Christophe Darras
email: darras@univ-corse.fr

G Bastien

M Muselli
email: marc.muselli@univ-corse.fr

P Poggi

B Champel

P Serre-Combe
email: pierre.serre-combe@cea.fr

Techno-economical analysis of PV/H 2 system

Keywords: Renewable energy, Photovoltaic hybrid system, Fuel cell, Electrolyzer, Energy cost

The purpose of this paper is to analyze the costs of PV/H 2 systems connected to the grid and sized to satisfy a power setpoint, using the ORIENTE® software. The aim is todetermine the optimal components' size to minimize grid injection cost.

We study the case of guaranteed PV production, where the PV array is coupled to a hydrogen chain in order to be able to inject a constant power setpoint to the grid from 8am to 5pm (local time) all year long without failure (100 % level of satisfaction). Four power setpoints are studied: 100, 200, 300 and 400 kW.

The results show that regardless tothe power setpoint, the optimum system to be installed complies with the following ratios: installed PV power / power setpoint = 2.4-2.5, Electrolyzer nominal power / power setpoint = 0.72-0.75 and Fuel cell nominal power / power setpoint = 1. We have also found that the capacity of the H 2 tanks corresponds to around 48-50days of autonomy (number of days allowingonlythe fuel cells to fully supply the power setpoint).

Introduction

In times of global economical crisis, we will have to face one of the biggest technology shifts of the modern era. Indeed, in a world where energy demands continue to increase, the current generation means are problematic. Fossil fuels (main primary source [START_REF]Renewable Energy Policy Network for the 21st Century (Ren21). 2011-Global Status Report[END_REF]) pollute and are running out, affecting the health of the planet and its inhabitants [START_REF]GEO-4 report[END_REF][START_REF] Schlink | Statistical models to assess the health effects and to forecast ground-level ozone[END_REF]. Concerning nuclear power, although not polluting during generation, it generates radioactive wastes that are difficult to manage, and the confidence in the nuclear sector is shrinking with each new disaster, such as recently in Fukushima. So it seems necessary to find other ways to produce energy we need.

Inexhaustible and non-polluting (during the production), renewable energy seems to be the ideal solution. However, there are still some problems to be solved such as intermittent generation, the fit between production and consumption [START_REF] Ahmed | Power fluctuations suppressions of stand-alone hybrid generation combining solar photovoltaic/wind turbine and fuel cell systems[END_REF], or cost due to their young technological [START_REF] Rissanen | Simulation for Technical and Economic Analysis of a Hybrid-Fuel-Cell System[END_REF]. The research community is working actively on resolving these problems.

The intermittence and the lack of adequacy between consumption and power production of renewable energies, limit their integration into the grids (30 % in France for example [6]), as it threatens their stability [START_REF] Molina | Stabilization and control of tie-line power flow of microgrid including wind generation by distributed energy storage[END_REF]. Storage is one of the solutions to settle these problems. That is why many studies have been conducted to see if it is technically and economically possible to store electrical energy from renewable energy sources [START_REF] Kaldellis | Techno economic comparison of energy storage systems for island autonomous electrical networks[END_REF][START_REF] Zoulias | Techno economic analysis of the integration of hydrogen energy technologies in renewable energy based stand-alone power systems[END_REF][START_REF] Shabani | Energy and cost analysis of a solar hydrogen combined heat and power system for remote power supply using a computer simulation[END_REF][START_REF] Nelson | Unit sizing and cost analysis of stand-alone hybrid wind/PV/fuel cell power generation systems[END_REF][START_REF] Dufo-Lopez | Optimization of control strategies for standalone renewable energy systems with hydrogen storage[END_REF][START_REF] Jallouli | Sizing, techno-economic and generation management analysis of a stand alone photovoltaic power unit including storage devices[END_REF][START_REF] Budischak | Cost-minimized combinations of wind power, solar power and electrochemical storage, powering the grid up to 99.9% of the time[END_REF][START_REF] Andrews | Dimensionless analysis of the global techno-economic feasibility of solarhydrogen systems for constant year-round power supply[END_REF][START_REF] Tzamalis | Techno-economic analysis of RES & hydrogen technologies integration in remote island power system[END_REF][START_REF] Paola | Techno-economic optimisation of hydrogen production by PV e electrolysis: "RenHydrogen" simulation program[END_REF][START_REF] Bernal-Agustin | Techno-economical optimization of the production of hydrogen from PV-Wind systems connected to the electrical grid[END_REF][START_REF] Genç | Cost analysis of wind-electrolyzer-fuel cell system for energy demand in Pınarbası-Kayseri[END_REF]. J.K. Kaldellis et al. [START_REF] Kaldellis | Techno economic comparison of energy storage systems for island autonomous electrical networks[END_REF] haveconducted a techno-economic comparison of different means of energy storage (water pumping, compressed air, batteries, FC, ...) in insular conditions, on a real application: Greek Islands (Aegean Islands Archipelago). Due to the diversity in the sizes of these islands, different cases (from very small electrical networks to big island electrical networks) have been studied. These cases lead to different conclusions, but it appears from this study that with an existing electricity generation cost which is very expensive (due to the insular situation), renewable energies become profitable, and allow reducing the use of thermal power plants. E.I. Zoulias et al. [START_REF] Zoulias | Techno economic analysis of the integration of hydrogen energy technologies in renewable energy based stand-alone power systems[END_REF] haveconducted a techno-economical study with HOMER® [20] on Kythnos platform (8.8 kWp of PV array), by comparing a PV/diesel system and a PV/H 2 system. They concluded that the PV/H 2 system will be profitable by 2020 (assuming a hypothetical reduction of H 2 storage costs). M. Rissanen et al. [START_REF] Rissanen | Simulation for Technical and Economic Analysis of a Hybrid-Fuel-Cell System[END_REF] have developed a software for sizing and estimating the cost of heat and electricity generation from PV/biogas/FC in a small office building (300 m²). They concluded that this technology is currently not competitive but different scenario consideredabout future costs can change that.

B. Shabani et al. [START_REF] Shabani | Energy and cost analysis of a solar hydrogen combined heat and power system for remote power supply using a computer simulation[END_REF] have sought to optimize by a techno-economical way, a PV/H 2 plant combined with heat recovery, with a program on Visual Pascal (Delphi). The simulation was tested under the conditions of Melbourne (Australia), with a load amounting 5 kWh/day. In addition to the significant gain in efficiency by recovering the heat generated by the FC, this optimization shows interesting results. The fuel cell in the system is conventionally sized to meet the peak of the load profile. However, an economic analysis illustrates that installing a larger fuel cell could lead to up to a 15 % reduction in the unit cost of the electricity to an average of just below 90 c€/kWh over the assessment period of 30 years. This result is explained by the fuel cell yield curve, which exhibits the highest efficiency at an intermediate operating point. D.B. Nelson et al. [START_REF] Nelson | Unit sizing and cost analysis of stand-alone hybrid wind/PV/fuel cell power generation systems[END_REF] have developed a Matlab® program to design the components and optimize the cost of electricity produced by a wind turbine/PV/H 2 system. This study was conducted at a remote site: a house in the northeast Pacific. They then compared this system to a battery storage system, and found that the batteries were, at present, more profitable due to the performance of the FC.

Several papers have studied the complementarity of battery and H 2 chain (see for example [START_REF] Dufo-Lopez | Optimization of control strategies for standalone renewable energy systems with hydrogen storage[END_REF] or [START_REF] Jallouli | Sizing, techno-economic and generation management analysis of a stand alone photovoltaic power unit including storage devices[END_REF]), and have shown that the combination of both storage means, coupled to an optimized strategy determining which storage technology to activate at each time is interesting from an economic point of view.

C. Budischaka et al. [START_REF] Budischak | Cost-minimized combinations of wind power, solar power and electrochemical storage, powering the grid up to 99.9% of the time[END_REF] have performed an optimization to minimize the total cost to power a large grid only from renewable energies, and see if it is technically and economically viable. The selected grid is PJM (Eastern U.S.), which represents one fifth of the United States (72 GW of generation, with an average load of 31.5 GW). The calculations were performed for a period of four years. The program has tested 28 billion combinations, and it appears that on the one hand, it seems more interesting to oversize generation capacity (mostly wind) rather than using storage means and on the other hand, renewable energies will be cheaper than fossil fuels by 2030. J. Andrews et al. [START_REF] Andrews | Dimensionless analysis of the global techno-economic feasibility of solarhydrogen systems for constant year-round power supply[END_REF] have conducted a dimensionless study (allowing easy adaptation to different situations) that gives an indicative evaluation of the economic viability of adding an hydrogen storage to a photovoltaic-based solar supply, either for a large-scale grid or small scale autonomous application. The study was applied to 78 big cities on five continents and in many different latitudes.It appears that the storage of H 2 could be profitable in half of them (all of them if prices fell). The places where H 2 technologies are less profitable are the equatorial areas where the sun is almost constant during the year.

