Pierre Bomel
email: pierre.bomel@univ-ubs.fr

Eric Martin
email: eric.martin@univ-ubs.fr

Emmanuel Boutillon Lester

Synchronization Processor Synthesis for Latency Insensitive Systems

In this paper we present our contribution in terms of synchronization processor for a SoC design methodology based on the theory of the latency insensitive systems (LIS) of Carloni et al[1]. Our contribution consists in IP encapsulation into a new wrapper model which speed and area are optimized and synthetizability guarantied.

The main benefit of our approach is to preserve the local IP performances when encapsulating them and reduce SoC silicon area.

Introduction

Modern integrated circuits (ICs), named ''systems on a chip'' (SoCs), are the composition of several subsystems exchanging data. SoC size increase is such that an efficient and reliable interconnection strategy is now necessary to combine sub-systems and preserve, at an acceptable design cost, the speed performances actual very deep sub-micron technologies provide [START_REF]International Technology Roadmap for Semiconductors[END_REF]. This communication requirement can be satisfied by a LIS communication network between IPs. The LIS methodology enables to build functionally correct SoCs by 1) promoting pre-developed IPs intensive reuse, 2) segmenting inter-IPs interconnects with relay stations to break critical paths and 3) bringing robustness to data stream latencies to IPs by encapsulating them into synchronization wrappers. These encapsulated IPs are called "patient processes".

Related Works

Patient processes [START_REF] Carloni | Coping with Latency in SoC Design[END_REF] are a key element in the LIS theory. They are suspendable synchronous IPs (named pearls) encapsulated into a wrapper (named shell) which function is to make them insensible to the IO latency and to drive the IP clock. The decision to drive or not the IP's clock is implemented very efficiently with combinatorial logic. Figure 1 illustrates a patient process' structure. Nevertheless, the LIS approach relies on a simplifying, but restricting, assumption: an IP is activated only if all its inputs are valid and all its outputs are able to store a result produced at next clock cycle. Now, it is frequent that only a subset of the inputs and outputs are necessary to execute one step of computation in a synchronous IP. So as to limit the patient process sensitivity to a subset of the inputs and outputs, Singh and Theobald [START_REF] Singh | Generalized Latency-Insensitive Systems for Single-Clock and Multi-Clock Architectures[END_REF] suggest to replace the combinatorial logic that drives the clock by a Mealy type FSM. This FSM tests the state of only the relevant inputs and outputs at each cycle and drives the IP clock only when they are all ready. This approach is an extension of the LIS original model and has the advantage to correspond to a more realistic communication behavior. It can be implemented if one disposes of input/output schedules that proves the IP communication behavior is cyclic and not data-dependent : i.e. it is statically predictible. The major drawbacks of FSMs are their synthezability and silicon size when communication scenarios are long and complex like in computing intensive digital signal processing applications. Finally, in order to reduce hardware cost, Casu and Macchiarulo [START_REF] Casu | A New Approach to Latency Insensitive Design[END_REF] prove that, if it is possible to determine a static scheduling of all the IPs activation, then the relay stations can be replaced by simple flip-flops and the synchronization upstream and downstream protocol signals can be definitely removed. The IP activation static schedule is implemented with shift registers which contents drive the IP's clock. This approach relies on the hypothesis that there are no irregularities in the data streams: it is never necessary to randomly freeze the IPs.

New Approach -a Smaller Wrapper

As 1) LIS methodology lacks the ability to dynamically sense IO subsets, 2) FSMs can become too large as communication bandwidth does, and 3) shift register based synchronization targets only extremely rapid environments, we propose to encapsulate IPs into a new synchronization wrapper model which area is much less than the FSM-based wrappers area, speed is enhanced (mostly thanks to area reduction) and synthesizability is guaranteed whatever the communication schedule is.

The solution we suggest is functionally equivalent to the FSMs. This is a specific processor that reads and executes cyclically operations stored in a memory. We name it a "synchronization processor" (SP). Figure 2 The SP communicates with the LIS ports with FIFOlike signals. These signals are formally equivalent to the voidin/out and stopin/out of [START_REF] Carloni | Theory of Latency-Insensitive Design[END_REF] and valid, ready and stall of [START_REF] Singh | Generalized Latency-Insensitive Systems for Single-Clock and Multi-Clock Architectures[END_REF]. Number of input and output ports can be any. It drives the IP's clock with the enable signal. The SP model is specified by a three states FSM: a reset state at power up, an operation-read state, and a free-run state. This FSM is concurrent with the IP and contains a data path: this a "concurrent FSM with data path" (CFSMD). Operation's format is the concatenation of an input-mask, an output-mask and a free-run cycles number. The masks specify respectively the input and output ports the FSM is sensible to. The run cycles number represents the number of clock cycles the IP can execute until next synchronization point. To avoid unnecessary signals and save area, the memory is an asynchronous ROM (or SRAM with FPGAs) and its interface with the SP is reduced to two buses : the operation address and operation word. The execution of the program is driven by an operation "readcounter" incremented modulo the memory size.

Implementation and Results

Our SP has been successfully applied to the high-level synthesis of Reed-Solomon (RS) and Viterbi decoder IP cores with GAUT [START_REF] Gaut | web site[END_REF]. Table 1 gives comparative results of FSM and SP physical synthesis and shows that very important gains in area (up to 99 % saved) and speed (up to 47% increase) can be obtained with our SP.

Conclusion

Our SP has an essential characteristic: its complexity does not depend on the number of cycles the IP needs for a whole computation but only on the number of ports. Consequently its frequency and area are constant, for a given number of ports. It allows to build SoCs with a LIS based methodology, preserves IP frequencies and saves silicon space to implement patient processes.

Figure 1 :

 1 Figure 1 : Carloni et al.'s Patient Process Model

Figure 2 :

 2 Figure 2 : Patient Process Model with Synchronization Processor

Table 1 : Applicative Results

 1

Proceedings of the Design, Automation and Test in Europe Conference and Exhibition (DATE'05) 1530-1591/05 $ 20.00 IEEE


Nazan Kalo

] Murat

Kalo Lu

] Muhammad

Nawaz Tahir

Cengiz Arıcı

Christian Bruneau

Henri Doucet

Pierre H Dixneuf

Bekir Çetinkaya

Smail Özdemir
email: ismail.ozdemir@inonu.edu.tr

Synthesis of N-Heterocyclic Carbene-Palladium-PEPPSI Complexes and Their Catalytic Activity in The Direct C-H Bond Activation

Keywords: N-heterocyclic carbene, benzimidazolium salts, PEPPSI-type palladium-NHC complexes, pyrroles, direct arylation

A series of benzimidazolium salts having their two nitrogen atoms substituted by bulky groups have been synthesized. The benzimidazolium salts were readily converted into the corresponding palladium-NHC-PEPPSI complexes with general formula [PdBr2(NHC)(Py)], (NHC = N-heterocyclic carbene; PEPPSI = pyridine-enhanced precatalyst preparation, stabilisation, and initiation). The structures of all new compounds were characterised by NMR, IR spectroscopy and microanalysis techniques, which support the proposed structures. The molecular structure of complex 2g was determined by single-crystal X-ray diffraction study. Next, the palladium-NHC-PEPPSI complexes were used as catalysts in the direct C5arylation of 1-methylpyrrole-2-carboxaldehyde by aryl halides. These complexes exhibited moderate to high catalytic activities and gave C-H activation selectively at the C5-position of 1-methylpyrrole-2-carboxaldehyde. Both electron-donating and electron-withdrawing substituents were well tolerated with catalytic systems based on these complexes, even non-activated aryl chlorides such as chlorobenzene or 4-chlorotoluene were coupled with pyrrole in moderate yields.

Introduction

In recent years, NHCs have become a very important class of ligands in organometallic chemistry and catalysis [1][2][3]. The strong -donating but poor -accepting ability of these NHCs lead to the formation of many stable metalcarbene complexes. For this reason, the metal-NHC complexes have been widely used as highly reactive and rather selective catalysts for numerous chemical transformations [4][5][6][7][8][9][10][11][12][13]. Metal-NHCs have also been utilized extensively in medicinal applications [14][15][16][17][18][19]. Nowadays, many research groups around the world are currently focusing on metal-NHC complexes for various purposes.

Palladium-catalyzed bond-forming reactions are one of the most common tools synthetic chemists employ, in industry as well as in academia [START_REF]N-Heterocyclic Carbenes in Synthesis[END_REF][START_REF] Guest | [END_REF][22][23]. The strong palladium-NHC bonds contribute to the high stability of the active species, even at low catalyst loading and high temperatures. With a number of commercially available, stable, userfriendly, and powerful palladium-NHC catalysts, the goal of a universal cross-coupling catalyst is within reach. In particular, palladium-catalyzed cross-coupling reactions are now widely used to form C-C and C-N bond formation. Such processes include the Suzuki- Miyaura [24], Mizoroki-Heck [START_REF] Bräse | Cross-coupling of organic halides with alkenes: The Heck reaction[END_REF], Negishi [26], Hiyama [27], Sonogashira [28], amination [29], and arylation of arenes reactions [START_REF]Modern arylation methods[END_REF][START_REF] Özdemir | [END_REF], often involving coupling of an aryl halide with a nucleophilic parthner.

Pyrrole is a very important backbone that is often present in the pharmaceuticals and biologically active compounds. As selected examples, Lipitor is the leading cholesterol lowering drug [32], Norbinaltorphimine is an opioid antagonist used in scientific research [33], Tanaproget is a progesterone-receptor agonist [34], Dopamine antagonist is a type of drug which blocks dopamine receptors by receptor antagonism [35], (Figure 1). Considering the importance of pyrroles in the pharmaceutical industry, straightforward and eco-friendly preparation of these compounds still represents a challenging task for chemists. Ohta and co-workers reported that the direct C2-or C5-arylation of several heteroaromatics with aryl halides through C-H bond activation proceeds in moderate to good yields using [Pd(PPh3)4] as catalyst [37]. Since these exciting results, the palladium-catalyzed direct arylation of various heteroaromatics with aryl halides has proved to be a powerful method for the synthesis of a wide variety of arylated heterocycles [38]. In this connection, recently we have prepared novel palladium-bis(NHC) complexes and have used they as effective catalysts in the direct arylation of pyrrole

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derivatives with aryl chlorides [39]. The general structure of palladyum-bis(NHC) complexes we used is shown in Scheme 1.

