to Eq. [START_REF] Glasser | Abel inversion applied to plasma spectroscopy -new interactive method[END_REF]. The Doppler width is typically of about a few picometers, and therefore much smaller than the apparatus spectral width w app of most LIBS systems. The accuracy of the line width ∆w sd depends thus on the ratio between Stark width w s and apparatus width [START_REF] Taleb | Measurement error due to self-absorption in calibration-free laser-induced breakdown spectroscopy[END_REF]. For strongly Stark-broadened lines, when w s ≥ w app , the line width can be measured. In that case, an accuracy of ∆w sd ≈ 5% can be reached. In the opposite case of weakly Stark-broadened lines, when w s < w app , the line width cannot be accurately measured. It is therefore computed, and ∆w sd is large due to the elevated uncertainties associated to the electron density and the Stark broadening parameter of most transitions [START_REF] Konjević | Experimental Stark widths and shifts for spectral lines of neutral and ionized atoms (A critical review of selected data for the period 1989 through 2000)[END_REF]. A typical value ∆w sd ≥ 25% is expected in that case.

Accuracy lowering due to the uncertainty of plasma size

The plasma size can be deduced from the intensity ratio of spectral lines having significantly different optical thicknesses [START_REF] Gerhard | Quantitative analyses of glass via laser-induced breakdown spectroscopy in argon[END_REF] or from fast plasma imaging [START_REF] Hermann | Evaluation of pressure in a plasma produced by laser ablation of steel[END_REF]. In both cases, the uncertainty of the plasma size is large, and a value of ∆L ≈ 10% is expected in the best case.

In summary, self-absorption lowers the accuracy of the analytical measurement. The loss of accuracy may be moderate if the line width w sd and the plasma size L are precisely known. For resonance lines, w sd is often small and therefore deduced from calculation. In that case, ∆w sd is generally large and the loss of accuracy due to self-absorption is high.

Other sources of uncertainty

The uncertainties associated with plasma temperature, electron density, and the failure of model validity are ignored in the evaluation of the analytical performance by means of Eq. [START_REF] Glasser | Abel inversion applied to plasma spectroscopy -new interactive method[END_REF]. The failure of model validity can be avoided by choosing the appropriate experimental conditions (see Section 2). Temperature and electron density are secondary error sources, as ∆T and ∆n e are resulting from the uncertainties associated with spectroscopic data and experimental parameters [START_REF] Hermann | Calibration-free laser-induced breakdown spectroscopy[END_REF]. Aside from the accuracy improvement related to the primary error sources, ∆T and ∆n e can be minimized statistically by deducing average values from a large number of measured lines. Thus, their influence on the analytical measurement uncertainty is expected to be small compared to the direct influence of the primary error sources expressed in Eq. ( 10).

Upcoming improvements

Although calibration-free laser-induced breakdown spectroscopy has been shown to enable accurate analyt-

Figure 12: Measured and computed spectra of steel displayed for spectral windows in the blue (a) and the green (b) ranges. The blue-labelled transitions were selected to deduce the ratio between measured and computed intensities. (Adapted from Ref. [START_REF] Taleb | Echelle spectrometer calibration by means of laser plasma[END_REF]) ical measurements when it is performed in appropriate experimental conditions [START_REF] Hermann | Analysis of multi-elemental thin films via calibration-free laser-induced breakdown spectroscopy[END_REF][START_REF] Gerhard | Quantification of surface contamination on optical glass via sensitivity-improved calibration-free laser-induced breakdown spectroscopy[END_REF][START_REF] Hermann | Simulation of emission spectra from nonuniform reactive laser-induced plasmas[END_REF], there are still obstacles that hinder its use in industrial applications.