In this context of research, we have investigated [START_REF] Darras | Modélisation de systèmes hybrides Photovoltaïque/Hydrogène: Applications site isolé, microréseau, et connexion au réseau[END_REF][START_REF] Darras | PV output power fluctuations smoothing: The MYRTE platform experience[END_REF][START_REF] Darras | Sizing of photovoltaic system coupled with hydrogen/oxygen storage based on the ORIENTE model[END_REF] the ability for hydrogen technologies coupled to renewable primary energy sources (PV, wind turbine,…) to smooth their fluctuating production and guarantee power injection level to the grid.

For that purpose, a software called ORIENTE® (Optimization of Renewable Intermittent Energies with Hydrogen for Autonomous Electrification, programmed with Matlab®) has been developed to model systems [START_REF] Darras | Modélisation de systèmes hybrides Photovoltaïque/Hydrogène: Applications site isolé, microréseau, et connexion au réseau[END_REF][START_REF] Darras | PV output power fluctuations smoothing: The MYRTE platform experience[END_REF][START_REF] Darras | Sizing of photovoltaic system coupled with hydrogen/oxygen storage based on the ORIENTE model[END_REF][START_REF] Darras | Dépôt UDC/CNRS sous le régime des droits d'auteurs de logiciel Propriété intellectuelle[END_REF] coupling a PV array (solar panels and electric inverters) to a H 2 chain (electrolyzer -H 2 , O 2 and H 2 O tanks -and fuel cell). This software allows simulating the operation of such systems, and optimizing their sizing and operating strategies based on technical considerations.

For the present study, we have developed an add-on to ORIENTE®. It allows performing a techno-economical analysis in order to assess the optimal sizes of PV/H 2 system components and the cost of electricity generation for a PV-guarantee application.

The operating of the PV/H 2 system we study is to inject a constant power to the grid from 8am to 5pm (local time) all year long without failure (100 % level of satisfaction: the power setpoint is always satisfied by the PV and or by the fuel cell, equivalent to LLP (loss of load probability) = 0 for a non-connected system to the grid). Four power setpoints are studied: 100, 200, 300 and 400 kW. We assess the optimal size of the sub-systems in order to minimize the grid injection cost. The system is located to Ajaccio in Corsica island (France).

The paper is organized as follows. Section 2 details the contents of the techno-economical model. Then, Section 3 presentsthe parameters used for the simulations. Section 4 describes the results and their explanations. Finally, Section 5 concludes our study.

Techno-economical model description

We have developed a techno-economical optimization algorithm (Figure 1), using nested loops.

The user defines a start and a final installed PV power, and for each value of this power, the program determines the most economical combination of other subsystems (inverters, electrolyzer, gas storage and fuel cell), which can supply the power setpoint. For this the user also defines the intervals and the steps in which the search is carried out to the size of the subsystems. Theses values are specified in Section 3.

Figure 1

Technical section

Concerning the technical part of the program, it consists in the calculation of the different flows (electric power, gas production and consumption,...). It is based on ORIENTE® which is described in previous paper [START_REF] Darras | Modélisation de systèmes hybrides Photovoltaïque/Hydrogène: Applications site isolé, microréseau, et connexion au réseau[END_REF][START_REF] Darras | PV output power fluctuations smoothing: The MYRTE platform experience[END_REF][START_REF] Darras | Sizing of photovoltaic system coupled with hydrogen/oxygen storage based on the ORIENTE model[END_REF][START_REF] Darras | Dépôt UDC/CNRS sous le régime des droits d'auteurs de logiciel Propriété intellectuelle[END_REF]. In this part of the program, the system is divided in five subsystems: PV, inverters, electrolyzer, gas storage and fuel cell.

From meteorological data (Gi: solar irradiation and T A : ambient temperature; hourly time-step on two years) and power setpoint (P Power_Setpoint : powerto send to grid), ORIENTE® calculates the flow of power and gas using technical parameters of the different elements of the system.

The management (step (t) by step until t Max ) of power flow of the simulation is as follows (see Figure 2):



The PV production (P PV ) supplies in priority the grid (P Power_Setpoint ). In the case of an excess of PV production (P Excess ), of course taking into account the electrolyzer power threshold (P Threshold_EL ) and the electrolyzer nominal power (P Nominal_EL ), it creates H 2 (Prod H2 ) via electrolysis (P EL ). The H 2 quantity (Q H2 ) is also checked to avoid exceeding the H 2 tanks size (Q H2_Max ). It is sometimes necessary to degrade the PV production to obtain the necessary production. For this, the MPPT (Maximum power point tracking) inverters is used. It requires the inverters to operate at a lower efficiency.



When the PV production is not able to satisfy the power setpoint (P Missing ), the fuel cell (P FC )

supplies the complementary power. The fuel cell consumes for this H 2 (Cons H2 ). If the missing power is lower than the fuel cell power threshold (P Thershold_FC ), the PV production is degrades.Thus the system is not faulty and the fuel cell is operating at its threshold. If the missing power is higher than the fuel cell nominal power (P Nominal_FC ), the system is faulty.

Figure 2

The size of the H 2 tanks is not fixed. For against, the H 2 quantity at the start of the simulation is fixed (Q H2_Max = 10 9 Nm 3 ). This value is used high threshold. To know the size of the H 2 tanks necessary to satisfy the power setpoint, simply subtract the value of this threshold with the H 2 quantity lowest during the simulation.

Each group of subsystem obtained during the optimization satisfies the power setpoint on 2 years. However, wheninstalled PV power is low, the system is not renewable. The H 2 quantity early and end of the 2 years is not the same (Figure 5). We will see again later that the optimum appears from the moment where the system is renewable.

Threshold of the fuel cell and of the electrolyzer are fixed to 5 % of their respective nominal power (fixed during the optimization).The inverter efficiency curve as function of the inverter nominal power is represented Figure 3 (left abscissa). It is found that when inverters operate at more than 50%

of their nominal power, efficiency is higher than 90%. The curve of production (electrolyzer) and consumption (fuel cell) of H 2 as function of respectively nominal power of the electrolyzer and the fuel cell is represented Figure 3 (right abscissa).

Economical section

The economical model is based on the principle of net present value. The costs and revenues will be discounted and summed to obtain the net present value of the project [START_REF] Jallouli | Sizing, techno-economic and generation management analysis of a stand alone photovoltaic power unit including storage devices[END_REF]. The simulation method is the following: computation of the required sale price of electricity to achieve a net present value of zero for the lifetime of the project. This parameter is noted S€llPr and is calculated as follows:

𝑆€𝑙𝑙𝑃𝑟 𝑖 = - 𝐷𝑅𝑉 𝑖 8 𝑖=1 + 𝐶_𝐼𝑁𝑉 + 𝐷𝐶_𝑅𝑒𝑝 𝑖 8 𝑖=1 + 𝐷𝐶_𝑂𝑀 𝑖 8 𝑖=1 + 𝐷𝐶_𝑆𝑒𝑐 𝑃𝑟𝑜𝑑_𝑒𝑙𝑒𝑐 • 1+𝐼𝑅𝑒𝑙𝑒𝑐 𝑗 1+𝐷𝑅 𝑗 𝐿𝑇𝑃 𝑗 =1 (1)
Table 1 lists the economical parameters used in the equation 1.

Table 1

The system is divided into 8 subsystems: the land (i will be equal to 1 for this element and so on for the other subsystems), the photovoltaic array (i = 2), the inverters (i = 3), the electrolyzer (i = 4), gas storage (i = 5), the fuel cell (i = 6), infrastructure and manpower (i = 7) and finally the high voltage transformation post (i = 8).