Scheme 1.

General structure of recently reported palladium-bis(NHC) complexes.

Up to now, examples of palladium-catalyzed direct arylations of pyrroles by using aryl halides have been reported [36], but no reactions were found using palladium-NHC-PEPPSI catalysts in the literature. For this reason, we synthesized a series of new palladium-NHC-PEPPSI complexes with the general formula [PdBr2(NHC)(Py)], (2a-h). All new palladium-NHC-PEPPSI complexes were characterized by 1 H NMR, 13 C NMR, IR spectroscopy and elemental analysis techniques. Next, palladium-NHC-PEPPSI complexes were tested for direct C5-arylation of 1-methylpyrrole-2carboxaldehyde with aryl bromides and aryl chlorides. The reactive C2-position of pyrrole was blocked in order to maximize the yields of the monoarylated products.

Results and discussion

Synthesis of benzimidazolium salts

The benzimidazolium salts (1a-h) have already been reported in the literature [40]. These salts were prepared according to general reaction pathway depicted in Scheme 2. The structures of the salts were determined by their characteristic spectroscopic data and elemental analyses. The results are consistent with the literature [40].

Scheme 2. Synthesis of the benzimidazolium salts (1a-h).

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Synthesis of palladium-NHC-PEPPSI complexes

The general procedure for the preparation of PEPPSI-type palladium-NHC complexes (2a-h) according to the method reported by Organ is shown in Scheme 3 [41]. Benzimidazolium salts (1a-h) were incorporated into the PEPPSItype palladium-NHC complexes (2a-h) by reaction with PdCl2 in refluxing pyridine in the presence of K2CO3 as a base and a large excess of KBr for 16 h. For the 1 H NMR spectra of the palladium-NHC-PEPPSI complexes (2a-h), sharp peaks in the lower field region belonging to the acidic imino proton of benzimidazolium salts (NCHN) were not observed between = 10.0-12.0 ppm. Similarly, in the 13 C NMR spectra, imino carbon of benzimidazolium salts (NCHN) were not observed between = 150-155 ppm. In the 1 H NMR and 13 C NMR spectra of the complexes 2a-h, loss of the the characteristic peak of the acidic imino proton (NCHN) and imino carbon (NCHN), signal suggests the formation of the palladium-NHC-PEPPSI complexes. In addition, the characteristic carbenic carbons in compounds 2a-h appeared in the 13 C NMR spectra as deshielded singlets at 163. 1, 162.7, 164.2, 164.1, 163.8, 162.8, 164.1 and 163.1 ppm, respectively. The IR data also clearly indicated the presence of (CN) vibration at 1412,1416,1411,1410,1410,1416,1414 and1411 cm -1 for 2a-h. The formation of carbenes is correlated by a shift of the (CN) vibration from 1551-1564 cm -1 in the benzimidazolium salts to 1410-1416 cm -1 in the coordinated carbenes [40].

Scheme 3. Synthesis of the palladium-NHC-PEPPSI complexes (2a-h).

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Structural characterization of complex 2g

Single crystals of complex 2g suitable for diffraction study were obtained by slow diffusion of n-pentane into a dichloromethane solution of complex 2g. The molecular structure of complex 2g was confirmed by single-crystal X-ray diffraction analysis. This complex crystallizes in a centrosymmetric monoclinic P21/n system and adopts a square-planar geometry. The carbene and the pyridine ligands are in the trans-position, with respective distances to the palladium center of 1.952 (7) and2.087(6) Å. Cell volume and density of the single crystal was calculated as 3853.1(6) Å 3 and 1.400 g.cm -3 . The molecular structure of complex 2g is shown in Figure 2, and selected bond lengths and angles are summarized in Table 1.

Direct C5-arylation of 1-methylpyrrole-2-carboxaldehyde

In order to screen the experimental conditions, we selected the complex 2h as the catalyst. We also selected the 4-bromoacetophenone and 4-chloroacetophenone as the model coupling partner. The results of the reaction parameters including solvent, base, temperature and catalyst loading are gathered in Table 2. The chemical characterizations of the products were made by GC and GC-MS. The conversions were based on the aryl halide by GC.

Table 2. Influence of the reaction conditions for palladium-NHC-PEPPSI catalyzed direct C5-arylation of 1-methylpyrrole-2carboxaldehyde with 4-bromoacetophenone and 4-chloroacetophenone. [a] Entry Pd-NHC-PEPPSI 2h

[mol-%] X Solvent Base Time [h] Temp. [ o C]
Yield [b] [%] When NMP or toluene were used as solvent, the reaction gave low yield of only 21-70% with Cs2CO3, KOPiv.

or KOAc as base after 2 h at 120-150 o C (Table 2, entries 2, 3,5,6,8,9). DMAc proved to be the best tested solvent with all bases after 2 h at 150 o C (Table 2, entries 1,4,7). In this solvent decreasing the reaction temperature from 150 °C to 120 °C had no effect on the yield (Table 2, entry 10), but decreasing the reaction temperature from 120 °C to 90 °C had detrimental effect on the yield (Table 2, entry 11). When the reaction time was reduced from 2 h to 1 h no noticeable effect on the yield was observed (Table 2, entry 12), but when the reaction time was reduced to 0.5 h, the yield dropped to 53% (Table 2, entry 13). In the presence of 1 mol% of 2h as the catalyst, KOAc as the base, DMAc as the solvent and 4-bromoacetophenone as the coupling partner at 120 °C for 1 h, the C5-arylated product was obtained in 90% isolated yield (Table 2, entry 14), but when the catalyst loading was reduced from 1 mol% to 0.5 mol%, the yield decreased to 69% (Table 2, entry 15). Finally, the best conditions leading to 90% yield of 4-bromoacetophenone with high selectivity in favor of the C5-arylated product were obtained when the reaction was carried out in DMAc in the presence of 1.5 equiv. of KOAc at 120 o C for 1 h (Table 2, entry 14). The DMAC / KOAc combination is commonly used for C-H arylation of 5 membered heterocycles. For this reason, the conditions are also consistent with the literature [36 b-d ].

When the less reactive 4-chloroacetophenone was used as substrate in the presence of 1 mol% of catalyst 2h and KOAc as the base at 120 o C, almost no conversion was observed after 2 h (Table 2, entry 16 and 17). When the reaction time was further increased to 20 h, the conversion was improved indicating that the catalysts were still active reaching a yield of 76% after 20 h.

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Finally, the scope of the direct C5-arylation of 1-methylpyrrole-2-carboxaldehyde was investigated with various aryl halides, including five aryl bromides and five aryl chlorides (Table 3) applying our best experimental conditions. Only a minor effect of the nature of the carbene ligand on the palladium-NHC-PEPPSI complex was observed for the coupling of aryl bromides with 1-methylpyrrole-2-carboxaldehyde. At elevated temperature, the oxidative addition of the aryl bromide to palladium is generally easy and does not require the use of very specific ligands. A more important effect of the nature of the ligand was expected for the coupling of aryl chlorides with the 1-methylpyrrole-2-carboxaldehyde. Indeed, again in all cases the coupling products were produced in good yields. Surprisingly, similar yields were obtained for the coupling of electron-deficient aryl chlorides (Table 3, entries 9-24) as with chlorobenzene (Table 3, entries 1-8). In this case, the presence of the NHC-ligands on palladium likely favors the oxidative addition step.

Table 3. Direct C5-arylation of 1-methylpyrrole-2-carboxaldehyde by using aryl halides. [a,b] Entry Aryl halide Catalyst Product Yield [c] [%] X = Br X = Cl

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Conclusion

In conclusion, we have synthesized and characterized eight benzimidazolium salts according to literature [40],

and their corresponding new Pd-NHC-PEPPSI complexes. The catalytic activity of the complexes was investigated in the direct C5-arylation of 1-methylpyrrole-2-carboxaldehyde by aryl halides. The Pd-NHC-PEPPSI complexes were all found to be suitable catalysts for the direct C5-arylation of 1-methylpyrrole-2-carboxaldehyde by aryl bromides. With Pd-NHC-PEPPSI complexes, even non-activated aryl chlorides (such as chlorobenzene or 4-chlorotoluene) were coupled with 1-methylpyrrole-2-carboxaldehyde in moderate to good yields. This good performance possibly arises from strong steric interactions between the bulky 3,5-di-tert-butylbenzyl substituted NHC ligand and the catalytic centre, which favours reductive elimination. We can say that there is no significant difference between these complexes on the catalytic activity of C5-arylation of 1-methylpyrrole-2-carboxaldehyde by aryl halides. The significant difference between 2d and 2c indicates that electronic effects are also playing some role in these processes. Finally, as the major by-products are KOAc associated to HBr instead of metallic salts, this procedure is environmentally more attractive than the classical coupling procedures.

Experimental

General methods

All manipulations were carried out under argon using standard Schlenk line techniques. Chemicals and solvents were purchased from Sigma-Aldrich Co. (Poole, Dorset, UK). The solvents used were purified by distillation over the drying agents indicated and were transferred under argon. Pd-NHC-PEPPSI complexes were prepared according to known methods in the literature [41]. DMAc analytical grade (99%) was not distilled before use. KOAc (99%) was employed. Elemental analyses were performed by nönü University Scientific and Technological Research Center (Malatya, Turkey). Melting points were measures in open capillary tubes with an Electrothermal-9200 melting points apparatus. IR spectra were recorded on ATR unit in the range of 400-4000 cm -1 with Perkin Elmer Spectrum 100 Spectrofotometer. 1 H NMR and 13 C NMR spectra were recorded using a Bruker Avance AMX spectrometer operating at 400 MHz ( 1 H NMR) and at 100 MHz ( 13 C NMR) in CDCl3. The NMR studies were carried out in high-quality 5 mm NMR tubes. The chemical shifts ( ) are reported in ppm relative to CDCl3. Coupling constants (J values) are given in hertz.