Compensation of the apparatus response

To compare the measured spectrum to emission of a plasma in LTE, the intensity of the recorded spectrum must be corrected by the apparatus response function. The apparatus response is generally measured with radiation standards such as a deuterium arc and a tungsten filament lamp for the ultraviolet and visible/near infrared spectral ranges, respectively. The frequency of such measurements depends on the mechanical stability of the optical system. Calibration-free LIBS is mostly operated with echelle spectrometers to record broadband spectra in which lines of all sample-composing elements can be observed. Echelle spectrometers exploit dispersion of light into two orthogonal directions, and both their spectral and intensity calibration are therefore sensitive to temperature variation [START_REF] Becker-Ross | Echelle-spectrograph as a tool for studies of structured background in flame atomic absorption spectrometry[END_REF]. Thus, a method for rapid checking and correcting the apparatus response R app (λ) was proposed [START_REF] Taleb | Echelle spectrometer calibration by means of laser plasma[END_REF].

Checking R app (λ) using LIBS emission

The method is based on the simulation of the emission spectrum of a plasma produced by laser ablation of steel as shown in Fig. 12. Reliable transition probabilities are available for numerous lines of atomic and ionic iron [START_REF] Kramida | NIST Atomic Spectra Database (version 5.8)[END_REF], and the ratio between measured and computed line intensities can be used to verify the apparatus response as shown in Fig. 13 for hundreds of transitions, distributed all over the recorded emission spectrum. Here, the experimental conditions for the generation of a uniform LTE plasma (see Section 2) are exploited to enable the most accurate simulation of the plasma emission spectrum. When the apparatus response of the measured spectrum was properly compensated, the ratio between measured and computed line intensities is a fluctuating function with an average value equal to unity all over the measured spectral range [see Fig. 13(a)]. In the opposite case, the average value of the intensity ratio varies as a function of wavelength as shown in Fig. 13(b). The wavelength-dependent change of the apparatus response is obtained by dividing the intensity ratio of the uncalibrated apparatus by that of the calibrated one (c). The approximation of the apparatus response change by a cosinus-type function is then used to correct the apparatus response (d). Although the method for rapid checking and correcting the apparatus response presents a progress in calibration-free LIBS, it still requires an initial measurement of R app (λ) with classical radiation standards. The problem with these sources is that their emission spectra should be measured using the same optical path as the one used for the LIBS spectra recording. This is generally difficult or even impossible, in particular when commercial LIBS systems are used. Other drawbacks of classical radiation standards are the limited lifetime, typically of about 50 hours of use or 2 years of storage, and the high costs of their acquisition and calibration. The difficulty associated to the measurement of the apparatus response function presents therefore a major drawback that hinders the industrial development of calibration-free LIBS. To overcome this technological issue, the ideal solution would be to use the laserinduced plasma itself as a radiation standard. This could be achieved by exploiting the continuum radiation that dominates the plasma emission during the early stage of plume expansion. The operator would then be able to measure the apparatus response by simply changing the sample material and setting the appropriate delay of the gated detector.

Need of T -stabilized echelle spectrometer

Echelle spectrometers are extremely sensitive to temperature variation and even small changes of the order of 1 K may alter the spectral calibration and the apparatus response function [START_REF] Becker-Ross | Echelle-spectrograph as a tool for studies of structured background in flame atomic absorption spectrometry[END_REF]. The loss of calibration may even occur when the spectroscopic apparatus is located in a temperature-stabilized laboratory environment, because the heat generated by the detector may cause temperature variation of the optical system [START_REF] Taleb | Echelle spectrometer calibration by means of laser plasma[END_REF]. Calibrationfree LIBS measurements should therefore be performed with T -stabilized echelle spectrometers to ensure highest accuracy.

Need of accurate spectroscopic data

The analytical performance of calibration-free LIBS measurements depends on the quality of the available spectroscopic data. According to Eq. ( 10), accurate Einstein coefficients of spontaneous emission are required for all lines, and Stark broadening parameters must be accurately known for transitions for which selfabsorption is significant.