The three sums in the numerator of the equation ( 1) are calculated as follows:

𝐷𝑅𝑉 𝑖 = 𝐶(𝑖) • 1 -𝑁𝑅(𝑖) • 𝐿𝑇(𝑖) 𝐿𝑇𝑃 • 1 + 𝐼𝑅(𝑖) 𝑇𝐼𝐿(𝑖) • 1 + 𝐼𝑅𝑔𝑒𝑛 𝐿𝑇𝑃-𝑇𝐼𝐿(𝑖) 1 + 𝐷𝑅 𝐿𝑇𝑃 (2)
𝐷𝐶_𝑅𝑒𝑝(𝑖) = 𝐶 𝑖 • 1 + 𝐼𝑅 𝑖 𝑗 •𝐿𝑇 𝑖 1 + 𝐷𝑅 𝑗 •𝐿𝑇 𝑖 𝑁𝑅𝑆 𝑖 𝑗 =1 + 1 + 𝐼𝑅 𝑖 𝑇𝐼𝐿 𝑖 + 1 + 𝐼𝑅𝑔𝑒𝑛 𝑗 •𝐿𝑇 𝑖 -𝑇𝐼𝐿 𝑖 1 + 𝐷𝑅 𝑗 •𝐿𝑇 𝑖 𝑁𝑅 𝑖 𝑗 =𝑁𝑅𝑆 𝑖 +1 (3)
𝐷𝐶_𝑂𝑀 𝑖 = 𝐶_𝑂𝑀(𝑖) + 𝐶𝑜𝑛𝑠(𝑖) • 1 + 𝐼𝑅𝑔𝑒𝑛 𝑗 1 + 𝐷𝑅 𝑗 𝐿𝑇𝑃 𝑗 =1 (4)
For this, we need to determine the following six parameters (where int represents the integer part):

𝐶 𝑖 = 𝐶_𝑢𝑛𝑖𝑡(𝑖) • 𝑁𝑢𝑚𝑏(𝑖) (5)
𝑁𝑅 𝑖 = 𝑖𝑛𝑡 𝐿𝑇𝑃 𝐿𝑇(𝑖) (6)
𝑇𝐼𝐿 𝑖 = 𝑙𝑜𝑔 1 + 𝐼𝐿(𝑖) 𝑙𝑜𝑔 1 + 𝐼𝑅(𝑖) (7)
𝑁𝑅𝑆 𝑖 = 𝑖𝑛𝑡 𝑇𝐼𝐿(𝑖) 𝐿𝑇(𝑖)

𝐶_𝑂𝑀(𝑖) = 𝐶 𝑖 • 𝐶_𝑂𝑀_𝑢𝑛𝑖𝑡(𝑖) (8)
𝐶𝑜𝑛𝑠 𝑖 = 𝑁𝑢𝑚𝑏(𝑖) • 𝐶𝑜𝑛𝑠_𝑒𝑙𝑒𝑐𝑡(𝑖) + 𝐶𝑜𝑛𝑠_𝑤𝑎𝑡(𝑖)

Table 2 lists the economical parameters used in the equations 2 to 10.

Table 2

The two parameters in the numerator of the equation ( 1) are calculated as follows:

𝐶 𝐼𝑁𝑉 = 1 + 𝑖𝑛𝑠𝑡 • 𝐶 𝑖 8 𝑖=1 + 𝑆𝑎𝑓𝑒_𝑆𝑡𝑢𝑑𝑦 + 𝑇𝐺𝐴𝑃_𝑎𝑢𝑡 (11)
𝐷𝐶_𝑆𝑒𝑐 = 𝐶𝑜𝑛𝑠𝐹 + 𝐶_𝐼𝑛𝑒𝑟𝑡 + 𝐶_𝑆𝑡𝑎𝑓𝑓_𝑡𝑟𝑎𝑖𝑛 + 𝐶_𝑇𝐺𝐴𝑃_𝑎𝑛𝑛 • 1 + 𝐼𝑅𝑔𝑒𝑛 𝑗 1 + 𝐷𝑅 𝑗 𝐿𝑇𝑃 𝑗 =1 (12)
For this, we need to determine the following four parameters:

𝐶𝑜𝑛𝑠𝐹 = 𝐶𝑜𝑛𝑠𝐹_𝑒𝑙𝑒𝑐 + 𝐶𝑜𝑛𝑠𝐹_𝑊𝑎𝑡 (13)
𝐶_𝑖𝑛𝑒𝑟𝑡 = 𝐼𝑛𝑒𝑟𝑡 • 𝑁𝑢𝑚𝑏(5) (14)
𝐶_𝑆𝑡𝑎𝑓𝑓_𝑡𝑟𝑎𝑖𝑛 = 𝑆𝑡𝑎𝑓𝑓_𝑡𝑟𝑎𝑖𝑛 • 𝑁𝑢𝑚𝑏(2) (15)
𝐶_𝑇𝐺𝐴𝑃_𝑎𝑛𝑛 = 𝐶𝑜𝑒𝑓𝑓_𝑇𝐺𝐴𝑃 • 𝑇𝐺𝐴𝑃_𝑎𝑛𝑛 (16)
Table 3 lists the economical parameters used in the equations 11 to 16. The model also takes into account the specific hydrogen safety requirement in France (see parameters highlighted in grey in this table) [START_REF] Dubois | Safety cost of a large scale hydrogen system for photovoltaic energy regulation[END_REF].

Table 3

The choice of the different parameters has been done by using recent years works [START_REF] Kaldellis | Techno economic comparison of energy storage systems for island autonomous electrical networks[END_REF][START_REF] Zoulias | Techno economic analysis of the integration of hydrogen energy technologies in renewable energy based stand-alone power systems[END_REF][START_REF] Shabani | Energy and cost analysis of a solar hydrogen combined heat and power system for remote power supply using a computer simulation[END_REF][START_REF] Nelson | Unit sizing and cost analysis of stand-alone hybrid wind/PV/fuel cell power generation systems[END_REF][START_REF] Dufo-Lopez | Optimization of control strategies for standalone renewable energy systems with hydrogen storage[END_REF][START_REF] Jallouli | Sizing, techno-economic and generation management analysis of a stand alone photovoltaic power unit including storage devices[END_REF][START_REF] Budischak | Cost-minimized combinations of wind power, solar power and electrochemical storage, powering the grid up to 99.9% of the time[END_REF][START_REF] Andrews | Dimensionless analysis of the global techno-economic feasibility of solarhydrogen systems for constant year-round power supply[END_REF][START_REF] Tzamalis | Techno-economic analysis of RES & hydrogen technologies integration in remote island power system[END_REF][START_REF] Paola | Techno-economic optimisation of hydrogen production by PV e electrolysis: "RenHydrogen" simulation program[END_REF][START_REF] Bernal-Agustin | Techno-economical optimization of the production of hydrogen from PV-Wind systems connected to the electrical grid[END_REF][START_REF] Genç | Cost analysis of wind-electrolyzer-fuel cell system for energy demand in Pınarbası-Kayseri[END_REF][START_REF] Dubois | Safety cost of a large scale hydrogen system for photovoltaic energy regulation[END_REF],

and we have used the most representative values.

As our program aims at testing different combinations of parameters, it was necessary to define unit costs and to include links with the varying parameters.

The variables Prod_elecand Coeff_TGAP are respectively calculated in the technical part and according to the French regulation [START_REF] Dubois | Safety cost of a large scale hydrogen system for photovoltaic energy regulation[END_REF].

After having explained the theoretical background, we will now present the simulations performed.

Simulation parameters

In order to control more easily, for the grid manager, the balance between supply and demand, it is necessary for it to know the power that will be sent over the grid by the RES type power generation

Table4-b

As can be seen in the table4, we assume a decrease in the investment cost of the PV, the inverters, the electrolyser, the Fuel Cell and the gas storage, due to the learning curve.

Results and discussion

Now, we have to observe and analyse the results obtained from our simulations.

Figure 4

Figure 4 shows, for the different power setpoints, the evolution of the required sale price of electricity to achieve a zero NPV, as a function of the ratio installed PV power / power setpoint. We separate the figure 4 into two parts, an overview (Figure 4 (A)) and zoom (Figure 4 (B)) on the most interesting part. The results are close whatever the power setpoint and the minimum sale price is obtained for a ratio between 2.4 and 2.5.



When installed PV power is low (ratio < 2), the electrolyser rarely operates (for ratios lower than 1, there is even no electrolysis (see figure 6-a and 6-b) since it does no thave excess power). On the contrary, the fuel cell must constantly provide additional power to supply the setpoint. As a consequence, the required H 2 gas tank size is huge and H 2 content in the storage decreases throughout the simulation (example Figure 5,'__'). The cost for this sizing is very high.



Increasing the installed PV power (up to 2.4-2.5 ratio) reduces the cost. Indeed, the PV power is large enough to supply the majority of setpoint and the electrolyzer (see figure 6-a and 6-b). The fuel cells is less operating which reduced the H 2 gas tank size (example Figure 5, '---').

 Beyond the 2.4-2.5 ratio, an increase of PV power only increases the cost. Indeed, much of the PV power is not used by the system (degraded power), it is limited by the inverter (see figure 6a and 6-b). It provides more power to the electrolyzer and demand less of the fuel cell, which has the effect of regularly cap gas tanks to their maximum (example Figure 5,'…').