NMR multiplicities are abbreviated as follows: s = singlet, d = doublet, t = triplet, m = multiplet. 1 H NMR spectra are referenced to residual protiated solvents ( = 7.26 ppm for CDCl3), 13 C chemical shifts are reported relative to deuteriated solvents ( = 77.16 ppm for CDCl3). All catalytic reactions were monitored on an Agilent 6890N GC and Schimadzu 2010 Plus GC-MS system by GC-FID with an HP-5 column of 30 m length, 0.32 mm diameter, and 0.25 m film thickness.

General procedure for the preparation of benzimidazolium salts (1a-h)

For the preparation of benzimidazolium salts (1a-h), N- (3,5-di-tert-butylbenzyl)benzimidazole, (1.0 mmol) was dissolved in anhydrous dimethylformamide (DMF), (3 mL) and alkyl bromide (1.0 mmol) was added at room temperature.

The reaction mixture was stirred at 80 °C for 24 h under argon. After completion of the reaction, the DMF was removed by vacuum and diethyl ether (15 mL) was added to obtain a white crystalline solid, which was filtered off. The solid was washed with diethyl ether (3×10 mL) and dried under vacuum. The crude product was recrystallized from ethanol/diethyl ether mixture (1:2, v/v) and completely dried under vacuum. These known compounds were synthesized and characterized by 1 H NMR, 13 C NMR, IR and elemental analysis techniques. The results we found, are consistent with the literature [40].

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General procedure for the preparation of palladium-NHC-PEPPSI complexes (2a-h)

Benzimidazolium salts (1a-h) (1.0 mmol) were converted, with moderated yields, into the palladium-NHC-PEPPSI complexes (2a-h) by reaction with PdCl2 (1.0 mmol) in refluxing pyridine in the presence of K2CO3 (5.0 mmol) as a base and a large excess of KBr and the mixture heated with vigorous stirring at 80 o C for 16 h. The reaction mixture was diluted with CH2Cl2 then filtered through a pad of celite and silica gel to remove the unreacted PdCl2 and benzimidazolium salt. The solvent was removed under vacuum. The pure complexes were washed with n-pentane (3×5 mL) and completely dried under vacuum. All complexes were isolated as air-stable yellow solids and were isolated in 58-

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General procedure for the direct arylation of 1-methylpyrrole-2-carboxaldehyde

An oven dried Schlenk flask was charged with aryl halide (1.0 mmol), 1-methylpyrrole-2-carboxaldehyde (2.0 mmol), KOAc (1.5 mmol), palladium-NHC-PEPPSI complex (0.01 mmol) and DMAc (2 mL). The Schlenk tube was purged several times with argon and it was placed in a preheated oil bath at 120 o C, and the reaction mixture was stirred for different durations given in the specific tables. At the end of the reaction, the solvent was removed under vacuum, and the residue was charged directly onto a silica gel column. The products were eluted by using n-hexane/diethyl ether mixture (5:1, v/v). The yields were established by GC and GC-MS based on aryl halides.

X-ray crystallographic data

Single crystal of complex 2g suitable for X-ray analysis were obtained by slow diffusion of n-pentane into a dichloromethane solution of the complex 2g. Empirical formula = C38H47Br2N3Pd; Mr = 811.99 g.mol -1 ; crystal system = monoclinic; space group = P21/n system; cell lengths, a = 17.8083( 16), b = 12.3036 (10), c = 18.2077(15) Å; cell angles, = 90 o , = 111.745(2)°, γ = 90 o ; cell volüme = 3853.1(6) Å 3 . Z = 4, d = 1.400 g.cm -3 , = 2.583 mm -1 . F(000) = 1648; T = 296 K. The crystal size (0.36 × 0.28 × 0.18 mm) was studied with a Bruker Kappa ApexII CCD diffractometer using Mo-K radiation ( = 0.71073 Å). The number of measured reflections are 22908 with 8267 independent reflections. The observed [I > 2 (I)] reflections are 3325. The structure was solved by direct methods with the SIR97 program, [42] and then refined with full-matrix least-square methods based on F 2 SHELXL2014/6 [START_REF] Sheldrick | Table 1. Selected bond lengths [Å] and angles [°] for complex 2[END_REF] [43]. All non-hydrogen atoms were refined isotropically first and then with anisotropic atomic displacement parameters. The large anistropic thermal parameters and wrong C-C bond distances etc for tert-butyl lead to consider disorder. The terminal C-atoms of tert-butyl were treated as disordered over two set of sites with occupancy ratio of 0.565(6): 0.435 (6) CCDC 1567377 (for 2g) contains the supplementary crystallographic data for this paper. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre.

Supporting Information (see footnote on the first page of this article): for the 1 H NMR, 13 C NMR and IR spectra of palladium-NHC-PEPPSI complexes (2a-h).

Figure 1 .

 1 Figure 1. Examples of biologically active pyrrole derivatives.

Figure 2 .Table 1 .

 21 Figure 2. Perspective view of the molecular structure of 2g.

  Conditions: 1-methylpyrrole-2-carboxaldehyde (2.0 mmol), aryl halide (1.0 mmol), base (1.5 mmol), solvent (2 mL).[b] Yields were calculated according to aryl halide by GC and GC-MS.

  Conditions: Pd-NHC-PEPPSI 2a-h (0.01 mmol), 1-methylpyrrole-2-carboxaldehyde (2.0 mmol), aryl halide (1.0 mmol), KOAc (1.5 mmol), DMAc (2 mL), 120 °C. [b] Reaction time: 1 h for aryl bromides, 15 h for aryl chlorides. [c] Yields were calculated according to aryl halide by GC and GC-MS.

  and having equal anistropic thermal parameters within the groups. The number of restraints used were 57. The H atoms were finally included in their calculated positions. A final refinement on F 2 , with 8267 unique intensities and 389 parameters, converged at R(F 2 ) = 0.1546, [R(F 2 > 2 (F 2 )] = 0.059 for 3325 observed reflections with I > 2 (I). The value of weight parameter is w = 1/[ (Fo 2 ) + (0.0546P) 2 ], where P = [Fo 2 + 2(Fc) 2 ]/3 and goodness of fit S = 0.946.

Figure 1 .Figure 2 .

 12 Figure 1. Examples of biologically active pyrrole derivatives.

Acknowledgements

This work was financially supported by the Technological and Scientific Research Council of Turkey TÜB TAK-BOSPHORUS (109T605) and the nönü University Research Fund ( .Ü. B.A.P. 2015/41). BC thanks to TÜBA for support.

 2. Influence of the reaction conditions for palladium-NHC-PEPPSI catalyzed direct C5-arylation of 1-methylpyrrole-2carboxaldehyde with 4-bromoacetophenone and 4-chloroacetophenone. [a] Entry Pd-NHC-PEPPSI 2h

Yield [b] [%] 3. Direct C5-arylation of 1-methylpyrrole-2-carboxaldehyde by using aryl halides. [a,b]

Entry

Aryl halide Catalyst Product Yield [c] [%] X = Br X = Cl


Mohamed El May

Nicolas Saintier

Thierry Palin-Luc

Olivier Devos

Olivier Brucelle

Corrosion Science

This paper presents an analytical model for assessing the corrosion fatigue crack initiation life on a martensitic stainless steel X12CrNiMoV12-3 in high cycle fatigue regime (between 10 5 and 10 7 cycles). Based on in-situ electrochemical measurements during corrosion fatigue tests in NaCl aqueous solution, the corrosion fatigue crack initiation mechanism was identified. Two main stages were investigated: (i) the fracture of the passive film by slip bands and (ii) the free dissolution of the metal developing fatigue crack initiation from a critical corrosion defect. The depassivation stress threshold corresponds to the median fatigue strength at 10 7 cycles for fatigue corrosion tests. For an applied stress range less than this threshold, the depassivation phenomenon was not observed at 10 7 cycles and no crack initiation occurred. The proposed model takes into account the depassivation process induced by the slip bands emergence at the specimen surface and the corrosion rate under cyclic loading. The experimental results are compared to the proposed model taking into account mechanical and electrochemical material parameters.

Introduction

The practical importance of fatigue failure in structural materials has motivated many works towards assessing the physical reasons for material sensitivity to corrosion fatigue (CF) and providing adequate damage models for engineers. In particular, developing an analytical methodology for predicting corrosion fatigue crack initiation (CFCI) [START_REF] Kondo | Prediction of fatigue crack initiation life based on pit growth[END_REF][START_REF] Chen | Transition from pitting to fatigue crack growthmodeling of corrosion fatigue crack nucleation in a 2024-T3 aluminium alloy[END_REF][START_REF] Xu | Crack initiation mechanisms for low cycle fatigue of type 316Ti stainless steel in high temperature water[END_REF][START_REF] Laird | Mechanisms of fatigue crack nucleation, corrosion fatigue: chemistry, mechanics and microstructure[END_REF][START_REF] Muller | Theoretical consideration on corrosion fatigue crack in initiation[END_REF][START_REF] Magnin | Corrosion fatigue mechanisms in BBC stainless steels[END_REF][START_REF] Stoltz | Mechanisms of corrosion fatigue crack propagation in Al-Zn-Mg alloys[END_REF][START_REF] Hu | A continuum damage mechanics approach coupled with an improved pit evolution model for the corrosion fatigue of aluminum alloy[END_REF] and corrosion fatigue crack growth [START_REF] Stoltz | Mechanisms of corrosion fatigue crack propagation in Al-Zn-Mg alloys[END_REF][START_REF] Zhou | Small and short crack growth and the solution-conductivity dependent electrochemical crack size effect[END_REF][START_REF] Schnbauer | The influence of corrosion pits on the fatigue life of 17-4PH steam turbine blade steel[END_REF] for structures is a key issue. A model to compute the fatigue life of material (to crack initiation) under corrosion fatigue should takes many variables into consideration: chemical nature of the environment, stress/strain levels, stress ratio and loading frequency. Most investigators agree that CFCI occurs at the surface around corrosion defects [START_REF] El-May | Effect of corrosion on the high cycle fatigue strength of martensitic stainless steel X12CrNiMoV12-3[END_REF][START_REF] Huang | Pit to crack transition behavior in proportional and non-proportional multiaxial corrosion fatigue of 304 stainless steel[END_REF][START_REF] Co | Effects of macro-scale corrosion damage feature on fatigue crack initiation and fatigue behavior[END_REF][START_REF] Amiri | A continuum damage mechanics model for pit-to-crack transition in AA2024-T3[END_REF]. The effect of corrosion on fatigue life of stainless steels is most closely related to the passive film resistance [START_REF] El-May | Effect of corrosion on the high cycle fatigue strength of martensitic stainless steel X12CrNiMoV12-3[END_REF][START_REF] Tan | Strain-rate dependent fatigue behavior of 316L stainless steel in high-temperature water[END_REF]. Much study has been focused on the effect of the microstructure, the thickness and the chemical composition of the passive films, on corrosion behaviour of metallic alloys, using surface analyses techniques [START_REF] Asami | An XPS study of the passivity of a series of iron-chromium alloys in sulphuric acid[END_REF][START_REF] Marcelin | Electrochemical characterisation of a martensitic stainless steel in a neutral chloride solution[END_REF][START_REF] Jung | XPS characterization of passive films formed on type 304 stainless steel in humid atmosphere[END_REF]. But the mechanical behaviour of passive layer (thickness lower than 10 nm) stay to be difficult to characterise experimentally.