Available spectroscopic data

Transition probabilities. The Einstein coefficients of spontaneous emission are tabulated in several atomic spectral line databases. Among them, the most complete lists of A ul -values are given by the databases NIST [START_REF] Kramida | NIST Atomic Spectra Database (version 5.8)[END_REF] and Kurucz [START_REF] Smith | Atomic spectral line database built from atomic data files from[END_REF]. The NIST database is regularly updated. Compared to the Kurucz database, less but more reliable data are tabulated by NIST. Thus, A ulvalues are given by NIST for many transitions with their estimated accuracy. The data tabulated by Kurucz have been obtained by atomic structure calculations for which the evaluation of uncertainty is difficult. The A ulvalues are therefore given without any uncertainty interval. Accurate Einstein coefficients are missing for many transitions and for many elements. As an example, the A ul -values of the most intense lines of atomic and ionic calcium are listed in Table 1. Except the resonance transition of neutral calcium, the transition probabilities of lines are reported with an accuracy of 25% or 40%. The situation is simular for many other elements. More accurate data are tabulated for a few elements only. Iron is one of these elements, and the method for checking and correcting the apparatus response (see Section 5.1.1) was therefore implemented with Fe and Fe + transitions.

Stark broadening parameters. Stark broadening parameters of many transitions have been tabulated in the book of Griem [START_REF] Griem | Spectral line broadening of plasmas[END_REF] and in multiple review papers by Konjević and collaborators [START_REF] Konjević | A critical review of the Stark widths and shifts of spectral lines from non-hydrogenic atoms[END_REF][START_REF] Konjević | Experimental Stark widths and shifts for spectral lines of neutral and ionized atoms (A critical review of selected data for the period 1989 through 2000)[END_REF][START_REF] Konjević | Experimental Stark widths and shifts for spectral lines of neutral atoms (A critical review of selected data for the period 1976 to 1982)[END_REF][START_REF] Konjević | Plasma broadening and shifting of non-hydrogenic spectral lines: present status and applications[END_REF][START_REF] Lesage | Experimental Stark widths and shifts for spectral lines of neutral and ionized atoms. A critical review of selected data for the period 2001-2007[END_REF]. The accuracy of these data is low for most tansitions, and estimated to about 20% to 30% in the best case. Data of higher quality can be found for a few spectral lines only. Among them, the hydrogen Balmer alpha transition enables electron density measurements with 10% accuracy in typical conditions of LIBS experiments, where the electron density is of the order of 1×10 17 cm -3 [START_REF] Griem | Stark broadening of the hydrogen Balmer-α line in low and high density plasmas[END_REF][START_REF] Gigosos | Computer simulated Balmer-alpha, -beta, and -gamma Stark line profiles for non-equilibrium plasma diagnostics[END_REF]. The temperature dependence is reported for a few transitions only [START_REF] Cirisan | Stark broadening measurement of Al II lines in a laser-induced plasma[END_REF][START_REF] Gigosos | Computer simulated Balmer-alpha, -beta, and -gamma Stark line profiles for non-equilibrium plasma diagnostics[END_REF][START_REF] Zielinska | Measurement of atomic Stark parameters of many Mn I and Fe I spectral lines using GMAW process[END_REF]. For most transitions, the Stark broadening parameters are reported for given temperature values [START_REF] Konjević | A critical review of the Stark widths and shifts of spectral lines from non-hydrogenic atoms[END_REF][START_REF] Konjević | Experimental Stark widths and shifts for spectral lines of neutral and ionized atoms (A critical review of selected data for the period 1989 through 2000)[END_REF][START_REF] Konjević | Experimental Stark widths and shifts for spectral lines of neutral atoms (A critical review of selected data for the period 1976 to 1982)[END_REF][START_REF] Konjević | Plasma broadening and shifting of non-hydrogenic spectral lines: present status and applications[END_REF][START_REF] Lesage | Experimental Stark widths and shifts for spectral lines of neutral and ionized atoms. A critical review of selected data for the period 2001-2007[END_REF].