Figure 5

Figure 5 shows the H 2 quantity evolution for the ratio 1, 2.5 and 4 of the simulation 1, as a function of time. The behaviour of these curves is the same when the power setpoint is 100 kW, 200 kW, 300 kW or 400 kW.

 '__': The H 2 quantity steadily decreases, H 2 is consumed but not produced.Indeed, the installed PV power is equal to the power setpoint (ratio = 1) therefore in the best case the PV production is equal to the power setpoint, it is no H 2 production (see figure 6-a and 6-b).

 '---':

The production and consumption of H 2 during the whole simulation are almost balanced.

The sizes of the PV, the inverter, the electrolyzer and the fuel cell are therefore adequatelysizedcompared to the power setpoint. The system is said to be renewable since there is as much H 2 at the beginning and the end of the simulation.

 '…':

The H 2 tanks level is regularly atits maximum, which reduces the profitability of the system (Figure 4). The H2 production is much greater than H 2 consumption, the size of tank is limited, so it has a lot of H 2 losses.  When the ratio is less than 2, inverter size increases when increasing PV power. The electrolyzer power increases when the ratio is greater than 1, otherwise it is zero since there isnot PV production excess.



When the ratio is greater than 2, the size of the inverter and the electrolyzer increases much less with increasing PV power. Is imposed as part of the optimization, a linear increase of the installed PV power. Nevertheless, technical-economic calculations limit the size of the inverter and the electrolyzer. Indeed, although the excess PV production increases, it is not economically viable to store it.

We can also see that the optimal power of the fuel cell is always equal to the power setpoint.

Indeed a lower fuel cell power could generatea system failure (level of satisfaction inferior to 100%) in the absence of sunlight. The optimization doesnot choose fuel cell power greater than the power setpoint.The potential performance gain with greater fuel cell nominal power doesnot generate sufficient gain on storage (less consumption for the same power supply, therefore a smaller and cheaper tank).

Figure 7

Figure 7 shows, for the different power setpoints, the share of PV cost in the total cost, as a function of the ratio installed PV power / power setpoint. We separate the Figure 7 into two parts, an overview (Figure 7 (A)) and zoom (Figure 7 (B)) on the most interesting part. The results are close whatever the power setpoint.

The share of PV (include the inverters) in the total cost is logically 0 % when PV is not present (ratio zero) and reaches 24% for a 5-fold ratio. At the optimal PV size, this share is between 16% and 17%.

Figure 8

Figure 8 shows, for the different power setpoints, the share of H 2 chain cost (include electrolyzer, fuel cell and gas storage) in the total cost, as a function of the ratio installed PV power / power setpoint. We separate the figure 8 into two parts, an overview (Figure 8 (A)) and zoom (Figure 8 (B)) on the most interesting part. The results are close whatever the setpoint.

The share of H 2 chain cost in the total cost is logically high (52 %) for low capacities of PV, and decreases down to 10 % for a 5-fold PV/setpoint ratio, with a value between 25% and 26% for the optimal PV size.

Figure 9

Figure 9 shows, for the different power setpoints, the share of the remaining cost in the total cost (includeland, infrastructure and manpower), as a function of the ratio installed PV power / power setpoint. We separate the figure 9 into two parts, an overview (Figure 9 (A)) and zoom (Figure 9 (A))

on the most interesting part. The results are close whatever the setpoint.

The share of the remaining cost in the total cost increases from 48 % when a zero PV/setpoint ratio to 67 % for a 5-fold ratio. At the optimal installation size, this share is between 56.5% and 58%.

Tabs. 5 and 6 summarize,for the different power setpoints, the optimization results of the system. Table5presents the optimal sizes of the subsystems and table6 gives the annual energies ratio and the system cost.

Table5 Table6

We can see in table 6, that whatever the power setpoint, the optimum system to be installed complies with the following ratios: installed PV power / power setpoint = 2.4-2.5,inverter power / power setpoint = 1.8-1.9 and electrolyzer nominal power / power setpoint = 0.72-0.75. We can also see that the fuel cells optimal power is equal to the power setpoint. Indeed the level of satisfaction being 100%, the electricity injected to the grid in non-sunny days must be full provided by the only fuel cells. One can note that the size of the converters is always lower than the PV power. A portion of the PV production is lost (degraded with MPPT inverters), but the system sizing is economically more interesting.This degraded PV power being quantified it will be used for a comparison in cost in Table 6 (cost without or with selling degraded PV production).Indeed, it normally sends on the grid only the power corresponding to the setpoint, however, it will look at the impact on the cost if the power was not degraded but sent surcharge on the grid.

We have also found that the capacity of the H 2 tanks corresponds to around 48-50 days of autonomy (number of days to allow the only fuel cells to fully supply the power setpoint profile).For example, for simulation 1, the fuel cell consumes 40 Nm 3 .h -1 to supply 100 kWh. If the fuel cell is working alone on the day (during 9 hours), it must provide 900 kWh, which requires 320 Nm 3 of H 2 .

Simulation 1 gives a size of tank of 17248 Nm 3 is 48 days of autonomous.

We can see in table 6, that whatever the power setpoint, around 77% of the energy is guaranteed by the PV and the remaining around 23% are supplied by the fuel cell. We also observed similar results in the distribution of PV production: 66-68% PV to power setpoint, 29-30% PV to electrolyzer and 1-3% degraded PV (not used). It is observed that the PV production is mainly used in the direct supplying of the power setpoint in order to limit the use of the H 2 chain and losses. The cost without selling degraded PV production ranges from 1.50 €/kWh to 1.60 €/kWh and ranges from 1.50 €/kWh to 1.56 €/kWh with selling degraded PV production. The sale of the degraded PV power reduces the cost of kWh around 2-4%. … Table 5: Optimal sizes of the subsystems according to the power setpoint Table 6: Ratio of annual energies from the different subsystems and system cost to the power setpoint

Conclusion

…

Figures and tables captions:

Figure 6 Figures 6 -

 66 Figure6-a

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1 : Techno-economical optimization algorithm

Figure 4 :Figure 5 :Figure 7 :

 457 Figure 4: Required sale price of electricity (€/kWh) to achieve zero NPV, as a function of the ratio

Figure 8 :

 8 Figure 8: Share of H 2 chain cost in the total cost, as a function of the ratio installed PV power / power setpoint; --: Area having the optimum point

Figure 9 :

 9 Figure 9: Share of remaining cost in the total cost, as a function of the ratio installed PV power /

(PV array, wind turbine,…). This will also increase the rate of integration of sources on the grid. In France, the solutions proposed by the CRE (French energy regulation commission) to resolve this problem are stipulated in the call for tender of the PV energy and the WT energy [26]. These solutions consist globally in coupling the renewable energies with a storage method and a meteorological data prediction method. The power sent by this system on the grid is also subject to certain constraint. For simple summary, these profiles must have a trapeze appearance, which is a power upslope, a tray with constant power, and finally a power downslope. The power producer must adhere to profile (whose power is bought by the grid Manager) on pain of financial penalty. For our study we will study four power setpoint profiles for which we will limit ourselves to the tray.

The annual power setpoint is as follows:



From 8am to 5pm: constant at 100 kW (simulation 1), constant at 200 kW (simulation 2), constant at 300 kW (simulation 3), constant at 400 kW (simulation 4).



From 0am to 8am & from 5pm to 12pm: constant at 0 kW.

Meteorological data were measured at the site of the MYRTE platform in Corsica. We used two years of data at one hourly time-step, and restricted the data to the daily period (8am to 5pm each day).This choice of restriction is due to the fact that we do not consider the phases of power upslope and power downslope in the profile to be sent tothe grid.

The intervals and the steps used in the techno-economical optimization algorithm are the following:

 PV: from 0 to 5 times the power setpoint by step of 10 kWp; Table4-a


Robert Kammel

Klaus Bergner

Jens U Thomas

Roland Ackermann

Stefan Skupin

Stefan Nolte

Simultaneous spatial and temporal focusing: A route towards confined nonlinear materials processing

Keywords: Simultaneous spatial and temporal focusing, femtosecond laser materials processing, laser-induced optical breakdown, Kerr self-focusing, filamentation, femtosecond laser surgery

Ultrashort pulse lasers enable reliable and versatile high precision ablation and surface processing of various materials such as metals, polymers and semiconductors. However, when modifications deep inside the bulk of transparent media are required, nonlinear pulse material interactions can decrease the precision, since weak focusing and the long propagation of the intense pulses within the nonlinear media may induce Kerr self-focusing, filamentation and white light generation. In order to improve the precision of those modifications, simultaneous spatial and temporal focusing (SSTF) allows to reduce detrimental nonlinear interactions, because the ultrashort pulse duration is only obtained at the focus, while outside of the focal region the continuously increasing pulse duration strongly reduces the pulse intensity. In this paper, we review the fundamental concepts of this technology and provide an overview of its applications for purposes of multiphoton microscopy and laser materials processing. Moreover, numerical simulations on the nonlinear pulse propagation within transparent media illustrate the linear and nonlinear pulse propagation, highlighting the differences between conventional focusing and SSTF. Finally, fs-laser induced modifications in gelatine are presented to compare nonlinear side-effects caused by conventional focusing and SSTF. With conventional focusing the complex interplay of self-focusing and filamentation induces strongly inhomogeneous, elongated disruptions. In contrast, disruptions induced by SSTF are homogeneously located at the focal plane and reduced in length by a factor >2, which is in excellent agreement with the numerical simulations of the nonlinear pulse propagation and might favor SSTF for demanding applications such as intraocular fs-laser surgery.