Several CFCI models were proposed and can be classified into:

1. competition between pit growth and short crack propagation [START_REF] Kondo | Prediction of fatigue crack initiation life based on pit growth[END_REF][START_REF] Chen | Transition from pitting to fatigue crack growthmodeling of corrosion fatigue crack nucleation in a 2024-T3 aluminium alloy[END_REF], 2. preferential dissolution of plastically deformed material [START_REF] Xu | Crack initiation mechanisms for low cycle fatigue of type 316Ti stainless steel in high temperature water[END_REF][START_REF] Cui | Effect of plastic deformation on the electrochemical and stress corrosion cracking behavior of X70 steel in near-neutral ph environment[END_REF][START_REF] Ahmed | Effect of surface and bulk plastic deformations on the corrosion resistance and corrosion fatigue performance of AISI 316L[END_REF][START_REF] Hu | A continuum damage mechanics approach coupled with an improved pit evolution model for the corrosion fatigue of aluminum alloy[END_REF][START_REF] Amiri | A continuum damage mechanics model for pit-to-crack transition in AA2024-T3[END_REF], 3. local rupture of the passive film by slip bands [START_REF] Laird | Mechanisms of fatigue crack nucleation, corrosion fatigue: chemistry, mechanics and microstructure[END_REF][START_REF] Muller | Theoretical consideration on corrosion fatigue crack in initiation[END_REF][START_REF] Magnin | Corrosion fatigue mechanisms in BBC stainless steels[END_REF][START_REF] Wu | Crack initiation mechanism of Z3CN20.09M duplex stainless steel during corrosion fatigue in water and air at 290 °C[END_REF][START_REF] Sudarshan | Fatigue processes in metals-role of aqueous environments[END_REF][START_REF] Zhao | Corrosion fatigue crack initiation and initial propagation mechanism of E690 steel in simulated seawater[END_REF][START_REF] Zhao | Study on the mechanism of high-cycle corrosion fatigue crack initiation in x80 steel[END_REF] and 4. lowering of surface energy due to adsorption of specific species from the environment [START_REF] Stoltz | Mechanisms of corrosion fatigue crack propagation in Al-Zn-Mg alloys[END_REF].

All of the above mechanisms involve a synergetic contribution of mechanical loading and electrochemical behaviour of tested material. In order to accurately predict the CFCI life, it is necessary to consider synergetic effects between cyclic loading (stress level, load ratio, loading frequency), material (passive layer) and corrosion parameters [START_REF] Sudarshan | Fatigue processes in metals-role of aqueous environments[END_REF][START_REF] Huang | Pit to crack transition behavior in proportional and non-proportional multiaxial corrosion fatigue of 304 stainless steel[END_REF][START_REF] Tan | Strain-rate dependent fatigue behavior of 316L stainless steel in high-temperature water[END_REF][START_REF] Hu | A continuum damage mechanics approach coupled with an improved pit evolution model for the corrosion fatigue of aluminum alloy[END_REF]. Recent advances in local electrochemical characterization techniques have allowed to develop research works at the microscopic scale [START_REF] Oltra | Recent advances in local probe techniques in corrosion research analysis of the role of stress on pitting sensitivity[END_REF]. These techniques are used in stress corrosion cracking (SCC) tests to capture, in-situ, the temporal electrochemical behaviour [START_REF] Devos | Simultaneous EIS and in situ microscope observation on a partially blocked electrode application to scale electrodeposition[END_REF][START_REF] Brewick | Microstructure-sensitive modeling of pitting corrosion: effect of the crystallographic orientation[END_REF]. Experimental evidence show that pit growth rate is influenced by static stress in SCC tests [START_REF] Staehle | Transient stability of passive films in aqueous solutions[END_REF][START_REF] Brewick | Microstructure-sensitive modeling of pitting corrosion: effect of the crystallographic orientation[END_REF][START_REF] Turnbull | Modelling the electrochemical crack size effect on stress corrosion crack growth rate[END_REF] and by cyclic stress [START_REF] Magnin | Corrosion fatigue mechanisms in BBC stainless steels[END_REF][START_REF] Murtaza | Empirical corrosion fatigue life prediction models of a high strength steel[END_REF][START_REF] Ishihara | Prediction of corrosion fatigue lives of aluminium alloy on the basis of corrosion pit growth law[END_REF]. In HCF tests, the magnitude of synergetic effects is of major influence the pit initiation (depassivation) stage. This phenomenon can be evaluated by the use of in-situ global free potential measurement during CF test [START_REF] Ebara | Corrosion fatigue crack initiation in 12% chromium stainless steel[END_REF][START_REF] El-May | Effect of corrosion on the high cycle fatigue strength of martensitic stainless steel X12CrNiMoV12-3[END_REF][START_REF] Klein | Corrosion fatigue assessment of creep-resistant magnesium alloys DieMag422 and AE42[END_REF]. That allows to characterize the depassivation and corrosion T activity at the surface of the specimen during cyclic loading.

In the case of stainless steels, CF crack initiation is promoted by cyclic mechanical damage of the protective passive layer induced by the emergence of slip band. Passive films can act as obstacles to dislocation activity [START_REF] Wu | Crack initiation mechanism of Z3CN20.09M duplex stainless steel during corrosion fatigue in water and air at 290 °C[END_REF][START_REF] Tan | Strain-rate dependent fatigue behavior of 316L stainless steel in high-temperature water[END_REF]. Consequently, the PSB spacing will be small because slip localization is inhibited [START_REF] Finney | Strain localization in cyclic deformation of copper single crystals[END_REF][START_REF] Aubert | A methodology to obtain data at the slip-band scale from atomic force microscopy observations and crystal plasticity simulations. Application to hydrogen-induced slip localization on AISI316L stainless steel[END_REF]. By raising the dissolution rate, or by making the oxide film thinner, the barriers to slip are reduced, slip localization is favoured and PSB spacing increases [START_REF] Sudarshan | Fatigue processes in metals-role of aqueous environments[END_REF]. The matrix dislocations tends to be transferred more easily into the PSB, further accelerating the formation of PSBs causing local ruptures of the passive layer [START_REF] Finney | Strain localization in cyclic deformation of copper single crystals[END_REF]. Once the passive film damaged, the metal is directly exposed to the aqueous environment, resulting in preferential anodic area for local corrosive dissolution. This results in geometric corrosion defects leading to corrosion fatigue crack initiation [START_REF] El-May | Non-local high cycle fatigue strength criterion for metallic materials with corrosion defects[END_REF][START_REF] Palin-Luc | Fatigue crack initiation and growth on a steel in the very high cycle regime with sea water corrosion[END_REF][START_REF] Co | Effects of macro-scale corrosion damage feature on fatigue crack initiation and fatigue behavior[END_REF]. The stress intensity will become high enough to initiate a crack in the metal [START_REF] Murakami | Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions[END_REF][START_REF] Amiri | A continuum damage mechanics model for pit-to-crack transition in AA2024-T3[END_REF][START_REF] Hu | A continuum damage mechanics approach coupled with an improved pit evolution model for the corrosion fatigue of aluminum alloy[END_REF]. An important requirement for this crack initiation scenario is that the material, in the given aqueous environment, must be in the passive domain without any cyclic loading. An efficient way to monitor passive film failure is to perform in-situ measurements of the free potential. Previous studies have shown that a sudden drop in the potential occurs when the mechanical cyclic loading (fatigue) is applied to the material [START_REF] El-May | Effect of corrosion on the high cycle fatigue strength of martensitic stainless steel X12CrNiMoV12-3[END_REF]. When considering the CF processes on stainless steel, it should be noticed that both characteristic times of the cyclic loading and passivation processes influence the overall mechanical-corrosion synergetic effects. Indeed, if the passivation process occurs at a higher rate than the mechanically induced passive film damage, no CF effect is expected. On the contrary if the passivation process is too slow, localized dissolution process can take place leading to fatigue crack initiation.

In this study we investigated the effect of aqueous corrosion (0.1 M NaCl + 0.044 M Na 2 SO 4 ) on the CFCI mechanism of a martensitic stainless steel X12CrNiMoV12-3 used for aeronautical applications. An analytical CFCI model is proposed considering the coupling between corrosion parameters and cyclic loading in the HCF regime.

Material and methods

Material

The material used in this study is a martensitic stainless steel X12CrNiMoV12-3 used in aeronautical structures. The chemical composition of this alloy, in weight percent, is given as follows: 0.12C, 12Cr, 3Ni, 1.6Mo, 0.3V and Fe balance. Microstructural investigations showed a typical equiaxed martensite lath structure with residual austenite (less than 1% in the volume). The martensite lath dimension is around 2-3 μm. Prior austenite average grain dimension is around 100 μm. The 0.2% offset yield strength, and elongation to failure of the alloy are 1040 MPa and 15% respectively.