Imminent progress of spectroscopic database

The amount and the quality of spectroscopic data are continuously increasing as shown by the regular updates of the NIST database [START_REF] Kramida | NIST Atomic Spectra Database (version 5.8)[END_REF]. Here, measurements using the laser-induced plasma constitue a rising contribution over the past years [START_REF] Aragón | Stark width measurements of Fe II lines with wavelengths in the range 260-300 nm[END_REF][START_REF] Cirisan | Stark broadening measurement of Al II lines in a laser-induced plasma[END_REF][START_REF] Alonso-Medina | A spectroscopic study of laser-induced tinlead plasma: transition probabilities for spectral lines of Sn I[END_REF]. The unique properties of the laser-produced plasma, that under appropriate experimental conditions combines spatial uniformity with local thermodynamic equilibrium (see Section 2), open new perspectives in the completion of spectroscopic databases. Accurate measurements of transitions probabilities and Stark broadening parameters are thus possible, without the requirement of space-resolved spectroscopic observations and complex data analysis that are necessary with other LTE plasma sources [START_REF] Belmonte | Theoretical background of optical emission spectroscopy for analysis of atmospheric pressure plasmas[END_REF][START_REF] Glasser | Abel inversion applied to plasma spectroscopy -new interactive method[END_REF]. Thus, simple space-integrated observations with echelle spectrometers can be used to measure accurately spectroscopic data of numerous transitions at the same time [START_REF] Burger | Stark broadening measurements in plasmas produced by laser ablation of hydrogen containing compounds[END_REF][START_REF] Hermann | Simulation of emission spectra from nonuniform reactive laser-induced plasmas[END_REF]. The potential of such measurements is illustrated by the fluctuation of the intensity ratio between measured and computed lines observed in Fig. 13. Indeed, the standard variation of this fluctuation equals the average uncertainty of the transition probabilities (∆A ul /A ul = 10%...15%). This indicates that A ul -values can be measured with an expected accuracy of 5% or even better.

Conclusion

Calibration-free laser-induced breakdown spectroscopy is a powerful analytical tool, when operated in appropriate experimental conditions. It enables quantitative elemental analyses of the entire material composition, including major, minor and trace elements. The often reported reduced accuracy for the quantification of low-concentration elements is mostly attributed to the LTE condition and the probe volume. Solutions to limit the loss of analytical performance due to both effects are given by a two-step method and indepth measurements presented in Sections 4.1.2 and 4.1.3. Accuracy lowering due to self-absorption depends critically on the knowledge of the line width. Most resonance transitions have line widths smaller than the apparatus spectral width. The line width can be assessed by calculation only, which has low accuracy due to the large uncertainties of Stark broadening parameters and electron density. In that case, the loss of analytical performance due to self-absorption is significant. The principal limitations that still hinder the use of calibration-free LIBS in industrial applications are related to the lack of accurate spectroscopic data and to the uncertainty of the apparatus response function. The further development of calibration-free LIBS analyses depends thus on the solutions to reduce the uncertainties associated to both error sources. Concerning the spectroscopic data, frequent improvements can be observed in literature and by the regular updates of the NIST database, with an increasing contribution of measurements using the plasma produced by laser ablation. The unique properties of the laserproduced plasma, that may combine spatial uniformity with the LTE state, facilitate such measurements, and an acceleration in the progress of spectroscopic databases is expected. To reduce the uncertainty of the apparatus response function, the use of temperature-stabilized echelle spectrometers is mandatory. Morevover, to overcome the difficulties associated to the measurement of the apparatus response with tradiational radiation standards, the development of a radiation standard based on the laser-induced plasma itself would remove the main obstacle that hinders the dissemination of compositional analysis via calibration-free laser-induced breakdown spectroscopy.

Figure 1 :

 1 Figure 1: (a) Image of the plasma produced by UV laser ablation under argon atmosphere. Schematics illustrating (b) the geometry of spectroscopic observation and (c) the radiation flux propagating along the z -axis in a medium of refractive index n.

Figure 3 :

 3 Figure 3: Illustration of the line shape of a strongly Stark-shifted transition emitted from a nonuniform plasma. The superposition of profiles with different widths and shifts results in an asymmetric line shape captured by the spectroscopic apparatus.

Figure 4 :

 4 Figure 4: Ba II 413.06 nm line profile recorded during laser ablation of barite crown glass for t = 500 ns. Comparison to the spectral radiance computed for a uniform LTE plasma. The blue line indicates the resonance wavelength of the transition. (Adapted from Ref. [25])

Figure 5 :

 5 Figure 5: Illustration of the line shape of a strongly self-absorbed resonance line emitted from a nonuniform plasma. The large line profile emitted from the hot core is reabsorbed by the cold border in a narrow spectral range close to the resonance wavelength, according to the reduced electron density. The observed line shape exhibits thus an absorption dip.