INTRODUCTION

Materials processing using ultrashort femtosecond (fs) laser pulses has gained great attraction within recent years, since new compact and reliable fs-laser sources allow precise and versatile micromachining of numerous materials such as metals or semiconductors. [START_REF] Cheng | A review of ultrafast laser materials micromachining[END_REF] Due to the short timescales of those pulses, side-effects such as heating of the processed material are hampered, allowing for well-located ablation, drilling or cutting on µm-scales. [START_REF] Chichkov | Femtosecond, picosecond and nanosecond laser ablation of solids[END_REF] Moreover, when focused into transparent materials extremely high intensities >10 [START_REF] Papagiakoumou | Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses[END_REF] W/cm² are achieved at the focal region triggering nonlinear absorption processes. [START_REF] Itoh | Ultrafast processes for bulk modification of transparent materials[END_REF] Dependent on the laser parameters and the processed material various types of material modifications can be induced, reaching from well-localized refractive index modifications to extensive disruptions. [START_REF] Sugioka | Femtosecond laser three-dimensional micro-and nanofabrication[END_REF] In order to gain maximum precision, ultrashort fs-laser pulses are commonly focused with comparatively high numerical aperture (NA) to achieve intensities well above the interaction threshold within a focal spot size of few µm. However, for certain applications more sophisticated focusing methods can be required. For instance, the µm-sized focal spot provides high spatial and axial resolution in laser-scanning multiphoton fluorescence microscopy, but also demands the timeconsuming scanning of the focal spot across the sample and, thereby, hampers fast image acquisition at video refresh rates. *robert.kammel@uni-jena.de; phone +49 3641 -9 47 821; fax +49 3641 -9 47 802; www.iap.uni-jena.de

f, f, H H f2 f2
On the other hand weaker focusing with lower NA enables wide-field imaging of the sample area, but suffers from low axial resolution and reduced contrast due to the enlarged Rayleigh length, which also causes the excitation of fluorescent molecules upstream and downstream from the focal plane. Moreover, for various applications such as intraocular fs-laser surgery or back-side processing of glass, ultrashort laser pulses are focused deeply into the material with low NA. [START_REF] Merker | An in vitro study on focusing fs-laser pulses into ocular media for ophthalmic surgery[END_REF][START_REF] Ackermann | Femtosecond laser treatment of the crystalline lens: a 1-year study of possible cataractogenesis in minipigs[END_REF] As a result, the extreme intensities are not only obtained at the geometrical focus, but may also be achieved already during the propagation towards the focal plane and may evoke nonlinear pulse-material interactions even extending the focal region such as Kerr self-focusing and self-phase modulation of the pulse. [START_REF] Couairon | Femtosecond filamentation in transparent media[END_REF] In general these nonlinear interactions are detrimental, since they reduce the precision of the applied modifications and induce side-effects such as strong supercontinuum generation. Moreover, the complex interaction of self-focusing and the defocusing refractive index diminution of the induced plasma can result in the formation of an extended filament, [START_REF] Kammel | Pump-probe investigation of fs-LIOB in water by simultaneous spatial and temporal focusing[END_REF] while intensity clamping inside the filament may impair the development of the material modification. [START_REF] Liu | Intensity clamping of a femtosecond laser pulse in condensed matter[END_REF] To overcome those limitations, simultaneous spatial and temporal focusing (SSTF) was proposed as a versatile tool to independently control the size of the focal spot and the corresponding axial resolution. [START_REF] Zhu | Simultaneous spatial and temporal focusing of femtosecond pulses[END_REF][START_REF] Oron | Scanningless depth-resolved microscopy[END_REF] In the concept of SSTF the spectral components of a broadband fs-laser pulse are spatially dispersed outside of the focal volume, which results in a significant increase of the pulse duration. After focusing, the entire pulse spectrum superposes locally at the focal plane and compresses the pulse to its bandwidth-limited pulse duration. Because the ultrashort pulse duration is restricted to the focal plane, the intensity outside of the focal region is strongly reduced, thereby reducing detrimental pulse-material interactions. Moreover, this additional degree of freedom allows shaping the focal intensity distribution. Thus, SSTF has not only raised great attraction for imaging purposes such as multiphoton microscopy, but has also been applied numerously for materials processing such as in-volume structuring of transparent materials. In this publication we provide an insight into the fundamentals of SSTF, present an overview on recent applications in microscopy and material processing, and outline potential future developments of this promising technology.

SIMULTANEOUS SPATIAL AND TEMPORAL FOCUSING IN MICROSCOPY

High quality multiphoton microscopy requires high lateral and axial resolution together with fast image acquisition. However, with conventional focusing methods it is challenging to address all these demands simultaneously. On the one hand, confocal laser-scanning microscopy provides high resolution and high signal-to-background ratio via excitation of fluorescent molecules within a small µm-sized focal volume, but requires a time-consuming scanning of the focal spot across the sample to generate a wide-field image. On the other hand, wide-field imaging with low NA enables imaging the entire sample area at video frame rates without the need of scanning, but suffers from detrimental out-of-focus signal since the increased Rayleigh length causes the excitation of fluorescent molecules outside of the focal plane. In 2005, Zhu et al. [START_REF] Zhu | Simultaneous spatial and temporal focusing of femtosecond pulses[END_REF] and Oron et al. [START_REF] Oron | Scanningless depth-resolved microscopy[END_REF] proposed SSTF to provide both wide-field imaging with improved depth-resolution and increased signal-to-background ratio. The experimental scheme of this concept is depicted in Figure 1. Like briefly introduced above, an incident fs-laser pulse is spectrally dispersed by means of a reflective grating, which is mounted in the back focal plane of a lens (L1). The spectral separation of the pulse results in a reduction of the local bandwidth within the 'rainbow-like' beam and strongly increases the pulse duration. Thereafter, the stretched pulse is focused by means of a second lens (L2) or a microscope objective, while the superposition of the entire pulse spectrum causes the compression to the ultrashort pulse duration at the focal plane. Because the ultrashort pulse duration is only achieved at the focus and rapidly increases upstream and downstream from this plane, the nonlinear multiphoton fluorescence signal is strongly restricted to the focal region, thus enabling depth-resolved microscopy with improved signal-to-background contrast. The setup provides wide-field imaging with an enlarged focal spot avoiding the need of scanning the focal spot across the sample, since the focal spot size is determined by the input beam diameter and the magnification of the optical system depicted in Figure 1. [START_REF] Durfee | Intuitive analysis of space-time focusing with double-ABCD calculation[END_REF] The large potential of this technique has been demonstrated in numerous publications. For example, in the initial paper on SSTF Zhu et al. demonstrated an increase of the pulse duration from 84 fs at the focus to > 1 ps at a distance of ±200 µm away from the focal plane, which was proposed for multiphoton imaging to reduce the out-of-focus background. [START_REF] Zhu | Simultaneous spatial and temporal focusing of femtosecond pulses[END_REF] Moreover, various innovative procedures made use of the spectrally separated SSTF beam in order to enhance the imaging process. Thus, numerous publications demonstrated the use of spatial light modulators (SLM) providing arbitrarily shaped, depth-resolved excitation patterns to control multiphoton processes with a depth-resolution of few µm even at video refresh rates. [START_REF] Papagiakoumou | Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses[END_REF][START_REF] Therrien | Wide-field multiphoton imaging of cellular dynamics in thick tissue by temporal focusing and patterned illumination[END_REF][START_REF] Papagiakoumou | Temporal focusing with spatially modulated excitation[END_REF] Similarly, temporally focused computer generated holograms were studied to provide a shift of the temporal focus plane aiming at adapted three-dimensional multiphoton excitation patterns. [START_REF] Leshem | When can temporally focused excitation be axially shifted by dispersion?[END_REF] Moreover, adding group delay dispersion to the input pulses be means of a deformable mirror device, z-scan with high repetition rate of 200 Hz for multiphoton microscopy was presented by Straub et al. [START_REF] Straub | High speed multiphoton axial scanning through an optical fiber in a remotely scanned temporal focusing setup[END_REF] In order to further improve the sectioning ability for multiphoton microscopy purposes, Song et al. recently extended the conventional one-dimensional spatial dispersion of SSTF towards a two-dimensional spectral disperser using a virtually imaged phased array. [START_REF] Song | Two-dimensional spatiotemporal focusing of femtosecond pulses and its applications in microscopy[END_REF] Due to the enhanced spectral separation ability, both the axial resolution and the image contrast could be significantly enhanced compared to conventional gratingbased SSTF.