Experimental methods

For crack nucleation tests, 8 mm cylindrical specimens were machined from rolled bar and mechanically polished to obtain a surface finish with arithmetic roughness of about 0.1 μm. Uniaxial fatigue tests were performed under load control, using a resonant electromagnetic fatigue testing machine at 120 Hz (Vibrophore type). This frequency was chosen so that failure between 10 5 and 10 7 cycles could be reached in a reasonable testing time. Moreover, this loading frequency is typical for HCF test, even if it is higher than what is usually used for corrosion fatigue studies [START_REF] Ebara | Corrosion fatigue crack initiation in 12% chromium stainless steel[END_REF]. The stop criterion was a loading frequency drop of 0.7 Hz, corresponding to a technical fatigue crack with a typical surface length of 5 mm and a depth of 2 mm. In order to investigate a possible frequency effect, additional CF tests were performed at 10 Hz on a servo-hydraulic fatigue testing machine.

CF testing were carried out at room temperature (≃20 °C) under an aerated 0.1 M NaCl and 0.044 M Na 2 SO 4 aqueous solution (A s ) (with pH ≃ 6 and [O 2 ] ≃ 7 ppm) at the free potential. This aqueous media was optimized to reproduce the same morphology of corrosion defect observed at the surface of real components using a thin-layer cell [START_REF] Marcelin | Electrochemical investigations on crevice corrosion of a martensitic stainless steel in a thin-layer cell[END_REF]. The addition of sulphate limited the increase of the anodic potentials with the inhibitive effect of SO - 4 2 anion on pitting corrosion [START_REF] Marcelin | Electrochemical characterisation of a martensitic stainless steel in a neutral chloride solution[END_REF][START_REF] Trompette | Influence of the anion specificity on the electrochemical corrosion of anodized aluminum substrates[END_REF]. To perform corrosion fatigue tests in the passive domain of the tested material, the (A s ) aqueous solution was selected. In this case, the prospective depassivation and pitting of the material under cyclic loading (fatigue test) is stress-assisted mechanisms. An electrochemical corrosion cell was developed for corrosion fatigue tests. This cell allows insitu fatigue testing in an aqueous corrosive environment and include three electrodes: a saturated calomel electrode (SCE) as the reference maintained at a constant distance (3 mm) from the specimen surface, a platinum counter electrode (CE) and the specimen as the working electrode (WE). Specimens were electrically insulated from the frame of the fatigue testing machine. The whole gauge section (area of interest) of the specimen was electrochemically monitored using a potentiostat device (Versastat4). During the CF tests, the free potential (OCP) value was monitored. Electrochemical impedance spectroscopy (EIS) analyses were carried out every 30 min (2.16 × 10 5 cycles at 120 Hz). Usual equivalent circuit modelling of the EIS data was used to extract physical properties of the electrochemical system by modelling the impedance data in terms of an equivalent electrical circuit. A solution resistance, R s , in series with a constant phase element, CPE, in parallel with a charge-transfer resistance, R ct , was considered, in particular to investigate the high frequency domain of electrochemical impedance measurements [START_REF] Marcelin | Electrochemical investigations on crevice corrosion of a martensitic stainless steel in a thin-layer cell[END_REF][START_REF] Monnot | Molybdenum effect on the sulfide stress corrosion of a super martensitic stainless steel in sour environment highlighted by electrochemical impedance spectroscopy[END_REF][START_REF] Lu | The effect of tempering temperature on the microstructure and electrochemical properties of a 13wt.% Crtype martensitic stainless steel[END_REF]. The high frequency range provides information on the passive film. burgers vector e corrosion defect thickness galvanostat mode. A small amplitude sine wave current modulation ( ± 6 μA/cm 2 ) was carried out with a scanning frequency from 100 Hz up to 0.1 Hz. All diagrams exhibit a capacitive response characterised by the first part of a half circle. A fitting program (Zsimwin) was used to provide the coefficient values of both the CPE, R ct and R s . The capacitance of passive film was calculated using the Q and α parameters of the CPE using the Brug [START_REF] Brug | The analysis of electrode impedances complicated by the presence of a constant phase element[END_REF] formula given by:

= × - C Q R α α α 1 s 1 (1)
The electrochemical investigation was carried out under both steady state and transient regimes. Two electrochemical techniques were used in this study: voltammetry and electrochemical impedance spectroscopy (EIS).

Results and discussion

Corrosion fatigue crack initiation mechanism

The effect of both the loading frequency and the aqueous corrosion on fatigue crack initiation life is detailed in [START_REF] El-May | Effect of corrosion on the high cycle fatigue strength of martensitic stainless steel X12CrNiMoV12-3[END_REF]. Testing results show that the loading frequency (10 Hz and 120 Hz) has a little effect on the HCF strength in the aqueous solution as shown in Fig. 1. For a given stress amplitude the number of cycles to CFCI at 10 Hz are lower than those obtained at 120 Hz. The S-N curves of the X12CrNiMoV12-3 stainless steel under fully reversed tension show a significant effect of aqueous corrosion on fatigue strength. The median fatigue strength at 10 7 cycles in aqueous solution decreases of 33% compared to the one in air (Fig. 1).

The open circuit potential (OCP) of the X12CrNiMoV12-3 steel without any mechanical loading is approximately +0.1 V/SCE which corresponds to the passivation domain. However, the free potential measurements during CF tests show that, for all the corrosion fatigue tests leading to crack initiation, a decrease in the free potential to the activated corrosion peak potential is observed systematically (Fig. 2-a). This phenomenon is attributed to the damage of the protective (passive) film in aqueous solution due to the cyclic loading. For specimens which did not fail after 10 7 cycles, under the stress amplitude equals to the median CF strength at this number of cycles, the free potential decrease was not observed (Fig. 2-b).

During corrosion fatigue tests Electrochemical Impedance Spectroscopy (EIS) were carried out to evaluate the passivation state of the tested specimen. For CF test leading to failure, two regimes were clearly observed: one without cyclic stress and one with cyclic stress. In the regime without cyclic stress, a low capacitance was measured. This result denotes the presence of a protective passive film. Once the cyclic stress is applied, the capacitance increases significantly. This result shows a progressive local rupture of the passive film induced by the applied cyclic loading (no generalised corrosion was observed after CF tests). Optical and SEM observations of the surface indicate that this OCP drop is associated to the development of corrosion pits at the surface. For not failed specimens, the capacitance increase was not observed, and the capacitance value was constant, less than 7 μF/cm 2 . This value corresponds to the capacitance of the passive layer of stainless steels measured without cyclic loading in the passive regime. This demonstrates that the aqueous media does not dissolve the passive film but that synergetic effects drive the process. AES Measurements were performed at ten different locations on the specimen surface in order to evaluate the thickness uniformity of the passive film. The half of the oxygen content in the passive film is considered [START_REF] Rezek | Structure and corrosion resistance of oxides grown on maraging steel in steam at elevated temperatures[END_REF][START_REF] Vignal | Influence of the microstructure on the corrosion behaviour of lowcarbon martensitic stainless steel after tempering treatment[END_REF] as the corresponding thickness of passive film (see Fig. 3) and was found to be uniform on the specimen surface. Its value is around 3 nm for the passive film formed in air.

We conclude that the passive film is broken by material strain due to the applied cyclic loading, causing increased local corrosive attack. The passive film rupture threshold corresponds to the median fatigue strength at 10 7 cycles of the material in A s aqueous solution; it is noted Δσ th . Exceeding this threshold stress range, the local deformations associated to the development of micro-plasticity induce strong incompatibilities between the bulk and the passive film that cannot be accommodated by the later. This results in local passive film failure that initiates the local corrosion process. The fractographic examination indicates that corrosion fatigue cracks essentially nucleated and grew from one corrosion defect [START_REF] El-May | Effect of corrosion on the high cycle fatigue strength of martensitic stainless steel X12CrNiMoV12-3[END_REF]. A schematic illustration of this CFCI process is illustrated in Fig. 4. The proposed CFCI scenario contains tow essential periods leading to crack initiation from geometrical defect:

1. the local ruptures of the passive film by PSB emergence at the These steps are identified on the free potential evolution curve (Fig. 2-a). The decrease of the free potential corresponds to local ruptures of the passive film and the plateau corresponds to the free metal dissolution around depassivation sites at the potential of the activated corrosion peak. In this case, the number of cycles to CFCI life N f can be expressed by:

= + N N N f d c (2)
where N d is the number of cycles necessary to the local rupture of the passive film by PSBs emergence at the surface and N c is the number of cycles needed to develop a critical corrosion defect for fatigue crack initiation.

3.2. Analytical model

3.2.1. Depassivation life (N d )
The time necessary for the depassivation of the metal under cyclic loading depends on both the kinematic of PSBs emergence, and the time needed to obtain PSB high enough to induce the passive film failure.