Figure 6 :

 6 Figure 6: In I 410.18 nm resonance line recorded during laser ablation of indium for t = 500 ns. The measurements in argon and air are compared to the spectral radiances of a uniform and a nonuniform LTE plasma, respectively. (Adapted from Ref. [25])

Figure 7 :

 7 Figure 7: Measured (black line) and computed (red line) shapes of an Ar transition toward the metastable state. The recording was performed during ablation of steel in argon for t = 500 ns. The simulation was obtained by adding two layers to the laser plasma, representing the heated gas at the vapor-gas contact front and the adjacent colder gas. The absorption dip was observed for each investigated material.

4. 1 .

 1 Accuracy of minor or trace element quantification 4.1.1

Figure 8 :

 8 Figure 8: Schematic illustrating the setting of detector gate delay and width for spectra recording in calibration-free and calibrated LIBS.

Figure 9 :

 9 Figure 9: Time scheme of two-step procedure for sensitivity-enhanced calibration-free LIBS: two spectra are recorded at different times. The early recording in conditions of full LTE serves to quantify major and minor elements. The late recording in conditions of partial LTE serves to measure minor and trace elements. The color scale from green to red indicates the degree of equilibrium. (Adapted from Ref. [65])

Figure 10 :

 10 Figure 10: Trace element fractions of heavy flint glass (Schott, SF5) versus depth measured via depth-resolved calibration-free LIBS. (Adapted from Ref. [29])

Figure 13 :

 13 Figure 13: Ratio between measured and computed intensities R i = I meas /I comp of spectral lines for spectra recordings (a) directly after the intensity calibration with radiation standards (day d 0 ), and (b) 18 months later (day d 1 ). (c) Change of apparatus response obtained by dividing the later R i -value by the previous one. (d) Apparatus response measured on day d 0 -1 with calibration standards (black lines) and corrected apparatus response obtained for day d 1 . The measurements were performed for UV and VIS/NIR ranges of an echelle spectrometer (LTB: model Aryelle Butterfly). (Adapted from Ref. [58])

Table 1 :

 1 Wavelength λ, transition probability A ul with accuracy ∆A ul (B+ := 10%; C := 25%; D := 40%), energies E of lower (index l) and upper (index u) levels of the most intense transitions of calcium atoms and ions according to NIST[START_REF] Kramida | NIST Atomic Spectra Database (version 5.8)[END_REF]. λ (nm) A ul (µs -1 ) ∆A ul E l (cm -1 ) E u (cm -1 )

	Ca II 315.883	310	C	25 191.51 56 839.25
	Ca II 317.933	360	C	25 414.40 56 858.46
	Ca II 318.127	58	C	25 414.40 56 839.25
	Ca II 370.602	88	C	25 191.51 52 166.93
	Ca II 373.690	170	C	25 414.40 52 166.93
	Ca II 393.366	147	C	0.00 25 414.40
	Ca II 396.846	140	C	0.00 25 191.51
	Ca II 409.709	9.9	D	60 533.02 84 933.65
	Ca II 410.981	12	D	60 611.28 84 936.41
	Ca II 422.007	8.5	D	60 611.28 84 300.89
	Ca I	422.672	218	B+	0.00 23 652.304
	Ca II 500.147	20	D	60 533.02 80 521.53
	Ca II 501.997	23	D	60 611.28 80 526.16
	Ca II 528.526	7.8	D	60 533.02 79 448.28
	Ca II 530.722	15	D	60 611.28 79 448.28

Preprint submitted to Spectrochimica Acta Part B December 14, 2022

Acknowledgements

The research leading to these results has received funding from European Union's Horizon 2020 research and innovation programme under grant agreement No. 284464 and 654148 Laserlab-Europe (projects CNRS-LP3 002148, 002491 and 002585).