SIMULTANEOUS SPATIAL AND TEMPORAL FOCUSING FOR NONLINEAR MATERIALS PROCESSING

For purposes of multiphoton microscopy, fs-laser pulses are conventionally applied at low energy and high repetition rates up to MHz, while wide-field imaging requires a focal spot size of several tens up to hundreds of µm. In contrast, requirements of materials processing in the bulk of transparent media commonly differ from these parameters. For high precision micromachining the pulse energy commonly ranges from few tens up to hundreds of µJ to provide intensities >10 [START_REF] Papagiakoumou | Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses[END_REF] W/cm² within a small focal spot size of only few µm in order to trigger nonlinear absorption mechanisms, while repetition rates of several kHz enable the fast processing of extended regions via scanning the focal spot. Hence, SSTF setups used for materials processing are commonly based on single pass parallel grating compressor configurations as shown in Figure 2, which enable restricting the pulse-material interaction to a focal volume of only few µm. After passing the grating configuration, the frequency components of the input pulse are linearly displaced by  along the x-direction depending on the frequency difference ∆𝜔 = 𝜔 -𝜔 0 . 12 Here, 0 denotes the carrier frequency of the pulse, Ω is the FWHM of the spectral intensity profile and  describes the spectral separating capability of the grating configuration. After passing the gratings, the pulse with the input beam width 2𝑤 𝑥,𝑖𝑛 is horizontally stretched to a width of 2𝑤 𝑥,𝑆𝑆𝑇𝐹 . The spectral separation causes an increase of the pulse duration within the collimated SSTF beam to

𝜏 𝑆𝑆𝑇𝐹 = 4 ln 2 Ω ⋅ √1 + ( 𝛼Ω 𝑤 𝑥,𝑖𝑛 ) 2 = 𝜏 0 ⋅ 𝛽 𝐵𝐴 . (1)
Hence, the factor 𝛽 𝐵𝐴 , by which the transform limited FWHM pulse duration 𝜏 0 is increased, raises with increasing spectral separating capability of the grating configuration and decreasing input beam width 2𝑤 𝑥,𝑖𝑛 , since both parameters affect the spectral separation. While the pulse propagates with the FWHM duration 𝜏 𝑆𝑆𝑇𝐹 due to the spectral separation, the superposition of the entire bandwidth at the focal plane recompresses the pulse to its bandwidth limited pulse duration 𝜏 0 like illustrated in the inset of Figure 2.

In contrast to the single grating configuration presented in Figure 1, the parallel grating setup induces a temporal chirp of the pulse, which is commonly compensated by pre-chirping the input pulse via adjusting the compressor gratings of the laser system or by means of an additional external pulse stretcher setup. To overcome disadvantages such as reduced transmission and the complex alignment of multiple grating configurations, recent efforts aim at incorporating the SSTF grating configuration into chirped pulse amplification (CPA) systems to provide spatially dispersed but temporally unchirped pulses. [START_REF] Squier | High average power Yb:CaF femtosecond amplifier with integrated simultaneous spatial and temporal focusing for laser material processing[END_REF][START_REF] Block | Integrated single grating compressor for variable pulse front tilt in simultaneously spatially and temporally focused systems[END_REF] Due to an adjustable grating configuration, the setups allow tuning the beam aspect ratio BA to vary the increase of the pulse duration outside of the focal region.

Linear and nonlinear propagation of spatially dispersed, focused pulses

Theoretical treatments of SSTF have shown that the pulse propagation differs from conventional focusing in particular around the focal region. [START_REF] Durfee | Intuitive analysis of space-time focusing with double-ABCD calculation[END_REF][START_REF] Durst | Simultaneous Spatial and Temporal Focusing in Nonlinear Microscopy[END_REF][START_REF] Zhang | Analysis of pulse front tilt in simultaneous spatial and temporal focusing[END_REF] To provide a deeper insight into the SSTF pulse propagation, Figure 3 presents x/t-intensity profiles nearby the focal plane comparing pulses focused conventionally and using SSTF. For this numerical simulation, a 'Forward Maxwell equation' (FME) was used to study the linear (Epulse ≈ 0 µJ) and nonlinear (Epulse = 4 µJ) pulse propagation within a water cell with a volume of 224 mm³. A detailed description of this study can be found elsewhere. [START_REF] Kammel | Enhancing precision in fs-laser material processing by simultaneous spatial and temporal focusing[END_REF] To compare SSTF with conventional focusing, a circular beam with a 1/e²-radius of 200 µm and an initial pulse duration of 0 = 50 fs, centered at 0 = 800 nm, was conventionally focused initially using an ideal lens (f = 2mm) into the center of the observed volume. In contrast, the SSTF pulse was composed of elliptical, pulsed beamlets with wx,in = 40 µm and wy = 200 µm. Since the spatial chirp rate BA was set to 5, the SSTF pulse duration SSTF is increased by a factor of 5 to 250 fs before focusing, while the beam is horizontally stretched to a radius of wx,SSTF ≈ 200 µm. Due to the similar beam size (wx ≈ wy ≈ 200 µm) in front of the ideal focusing lens, NA ≈ 0.1 for both regimes.

Regarding conventional focusing for a low pulse energy (linear pulse propagation), the intensity profiles presented in Figure 3a show the well-known symmetrically converging and diverging behavior with a focal spot size of ~5 µm, while the pulse duration of 0 = 50 fs is constant during the propagation. In contrast, the pulse propagation of a focused SSTF pulse (Figure 3b) strongly differs from this behavior: Outside of the focal plane, the spectral separation increases the pulse duration to SSTF = 250 fs ('global pulse duration'). Due to the spectral overlap of the full pulse spectrum nearby the focal plane, the pulse is re-compressed and a strong pulse front tilt (PFT) is induced. Since the PFT causes a fast sweeping of the pulse across the focal plane, the 'local pulse duration' is reduced to  = 220 fs at z = ± 200 µm and to the bandwidthlimited duration 0 = 50 fs at z = 0, respectively. In contrast, the 'global pulse duration' remains at a constantly increased level of SSTF ≈ 250 fs during the entire pulse propagation. Although the NA ≈ 0.1 is similar for both configurations, the horizontal focal spot size is increased by a factor of 5 in case of SSTF due to the elliptical input beamlets described above.

This property of spatially dispersed, focused pulses was experimentally verified using various measurement techniques such as nonlinear fluorescent samples, SEA TADPOLE devices and interferometric characterization. [START_REF] Coughlan | Parametric spatio-temporal control of focusing laser pulses[END_REF][START_REF] He | Characterization and control of peak intensity distribution at the focus of a spatiotemporally focused femtosecond laser beam[END_REF][START_REF] Wang | Interferometric characterization of pulse front tilt of spatiotemporally focused femtosecond laser pulses[END_REF] Moreover, numerical analysis of the pulse propagation has shown that the focal PFT is a characteristic feature of SSTF, which cannot be overcome by techniques such as applying an opposed PFT or pre-chirping the input pulse. [START_REF] Zhang | Analysis of pulse front tilt in simultaneous spatial and temporal focusing[END_REF] As shown by Kazansky et al. a tilted focal intensity distribution can be used for so-called 'nonreciprocal-' or 'quill writing'. [START_REF] Kazansky | Quill" writing with ultrashort light pulses in transparent materials[END_REF] When focused onto the material surface or into the bulk the orientation of the focal intensity tilt towards the scan direction influences the pulseplasma interaction and the resulting type of material modifications, e.g. for purposes of optical storage, waveguide writing and laser surgery. [START_REF] Sugioka | Femtosecond laser three-dimensional micro-and nanofabrication[END_REF] Since PFT is an intrinsic feature of spatially dispersed, focused pulses, the formation of wells and Chevron structures through the ablation of glass was demonstrated being dependent on the writing direction of focused SSTF pulses, respectively. [START_REF] Vitek | Spatio-temporally focused femtosecond laser pulses for nonreciprocal writing in optically transparent materials[END_REF] With increasing pulse energy, nonlinear interactions between the field of the incident laser pulse and the transparent medium occur not only at the focal plane, but can induce detrimental side-effects even outside of the focal volume. Namely, self-focusing and filamentation can cause the collapse of the beam far in front of the focal plane and trap the high intensity within a prolonged channel even exceeding the focal plane, while self-phase modulation may cause a self-steepening of the laser pulse and the generation of an intense broadband supercontinuum.