Harvey [START_REF] Harvey | Atomic force microscopy and modeling of fatigue crack initiation in metals[END_REF] and Gerberich [START_REF] Gerberich | Low and high cycle fatiguea continuum supported by AFM observations[END_REF] described, from atomic force microscopy (AFM) measurements, the fraction of plasticity emerging at the free surface of a cyclically stressed specimen. Considering slip deformation, as depicted in Fig. 5, the amount of emerging strain at the surface during one cycle can be written as:

= f δ h *Δϵ p s s (3)
where δ s cum is the surface upset created by one cycle, h s is the slip plane

Table 1

Calibrated parameter values in the depassivation life model Eq. ( 10). spacing. The δ s /h s parameter represents a strain and the quantity f * Δϵ p reflects the fact that only a fraction of the dislocations created during a given stress cycle can emerge at the surface; i.e. f * is the fraction of emerging plasticity at the free surface. For N cycles, the cumulated strain (displacement over gauge length) can be written as:

= δ h f N *Δϵ s cum s p (4)
where δ s cum is the cumulative surface upset. Gerberich et al. [START_REF] Gerberich | Low and high cycle fatiguea continuum supported by AFM observations[END_REF] determined the experimental evolution of the damage fraction, f * , versus the number of cycles, N, for the specimens under fully reversed fatigue loading in air and in 0.9% saline water under plastic strain ranges between 5 × 10 -4 and 2 × 10 -3 . These authors show that the fraction of emerging plasticity vs. the number of cycles can be expressed as:

= × f β N * κ (5)
where β and κ are two constants depending on the grain size of the material and the environment of the fatigue tests. Harvey et al. [START_REF] Harvey | Atomic force microscopy and modeling of fatigue crack initiation in metals[END_REF] show that the slip band spacing can continue to decrease under cyclic loading as long as the applied stress increases. The slip band spacing h s is inversely proportional to the effective cyclic stress range and it is expressed by:

= h α σ G b Δ s eff ( 6
)
where G is the shear modulus, b the Burgers vector, Δσ eff is the cyclic effective stress range and α is a fitting constant depending on the material grain size [START_REF] Harvey | Atomic force microscopy and modeling of fatigue crack initiation in metals[END_REF]. The definition of α needs AFM observations coupled with fatigue test at different stress levels to determine the empirical relation between h s and the applied cyclic stress. Δσ th is the median corrosion fatigue strength at 10 7 cycles determined experimentally by the stair case method. This corresponds to the threshold stress range of the passive film cracking under cyclic loading. Also, the cumulative surface upset δ s cum can be obtained by substituting Eq. ( 6) into Eq. ( 4).

= + δ β α ε σ N G b Δ Δ κ s cum p eff ( 1) (7)
Table 2 Calibrated parameter values in the corrosion growth life model Eq. ( 14). Fig. 12. Comparison between the number of cycles to develop a critical defect (experimental data) on the martensitic stainless steel X12CrNiMoV12-3 tested in the aqueous solution (A s ) at R =-1 and the corrosion growth life model (Eq. ( 14)). 15)).

The number of cycles N d necessary to provoke the failure of the passive film corresponds to the threshold value, δ s,th cum , of δ s cum to break the passive layer (Eq. ( 8)).

⎜ ⎟ = ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ + + N δ βα σ ε G b Δ Δ d κ κ s,th cum 1/( 1) eff p 1/( 1) (8)
The relation between Δσ eff and Δε p is experimentally determined for the X12CrNiMoV12-3 using tension-compression tests under various imposed plastic strain range as follows:

= σ k ε Δ (Δ ) m eff p (9)
where k et m are material parameters identified by fitting experimental cyclic hardening data.

Considering the depassivation stage for the formation of the fatigue crack, the number of cycles, N d , necessary to break the passive film can be obtained by substituting Eq. ( 9) into Eq. ( 8) so that: [START_REF] Schnbauer | The influence of corrosion pits on the fatigue life of 17-4PH steam turbine blade steel[END_REF] Note that G and b are material parameters. Furthermore, α, β and κ can be determined by AFM measurements.

⎜ ⎟ = ⎛ ⎝ × ⎞ ⎠ + - + N δ k βα σ G b (Δ ) m κ m κ m d s,th cum 1/ 1/( 1) eff 1 ( 1)

Number of cycle for corrosion defect growth (N c )

The development of corrosion defects constitutes the second period of the corrosion fatigue crack initiation mechanism on the X12CrNiMoV12-3. This step represents the anodic dissolution of the depassivated surface zones due to the local PSB emergence. The corrosion rate can be expressed by the Faraday equation [START_REF] Laird | Mechanisms of fatigue crack nucleation, corrosion fatigue: chemistry, mechanics and microstructure[END_REF][START_REF] Muller | Theoretical consideration on corrosion fatigue crack in initiation[END_REF][START_REF] Magnin | Corrosion fatigue mechanisms in BBC stainless steels[END_REF][START_REF] Duquette | Nace-2[END_REF]:

= V t MI z ρ d d F C c o r r (11)
where I corr is the mean value of the electrochemical current achieved during the corrosion growth period. The volume of the dissolved metal during corrosion growth period is given by

= V MI z ρ t F C corr (12)
The fractographic analysis of corrosion fatigue test specimens reveals the presence of a single crack initiation zone localised around a corrosion defect. We consider that the total volume corresponds to a single corrosion defect at the origin of the corrosion fatigue crack initiation. The shape of the corrosion defect can be estimated by the defect area (area) and the average thickness of slip bands (e) (Fig. 6).

The volume of the corrosion defect, V C , is expressed by Eq. ( 13) using the geometrical parameter of Murakami ( area ) [38].

= V e ( area) C 2 (13)
The fatigue crack initiation life at a number of cycles less than 10 7 cycles occurs when the size of the corrosion defect reaches a critical value for the initiation of a fatigue crack. This critical size of corrosion defect ( area ) th depends on the level of the effective stress range.

The number of cycles necessary to achieve the critical corrosion defect size can be obtained by:

= = N t f z ρe MI f F (( area ) ) c c corr th 2 (14)
where f is the loading frequency of the fatigue test.

Expression of number of cycles to CFCI (N f )

The fatigue life prediction model, based on the local depassivation by slip bands emergence and corrosion defect growth in depassivated area, yields to the relation between the number of cycles to crack initiation, in the HCF regime (between 10 5 and 10 7 cycles), N f , and material and mechanical parameters (Eq. ( 15)).

⎜ ⎟ = ⎛ ⎝ ⎞ ⎠ + + - + N δ k βα σ z ρe MI f G b (Δ ) F (( area ) ) m κ m κ m f s,d cum 1/ 1/ ( 1) eff 1 ( 1) corr th 2
(15)

Experimental identification of the CFCI model parameters

Tow sets of material parameters must be determined: one set in the depassivation life model and one set in the local corrosion life model. The CFCI model parameters were identified from experimental data on the martensitic stainless steel X13CrMoV12-3.

Parameters identification needed to compute the depassivation life (N d )

The actual determination of the constants: β, α and κ involves AFM measurements on the tested material. In this study, these parameters were estimated using literature data. Marsh et al. [START_REF] Marsh | Environmental effects on fatigue crack development in HSLA steel and fine grain α-Titanium[END_REF] showed that the mechanism which produces slip band emergence is identical for the two environments (air and saline water with 0.9 mass% NaCl). However, this phenomenon strongly depends on the average grain size of the material [START_REF] Harvey | Atomic force microscopy and modeling of fatigue crack initiation in metals[END_REF]. Based on this argument, the material parameters β, α and κ (in Eqs. ( 5) and ( 6)) were estimated using experimental data obtained in the literature (see Figs. 7 and8) [START_REF] Marsh | Environmental effects on fatigue crack development in HSLA steel and fine grain α-Titanium[END_REF][START_REF] Landgraf | ICM II American Society for Metals[END_REF] on fatigue crack nucleation in fine grain HSLA steel tested in 0.9 mass% NaCl aqueous solution (vs. 0.6 mass% NaCl in our study). The typical martensite lath dimension is around 2-3 μm in HSLA steel [START_REF] Marsh | Environmental effects on fatigue crack development in HSLA steel and fine grain α-Titanium[END_REF][START_REF] Landgraf | ICM II American Society for Metals[END_REF], consequently these data can be used to estimate these parameters on the material of the present study. The fitted curve and the experimental data are shown in Fig. 8.

Cyclic hardening tests were carried out at imposed cyclic strain on the martensitic stainless steel X12CrNiMoV12-3 to calibrate k and m parameters (Eq. ( 9)). The plastic deformation, Δε p , and the stress range, Δσ eff , were calculated using the stabilized cycle (at N f /2). The fitted curve and experimental data for unixial cyclic hardening are shown in Fig. 9.

The threshold value, δ s,th cum , of δ s to break the passive layer (Eq. ( 8))

corresponds to the passive layer thickness determined experimentally from AES measurements given in Fig. 2. The values of material parameters b and G were calibrated using experimentally data obtained from the literature. The calibrated values of the depassivation model and identification experimental techniques are given in Table 1.

Fig. 10 presents the evolution of the depassivation life, N d , in A s as a function of the effective stress range, Δσ eff . Experimental data of N d was obtained using the OCP measurement during corrosion fatigue tests on the martensitic stainless steel X12CrNiMoV12-3 (Fig. 2-a). The predicted depassivation life, N d , agrees with the experimental data.

Parameters identification needed to compute the corrosion growth life (N c )

In the case of the stainless steel, we consider the oxidation degree of Fe 2+ , to calculate the molar mass (M = 55.85 g/mol and z = 2). F is the Faraday constant and ρ is the density of the studied material equal to 7, 8 g/cm 3 . The corrosion current, I corr , is considered as a constant value independent on the effective stress range levels of the fatigue tests. The corrosion current density value is around i corr = 4.5 μA/cm 2 at the free potential equal to -0.044V/SCE. It was determined experimentally determined by in-situ electrochemical measurements during corrosion fatigue test. The relation between the threshold defect size ( area ) th and the effective stress range, Δσ eff , is given by:

= C n σ ( area ) exp( Δ ) th eff (16)
where C and n are two material parameters identified by fitting of fatigue data obtained on precorroded specimens (Fig. 11). A study of the effect of the corrosion defect size on the fatigue crack initiation, when no mechanical/corrosion coupling occurs, can be found in [START_REF] El-May | Non-local high cycle fatigue strength criterion for metallic materials with corrosion defects[END_REF].

The calibrated values of corrosion growth model and identification experimental techniques are given in Table 2. Fig. 12 shows the evolution of the corrosion growth life, N c , in saline water (A s ) as a function of the effective stress range Δσ eff . The predicted life, N c , agrees well with the experimental data obtained using the OCP measurement during corrosion fatigue tests on the martensitic stainless steel X12CrNiMoV12-3 (Fig. 2-a).

CFCI life model verification

The results of the proposed CFCI model are compared with corrosion fatigue data on martensitic stainless steel X12CrNiMoV12-3 tested in A s aqueous solution at loading frequency f = 120 Hz and stress ratio R =-1 (Fig. 13). As it can be seen, the calculated corrosion model (Eq. ( 15)) gives a good assessment of the experimental results on X12CrNiMoV12-3. It is worth noting that the predicted CFCI life, N f , fits well the experimental ones while none parameter of the model directly uses these specific data.