To illustrate this behavior, the nonlinear propagation of a 50 fs laser pulse with a pulse energy of 4 µJ conventionally focused into water is presented in Figure 4a. In front of the focal plane (z = -200 µm), the intensity profile is similarly distributed in the linear (Epulse ≈ 0 μJ) and the nonlinear regime (Epulse = 4 µJ) resulting in a high intensity of ~ 14 TW/cm². During the propagation towards the focal plane (z = 0 µm) self-steepening compresses the conventionally focused pulse to an even shorter duration of 30 fs. Downstream from the focal plane, the pulse is trapped within an elongated filament owing to the dynamical balance between Kerr self-focusing of the pulse and the defocusing refractive index diminution of the laser-induced plasma. Moreover, the defocusing effect of the plasma restricts the intensity to a maximum value of 24 TW/cm², referred to as 'intensity clamping'. [START_REF] Couairon | Femtosecond filamentation in transparent media[END_REF][START_REF] Liu | Intensity clamping of a femtosecond laser pulse in condensed matter[END_REF] Although ~65% of the incident pulse energy is absorbed or scattered by the plasma, filamentation hampers the divergence of the pulse and keeps the intensity at a comparatively high level of ~7 TW/cm² even 200 µm downstream from the focal plane.

In the nonlinear SSTF pulse propagation (Figure 4b), the intensity profile in front of the focal plane (z = -200 µm) is similar to the linear regime shown above (compare Figure 3b). At the focal plane (z = 0 µm) the intensity profile becomes distorted, whereas the fundamental characteristics of the SSTF pulse are conserved. Thus, the PFT remains similar to the linear case and the pulse is locally compressed to ~50 fs. Owing to the intensity clamping effect mentioned above, a maximum intensity similarly achieved by conventional focusing is detected. Downstream from the focus, the increase of the pulse duration impedes the filamentation of the pulse and enables its divergence. Consequently, the intensity downstream from the focal plane is significantly reduced by both the spatial divergence and temporal broadening of the SSTF pulse.

Owing to the intensity reduction outside of the focal region, SSTF is able to reduce detrimental nonlinear side-effects and, furthermore, provides an additional degree of freedom via shaping the variation of the pulse duration during the pulse propagation. Consequently, numerous publications have shown its potential for improved laser materials processing. For example, Vitek et al. demonstrated the ability of backside-processing of 6 mm thick fused silica avoiding adverse selffocusing, filamentation and supercontinuum generation. [START_REF] Vitek | Temporally focused femtosecond laser pulses for low numerical aperture micromachining through optically transparent materials[END_REF] Moreover, the ablation of bone with nearly parallel sidewalls under aqueous immersion was shown experimentally in this study, proposing this technique as a versatile tool for surgical purposes such as craniotomies. Furthermore, He et al. demonstrated theoretically and experimentally the formation of spherically shaped intensity distributions at the SSTF focus. [START_REF] He | Fabrication of microfluidic channels with a circular cross section using spatiotemporally focused femtosecond laser pulses[END_REF] Using this technique microfluidic channels with a circular cross-sectional shape were generated in fused silica. To overcome the effect of intensity clamping discussed above, SSTF was further applied to enhance the intensity inside the filament in air. [START_REF] Zeng | Enhancement of peak intensity in a filament core with spatiotemporally focused femtosecond laser pulses[END_REF] While the length of the filament was reduced by a factor of 25, this technique was proposed to provide high-order harmonic generation with improved conversion efficiency and higher cutoff photon energies. Furthermore, spatio-temporal focusing has been combined with patterned excitation for purposes of multiphoton microfabrication of polymer structures. [START_REF] Li | Fast multiphoton microfabrication of freeform polymer microstructures by spatiotemporal focusing and patterned excitation[END_REF] Using a deformable mirror device, arbitrarily shaped 2D-patterns were imaged into the focal spot of the SSTF setup, providing a fabrication resolution of ~ 1 µm and enabling a stacking of 2D-structures.

Next to manufacturing techniques, SSTF has also been studied for purposes of medical applications. Investigating tissue ablation by means of SSTF and conventional focusing, Block et al. showed a reduced focal plane upstream shift, reduced filamentation and weaker bubble formation using SSTF. [START_REF] Block | Simultaneous spatial and temporal focusing for tissue ablation[END_REF] Moreover, investigations realized by our research group showed the large potential of SSTF for intraocular fs-laser surgery, e.g. to provide minimally-invasive cuts within transparent tissue such as the crystalline lens or the vitreous body. [START_REF] Kammel | Enhancing precision in fs-laser material processing by simultaneous spatial and temporal focusing[END_REF] In this study, the nonlinear pulse propagation in aqueous media was observed comparing SSTF and conventional focusing using equivalent focusing conditions. To outline the main findings, Figure 5 presents microscope images of fs-laser-induced modifications within gelatine, which serves as a model system for human lens tissue. For conventional focusing, ultrashort laser pulses with a pulse duration of 50 fs centered at a wavelength of 800 nm are focused by means of a 25 mm-off-axis parabolic mirror into the sample. Owing to the approximately circular beam diameter with 2wx ≈ 2wy ≈ 5 mm, the NA was 0.1 as discussed in the numerical study above. Since for both conventional focusing and SSTF underfilling the aperture of the focusing lens can increase detrimental effects such as self-focusing and filamentation, [START_REF] Cheng | Onset of nonlinear self-focusing of femtosecond laser pulses in air: Conventional vs spatiotemporal focusing[END_REF] the focusing conditions are optimized similarly for both configurations. Thus, for SSTF a cylinder lens telescope was used to reduce the horizontal beam width in front of the grating configuration by a factor of 5 to an elliptical beam size of 2wx,in ≈ 1 mm and 2wy ≈ 5 mm. Thereafter, the elliptical beam is spectrally separated by means of a grating setup similar to the one shown in Figure 2. The grating configuration provides a spatial chirp rate BA ≈ 5. Thus, the spectral separation results in stretching the input beam horizontally to a width of 2wx,SSTF ≈ 5 mm, yielding a circular beam size and NA similar to the conventional focusing configuration discussed above. Because BA ≈ 5, the pulse duration outside of the focal region is increased to SSTF ≈ 250 fs. At the focal plane the pulse is compressed to the bandwidth-limited local pulse duration of ~ 50 fs (see inset of Figure 2). When the SSTF pulses are focused into the gelatine sample with a low pulse energy of 1 µJ, the laser-induced disruptions are well confined to the focal plane for both focusing conditions (Figure 5a and 5b), although disruptions induced with SSTF show a reduction in length by a factor of ~2. For a high pulse energy of 8 µJ, Kerr self-focusing and filamentation of the conventionally focused pulse cause the elongation of the disruption to a length of >900 µm (Figure 5c). Moreover, intense side-branches and luminance fluctuations within the central channel are observed, which may be attributed to multi-filamentation of the pulse. [START_REF] Skupin | Modeling ultrashort filaments of light[END_REF] Hence, strong fluctuations of the plasma density within the filaments cause the breakup of the central disruption into a multitude of separated disruptions as well as the chemical decomposition of the molecular bonds of the tissue. [START_REF] Vogel | Mechanisms of femtosecond laser nanosurgery of cells and tissues[END_REF] In agreement with the numerical analysis presented above, the modifications induced by SSTF pulses with 8 µJ are more homogeneous and are not showing side-branches or breakups (Figure 5d). Owing to the localization of the ultrashort pulse duration, detrimental nonlinear side-effects such as self-focusing, filamentation and self-phase modulation are avoided, which restricts the nonlinear interactions and the optical breakdown to the focal region.