Effect of the loading frequency on the CFCI life

The effect of the loading frequency in corrosion fatigue is related to the repassivation rate after local ruptures of the passive film and possibly modifies the corrosion growth period. However, it has been shown in a previous study [START_REF] El-May | Effect of corrosion on the high cycle fatigue strength of martensitic stainless steel X12CrNiMoV12-3[END_REF] that the repassivation rate of the martensitic stainless steel X12CrNiMoV12-3 in A s aqueous solution is significantly lower than the characteristic time of the slip band emergence induced by the applied cyclic stress at a loading frequency of 10 and 120 Hz. This finding is in agreement with the CFCI life model (Eq. ( 14)). The loading frequency effect on the experimental CFCI life is given in Fig. 14. It shows that the loading frequency (at 10 Hz and 120 Hz) has an effect on the CFCI life in aqueous solution (A s ): this effect is well captured by the proposed approach. This effect may become more important for very low frequencies and should be checked with corrosion fatigue tests at lower loading frequencies where the repassivation characteristic time is larger than the slip band emergence one.

Effect of the CFCI threshold

For the studied X12CrNiMoV12-3 martensitic stainless steel, the CFCI threshold corresponds to the stress range of non developing slip bands to fracture the passive film in HCF regime (10 5 to 10 7 cycles). The CFCI threshold depends on the alloy microstructure, the interaction between the alloy and environment, and on the fatigue loading conditions (stress range, stress ratio, etc.). The effect of stress ratio on CFCI threshold was investigated in this study. Tow stress ratio were considered R =-1 and R = 0.1. CF results show that the stress ratio affects the threshold stress, Δσ th , of CFCI. Considering the effective stress range, Δσ eff , the CFCI life, N f , at stress ratio R =-1 and R = 0.1 (given in Fig. 15), tend to be similar. Fig. 15 shows that the predicted CFCI life model gives good estimation of the experimental data at R =-1 and R = 0.1 using the threshold stress range experimentally identified for each stress ratio.

Conclusions

An analytical model able to compute the CFCI life, in HCF regime, has been proposed. This is containing a pit initiation model and corrosion growth one. It has been compared with experimental data on the martensitic stainless steel X12CrNiMoV12-3 and gives good results. The following key conclusions were obtained:

1. In-situ electrochemical measurements during corrosion fatigue tests show that the CFCI is due to (i) the local depassivation of the passive film and (ii) the corrosion pitting. 2. The depassivation period is associated to the slip bands emergence at the surface of the specimen. A depassivation life model, based on Gerberich and Harvey [START_REF] Harvey | Atomic force microscopy and modeling of fatigue crack initiation in metals[END_REF][START_REF] Gerberich | Low and high cycle fatiguea continuum supported by AFM observations[END_REF] was identified to predict the number of cycles needed to fracture the passive film. The calculated results agreed well with the experimental data.

3. The transition between corrosion defect and crack propagation is defined by the size of the critical defects, which was chosen considering fatigue tests on precorroded specimens. Using the Faraday law, the corrosion rate under cyclic loading was estimated to compute the corrosion growth period of the critical corrosion defect. The calculated results agree well with the experimental data. 4. The proposed CFCI life model in HCF regime (Eq. ( 15)) takes into account the coupling effect between the pit initiation (depassivation) and the effective stress range. The consistency between the experimental and analytical results suggests that the effective stress range has an important effect on the corrosion process during the fatigue tests. In the same corrosion media, the calculated CFCI lives for stress ratio R = 0.1 and R =-1 agree well with the experimental data. 5. The effect of the loading frequency mostly concerns corrosion growth, while the depassivation phenomenon is not affected in the frequency range investigated here. Tow loading frequencies 120 Hz and 10 Hz were studied. With the decrease of loading frequency the corrosion time increases for the same number of cycles and the fatigue life is extended. The proposed CFCI life model gives a good estimation of the experimental data at 120 Hz and 10 Hz.

Fig. 1 .

 1 Fig. 1. Fatigue test results and S-N curves of corrosion fatigue tests under tension R =-1, for two different loading frequencies: f = 120 Hz and f = 10 Hz.

Fig. 2 .

 2 Fig. 2. Free potential (OCP) and capacitance of passive film (EIS) evolutions during corrosion fatigue test in A s aqueous solution under tension R =-1: (a) CFCI at 9.4 × 10 5 cycles and (b) specimen without failure at 10 7 cycles.

Fig. 3 .

 3 Fig. 3. Results of Auger measurements (AES) at the surface of the X12CrNiMoV12-3 martensitic stainless steel with passive film formed in air.

Fig. 4 .

 4 Fig. 4. Scenario of corrosion fatigue crack initiation on the X12CrNiMo12-3 martensitic stainless steel under a 120 Hz or 10 Hz loading frequency in the HCF regime (10 5 -10 7 cycles).

Fig. 5 .

 5 Fig. 5. Schematic of slip band height δ s cum , accumulated after N cycles and average spacing between slip bands h s on a single crystal from referecnce [47].

Fig. 6 .

 6 Fig. 6. Schematic model of the corrosion defect developed during corrosion fatigue test on martensitic stainless steel X12CrNiMoV12-3.

Fig. 7 .

 7 Fig. 7. Relation between the damage fraction f * and the number of cycle on HSLA steel, experimental data issue from reference [49].

Fig. 8 .

 8 Fig.8. Relation between PSB spacing, h s , and the plastic deformation range, Δε p , on HSLA steel, experimental data issue from reference[START_REF] Landgraf | ICM II American Society for Metals[END_REF].

Fig. 9 .

 9 Fig. 9. Relation between the effective stress range, Δσ eff , and the plastic deformation range, Δε p , on the martensitic stainless steel X12CrNiMoV12-3.

FromFig. 10 .

 10 Fig.10. Comparison between experimental data on the martensitic stainless steel X12CrNiMoV12-3 tested in the aqueous solution (A s ) at R =-1 and the depassivation life model (Eq. (10)).

Fig. 11 .

 11 Fig. 11. Relation between the threshold defect size ( area ) th for fatigue crack initiation and the effective stress range (Δσ eff ).

  Material parameters M = 55.85 g/mol, z = 2 and ρ = 7.8 g/ cm 3 From optical microscope e = 2 μm From in-situ electrochemical measurement I corr = 4.5 μA; the exposed surface is 1 cm 2 From fatigue tests on precorroded specimens C = 70.45 μm and n =-0.006 MPa -1

Fig. 13 .

 13 Fig. 13. Comparison between the number of cycles to fatigue crack initiation (experimental data) on the martensitic stainless steel X12CrNiMoV12-3 tested in the aqueous solution (A s ) at the loading frequency f = 120 Hz and the CFCI life model (Eq. (15)).

Fig. 14 .

 14 Fig. 14. Comparison between experimental data on the martensitic stainless steel X12CrNiMoV12-3 tested in the aqueous solution (A s ) with two loading frequency f = 120 Hz and f = 10 Hz at stress ratio R =-1 and the CFCI life model (Eq. (15)).

Fig. 15 .

 15 Fig. 15. Comparison between experimental data on the martensitic stainless steel X12CrNiMoV12-3 tested in the aqueous solution (A s ) with two stress ratio R =-1 and R = 0.1 at f = 120 Hz and the CFCI life model (Eq. (15)).

  The EIS investigation was performed in Δσ = σ max -σ min Δσ th threshold value of Δσ for depassivation Δσ eff effective stress range: Δσ eff = Δσ -Δσ th

	Nomenclature AES Auger electron spectroscopy HCF high cycle fatigue CFCI corrosion fatigue crack initiation CF corrosion fatigue PSB persistent slip band N f number of cycles to fatigue crack initiation N d number of cycles to depassivation N c number of cycles to develop a corrosion critical defect with regard to fatigue crack initiation R loading ratio f loading frequency Δσ stress range: Δε p area SCE E free I corr i corr A s ρ F M δ s h s	plastic strain range: square root area of the defect projected onto a plane = -ε ε ε Δ Δ Δ p max p min p perpendicular to the maximum normal stress saturated calomel electrode free electrochemical potential current value achieved during activated corrosion period (N > N d ) current density, the exposed surface is 1 cm 2 aqueous solution with 0.1 M NaCl and 0.044 M Na 2 SO 4 volumetric mass Faraday constant molar mass slip band height slip band spacing
	b

Acknowledgements

This work was carried out in the framework of the ARCAM project, with the financial support of DGCIS, Aquitaine, Auvergne and Midi-Pyrenees French regions. The authors thankfully acknowledge the industrial partners for the project, including Ratier-Figeac, Aubert et Duval, Olympus, and the academic partners at the Material Department of ICAM and the CIRIMAT (Toulouse University, France).


Giang-Nam Nguyen
email: gnguyen1@visteon.com

Kevin Heggarty

Julien Le Meur

Andreas Bacher

Patrick Meyrueis

Diffraction symmetry of binary Fourier elements with feature sizes on the order of the illumination wavelength and effects of fabrication errors

When building spot array binary Fourier diffractive optical elements (DOEs) having feature sizes on the order of the wavelength, we noticed remarkable variations in the experimental diffraction efficiency compared to the simulation results. Even with the use of high-cost electron beam lithography and rigorous Fourier modal method simulations, there appear to be no publications, to the best of our knowledge, showing close agreement in diffraction efficiency between the simulation and experimental results. In this Letter, we show that the diffraction symmetry of binary Fourier DOEs can be an efficient and consistent metric for evaluating the limit of the thin-element approximation and the effects of fabrication errors. of America https://doi.org/10.1364/OL.42.005178 diffraction models have been used for decades [START_REF] Moharam | [END_REF]7], there appears to be little or no experimental work verifying the limits of the TEA in practice, as it is often difficult to separate the simulation errors from the fabrication errors [START_REF] O'shea | Diffractive Optics: Design, Fabrication, and Test[END_REF]. Knowledge of fabrication errors for the zeroth order has been used for precompensation in the design and fabrication process to increase the performance of the fabricated DOEs [START_REF] O'shea | [END_REF][START_REF] Kress | Applied Digital Optics: From Microoptics to Nanophotonics[END_REF]. However, the errors in the non-zero orders are less studied and understood. In this Letter, by comparing TEA-based scalar diffraction models with rigorous Fourier modal method (FMM) simulations and the experimental results, the practical limit of the TEA will be examined. We show that the symmetry of the DOE diffraction pattern can be used as a simple, yet efficient, metric for evaluating the accuracy of the TEA.