CONCLUSION

Owing to the opportunity of accurately tailoring the laser-material interaction, ultrashort fs-laser pulses have enabled materials processing with high precision and low side-effects. Moreover, based on the development of compact and reliable fs-laser sources, innovative applications such as multiphoton fluorescence microscopy, processing of transparent materials and medical surgery gain even more importance. However, for several applications the nonlinear material interactions caused by conventional focusing are limited, since the intensity distribution at the focal plane cannot be freely designed, e.g. owing to the physical correlation between numerical aperture, focal spot size and the Rayleigh length. Here, simultaneous spatial and temporal focusing provides an additional degree of freedom, since the evolution of the pulse duration during the pulse propagation towards the focal plane can be adjusted to the targeted nonlinear pulse-material interactions. Hence, SSTF has been widely applied in multiphoton microscopy due to its ability to combine wide-field illumination with a high spatial resolution up to few µm. Additionally, the localization of the region of maximum intensity towards the focal plane enables reducing detrimental nonlinear interactions such as Kerr self-focusing, self-phase modulation and filamentation in order to improve the confinement of laser-induced modifications. Finally, innovative procedures such as 2D-SSTF or incorporating SSTF into conventional chirped pulse amplifier systems aim at even stronger intensity confinement and simplified application of this technique to provide high-precision processing of transparent materials such as intraocular fs-surgery.

Figure 1 .

 1 Figure 1. Scheme of simultaneous spatial and temporal focusing applied for wide-field multiphoton microscopy.

Figure 2 .

 2 Figure 2. Scheme of simultaneous spatial and temporal focusing based on a single pass parallel grating compressor configuration. The inset illustrates the development of the pulse duration around the focal region depending on the spatial chirp beam aspect ratio BA.

Figure 3 .

 3 Figure 3. Simulation of the linear pulse propagation (Epulse ≈ 0 µJ) of pulses focused (a) conventionally and (b) using SSTF, showing x/t-intensity profiles nearby the focal plane (z = -200 µm, 0 µm, + 200 µm).

Figure 4 .

 4 Figure 4. Simulation of the nonlinear pulse propagation (Epulse ≈ 4 µJ) of pulses focused (a) conventionally and (b) using SSTF, showing x/t-intensity profiles nearby the focal plane (z = -200 µm, 0 µm, + 200 µm).

Figure 5 .

 5 Figure 5. Microscope images of fs-laser-induced modifications within the bulk of a gelatine sample induced by conventional (a, c) focusing and SSTF (b, d) for 1 μJ (a, b) and 8 μJ (c, d) pulses, respectively.

ACKNOWLEDGEMENTS

This study was supported by the Thuringian Ministry of Education, Science and Culture (OptiMi 2020-Graduate Research School 'Green Photonics', B514-10061), the German Research Foundation (Leibniz program) and the Carl Zeiss Foundation. The numerical simulations were performed at the Rechenzentrum Garching.


Illia Thiele
email: illia-thiele@web.de

Stefan Skupin

Rachel Nuter

Modelling Laser Matter Interaction with Tightly Focused Laser Pulses in Electromagnetic Codes

We propose an algorithm to introduce arbitrarily shaped laser pulses into electromagnetic codes. In contrast to the often used paraxial approximation our approach models accurately tightly focused laser pulses and their interaction with matter.

Investigation of laser matter interaction with electromagnetic codes requires to implement sources for the electromagnetic fields. A common practise to do so is to prescribe the fields at the numerical box boundaries in order to achieve the desired fields inside the numerical box. Very often, the paraxial approximation is used to calculate the required fields at the boundaries. However, the paraxial approximation is valid only if the angular spectrum of the laser pulse is sufficiently narrow. Thus, it is not possible to use this approximation for strongly focused pulses. We propose a simple and efficient algorithm for a Maxwell consistent calculation of the electromagnetic fields at the boundaries of the computational domain. We call them laser boundary conditions (LBCs).

In numerical studies of laser matter interaction, one usually defines the laser by its propagation in vacuum, for example, by position and shape of the pulse at focus. In this paper, we choose to prescribe the pulse in a plane P parallel to a boundary of the rectangular numerical box, i.e., typically in the focal plane [see Fig. 1 (a)]. The laser (red) is passing through the plane P, where the fields E 0 , B 0 are prescribed for all times t. The goal is to calculate the fields E B , B B at the boundary from E 0 , B 0 .

The fields E 0 , B 0 describe forward and/or backward propagation waves indicated by the superscript ± in the following. It is important to note that E ± 0 , B ± 0 cannot be chosen arbitrarily. In fact, only two out of six vector components (for forward and backward direction, respectively) are independent. For example, we can choose to prescribe E ± 0,⊥ in the plane P. Then, by exploiting the Maxwell's equations in vacuum, we get all the other fields in frequency ω and transversal Fourier space (k x , k y ) along the propagation direction z as

Ē± ⊥ (k ⊥ , z, ω) = Ē± 0,⊥ (k ⊥ , ω)e ±ik z (k ⊥ ,ω)(z-z 0 ) (1) Ē± z (k ⊥ , z, ω) = ∓ k ⊥ • Ē± ⊥ (k ⊥ , z, ω) k z (k ⊥ , ω) (2)
B± (k ⊥ , z, ω) = 1 ωk z (k ⊥ , ω) R ± (k ⊥ , ω) Ē± ⊥ (k ⊥ , z, ω) , (3)
with the matrix R ± and k z according to

R ± (k ⊥ , ω) =   ∓k x k y ∓ k 2 z (k ⊥ , ω) + k 2 y ± k 2 z (k ⊥ , ω) + k 2 x ±k x k y -k y k z (k ⊥ , ω) k x k z (k ⊥ , ω)   (4) k z (k ⊥ , ω) = ω 2 /c 2 -k 2 x -k 2 y .
(5) Thus, transforming the given fields E 0 , B 0 into the frequency and transversal Fourier space, evaluating Eqs. ( 1)-( 3) and performing the Fourier back transforms gives the required fields at the boundary. Adding them to the boundary adapted to the grid of the specific Maxwell solver introduces the desired fields in the numerical box.

To demonstrate the significance of Maxwell consistent LBCs, the distributions of the electron density n e after a tightly focused Gaussian pulse has passed an initially neutral gas is considered. We perform the computations with the PIC code OCEAN [START_REF] Nuter | Prepulse suppression and optimization of backward Raman amplification with a chirped pump laser beam[END_REF] using paraxial approximation and the Maxwell consistent approach. For both cases a linear

E 0,⊥ (x,t) = E 0 e -x w 0 2 -t t 0 2 cos(ω c t)e x , (6)
with center wavelength 2πc/ω c = λ c = 0.8 µm, pulse duration t 0 = 20 fs, peak intensity I 0 = ε 0 c|E 0 | 2 /2 = 5 × 10 14 W/cm 2 giving E 0 = 61.4 GV/m and beam width w 0 = 0.35 µm. The particular choice of the beam width w 0 implies that non-negligible parts of Ē0,⊥ (k x , ω) are evanescent. These evanescent fields are suppressed in the calculation of E B (r ⊥ ,t) and B B (r ⊥ ,t) at the boundary z = z B , fully compatible with Abbe's diffraction limit.

The resulting distributions of the electron density n e after the laser pulse has passed through the interaction region are shown in Fig. 1 (b -c). The electron density profiles are even qualitatively different for paraxial and Maxwell consistent LBCs: The paraxial LBCs give a fish-like shape, where before the focus (negative z) the peak electron density appears off-axis [see Fig. 1(a)], and only up to 60% of the argon atoms get ionized. In contrast, the Maxwell consistent LBCs produce a cigar like shape with the peak electron density on the optical axis [see Fig. 1(b)], and a fully ionized plasma is produced. We would like to stress that these deviations in the plasma profile are far from negligible, and may have significant impact on features like back-reflected radiation or energy deposition in the medium. The observed sensitivity towards the LBC for tight focusing is not limited to ultrashort low energy pulses interacting with gaseous media, but should be equally important for solid targets and higher pulse energies.

Beside an accurate handling of the tightly focused Gaussian laser pulse injection, our algorithm offers a simple way to simulate more complex pulse configurations or even sampled experimental beam profiles [START_REF] Thiele | Modelling arbitrarily shaped and tightly focused laser pulses in electromagnetic codes[END_REF]. Such "structured light" receives a lot of recent interest from various communities [START_REF]Structured Light in Structured Media: From Classical to Quantum Optics[END_REF]. Thus, we believe that our approach will be useful for a larger community working on electromagnetic simulation codes.

Fig. 1 .

 1 Fig. 1. Schematic picture (a) of the laser (red) injection problem into the computational domain: Electric and magnetic fields E 0 , B 0 are prescribed in the plane P [here the (x, y)-plane at z = z 0 ]. The fields E B , B B at the boundary (blue) are unknown and have to be calculated. Electron densities n e produced by tightly focused Gaussian laser pulses: The profile produced by paraxial LBCs (a) is even qualitatively different than the one produced by Maxwell consistent LBCs (b). Electron densities are scaled to the initial neutral density n 0 . The laser propagates from left to right.