When building spot array binary Fourier DOEs with feature sizes on the order of the wavelength, we noticed remarkable variations in the symmetry of diffraction efficiency. In this Letter, we will assess the limit of the TEA in predicting two types of symmetries that are frequently used in practice, which are Hermitian and reflection symmetry [START_REF] Goodman | Introduction to Fourier Optics[END_REF][START_REF] Morrison | [END_REF]. Figure 1(a) shows a reflection symmetric binary phase Fourier DOE, which As diffractive optical element (DOE) fabrication techniques have improved, it has become common to fabricate DOEs with feature sizes close to the limits of the scalar diffraction models generally used to design DOEs. More rigorous vector models are available, but their computational complexity often limits their application to relatively simple DOEs (e.g., 1D, periodic, small extent). Thus, there is an increasingly important need for ways to determine when designing a DOE whether scalar models are sufficiently accurate or whether vector models are necessary. Unfortunately, there appears to be no general agreement in the literature on this question. Several authors [1] and recent publications [2] indicate that since the Rayleigh-Sommerfeld diffraction integral is rigorous for free-space propagation [START_REF] Voelz | Computational Fourier Optics: A Matlab Tutorial[END_REF], the validity of the scalar diffraction model primarily depends on the thin-element approximation (TEA), which is used to model the DOE region [START_REF] Mellin | [END_REF][START_REF] Goodman | Introduction to Fourier Optics[END_REF]. Although rigorous vector Optics Letters, 2017 was designed to generate a 5 × 5 spot array, as illustrated in Fig. 1(b).

The experimental results of this DOE are shown in Table 1, where the DOE was fabricated at a pixel size of 1 μm using electron beam lithography [12], and the DOE illumination wavelength was 633 nm. At this wavelength, the refractive index of the substrate is n 1.4574, which means that the etching depth should be about d λ∕2n -1 ≈ 692 nm to obtain a π phase shift between binary levels [START_REF] O'shea | Diffractive Optics: Design, Fabrication, and Test[END_REF]. Figure 2 shows the atomic force microscope (AFM) measurement results of a test DOE, with an etching depth of about 700 nm.

As the designed DOE was a binary phase, we expected Hermitian symmetry [START_REF] Goodman | Introduction to Fourier Optics[END_REF] but, in fact, we observed reflection symmetry in the experimental diffraction pattern. For instance, the spot diffraction efficiency η1; -2 should be the same as η-1; 2 but, in fact, they are very different from each other (i.e., 1.77% and 2.90%, respectively). Instead, η1; -2 and η2; -1 are very similar (i.e., 1.77% and 1.81%, respectively). Further analysis shows, within experimental errors, a reflection symmetry of the diffraction pattern in the same axis as the reflection symmetry of the designed DOE, which is the dashed line shown in Fig. 1(a).

The far-field diffraction pattern of this DOE with refractive index n 1.4574 and feature size s 1 μm was then reconstructed in simulation at λ 633 nm using the TEA for the DOE region and our scalar non-paraxial propagator for the free-space propagation region [2]. As the rigorous Rayleigh-Sommerfeld diffraction integral had been shown to be equivalent to our scalar non-paraxial diffraction model [2], which is based on a single Fourier transform (FT), we would expect the symmetry of the far-field diffraction pattern to depend on the symmetry of the FT. At the optimum etching depth predicted by the TEA, the phase shift between the binary levels is π, which means that the diffracted field after a binary DOE can be represented in the TEA by a real function f x; y (i.e., e j•0 1, e j•π -1). Therefore, the diffraction pattern F X ; Y should have Hermitian symmetry [START_REF] James | A Student's Guide to Fourier Transforms: With Applications in Physics and Engineering[END_REF]:

f x; y f x; y → F X ; Y F -X ; -Y ; (1)
where F X ; Y is the FT of f x; y and the symbol * denotes the complex conjugate. This means that the diffracted powers of the spots should be Hermitian symmetric, as jF X ; Y j 2 jF -X ; -Y j 2 . Simulations show that for general binary DOEs (i.e., not at π phase shift), scalar theory also leads to Hermitian diffraction symmetry. Moreover, as the DOE is reflection symmetric, as shown in Fig. 1(a), the scalar theory predicts that, as it is based on a single FT, the far-field diffraction pattern should have the same reflection symmetry [START_REF] James | A Student's Guide to Fourier Transforms: With Applications in Physics and Engineering[END_REF]:

f x; y f -y; -x → F X ; Y F -Y ; -X : (2)
Table 2 lists the simulated diffraction efficiency of different diffraction orders ηm; n obtained by the TEA and our scalar model. Both Hermitian and reflection symmetry can clearly be seen, i.e., ηm; n η-m; -n and ηm; n η-n; -m, with m; n -2; …; 2.

For reference, the FMM, which is a rigorous vector diffraction method included in LightTrans VirtualLab [START_REF] Gmbh | [END_REF], was used, where the DOE structure was defined in the programmable interface. Table 3 shows the simulation results for linearly polarized illumination at 45°. Although the predicted diffraction angles are the same as in the scalar simulation, the predicted diffraction efficiencies differ considerably. In particular, the symmetry of the diffracted pattern changes, e.g., η1; -2 and η-1; 2 are different from each other (i.e., 2.13% and 2.91%, respectively). Instead, the diffraction efficiencies η1; -2 and η2; -1 are exactly the same (i.e., 2.13%). Further analysis shows that the diffraction pattern is no longer Hermitian symmetric, while the reflection symmetry remains. This is because the phase of the diffracted field after the structure is a continuous function instead of a binary phase shift between binary levels. As the result, the diffracted field after the structure is a complex function, so the far-field diffraction pattern is not Hermitian symmetric. In order to show the limit of the TEA in predicting the diffraction symmetry of binary Fourier elements, we define an average Hermitian symmetry factor as follows:

S H 1 N H X s; where s ηm; n η-m; -n ; with s ≤ 1: (3)
In this expression, N H is the number of Hermitian symmetric pairs of diffraction spots predicted by the TEA and the freespace propagator, e.g., N H 12 for the case of our 5 × 5 spot array DOE. The far-field diffraction patterns of the DOE at different feature sizes were reconstructed using the FMM. The average Hermitian symmetry factor was calculated at different feature sizes and shown in Fig. 3. It can be seen that Hermitian symmetry reduces significantly with smaller pixel sizes, in comparison with the scalar prediction S H 1. The experimental results show good agreement, within experimental errors, to the FMM simulation.

Similar to the average Hermitian symmetry factor, we define an average reflection symmetry factor of the diffracted spot powers as follows:

S R 1 N R X s; where s ηm; n η-n; -m ; with s ≤ 1: (4)
In this expression, N R is the number of reflection symmetric pairs predicted by the TEA and the free-space propagator, e.g., N R 10 for the case of our DOE. Figure 4 shows the average reflection symmetry factor at different feature sizes, where both the scalar theory and FMM simulation with linear polarization at 45°predict that S R 1. The measured reflection symmetry is very close to the theoretical value, except at a 0.25 μm pixel size.

Our FMM simulations assume a perfect fabrication with no proximity effect pixel rounding taken into account. When the structure size approaches the limits of the fabrication technology, such rounding effects will start to occur, and deviations between simulated and experimental performance are to be expected. This would appear to explain the increased differences observed between the FMM simulations and the experimental results for the 0.25 μm pixel size. Due to fabrication errors, the edge between the phase levels of the fabricated DOE is not perfectly sharp, but rather smooth, as shown in Fig. 2(b). The fabricated DOE becomes, in effect, a multiphase level element; the transmission function is no longer real and, therefore, the diffraction pattern is no longer Hermitian symmetric. If the optical setup of the lithography is perfectly aligned, the fabrication errors will be symmetric, and the fabricated DOE would still retain the symmetry of the designed DOE, as would the experimental diffraction pattern. To investigate this smoothing effect on diffraction efficiency in simulation, it is therefore necessary to oversample the DOEs. We sampled at 125 nm resolution and applied different smoothing filters [START_REF] Gonzalez | Digital Image Processing[END_REF] to the DOE pattern. These filters correspond to different models for the proximity effect in fabrication, as shown in Fig. 5.

Simulations using our scalar non-paraxial propagator show that the diffraction pattern of the smoothed structure is no longer Hermitian symmetric, but still reflection symmetric. The average Hermitian symmetry of the smoothed structures using 3 × 3 filters is added in Fig. 3 for comparison, where they are quite close to the FMM simulation and the experimental result. These results show that the Hermitian symmetry of the diffraction pattern of a binary phase DOE appears to be a useful indicator of the practical limit of the TEA model. If the diffraction pattern maintains Hermitian symmetry, the TEA is valid and can be used to design DOEs. If the diffraction pattern loses Hermitian symmetry, the TEA is no longer valid and more sophisticated models taking into account the electromagnetic fields inside the DOEs structure are necessary. The Hermitian symmetry factor we introduced allows this distinction to be made with a degree of independence from the precise values of the diffraction efficiency of the different diffracted orders and, hence, of the fabrication errors. In terms of reflection symmetry, we see that diffraction symmetry of a binary Fourier DOE can also be used to determine whether the lithographic fabrication setup is well aligned, i.e., the DOE structure is still reflection symmetric, despite the proximity effect at the resolution limit of the lithography system.

For general binary phase DOEs without reflection symmetry, the output pattern will always show Hermitian symmetry if the TEA is valid because the DOE transmission is real. The Hermitian symmetry factor introduced here, therefore, can be used to assess the validity of the TEA approximation for all such binary phase DOEs. For DOEs to be used with different polarized illuminations, more detailed investigation using, for example, FMM simulations would be required to assess the polarization effects, but the approach indicated here would still be a useful guide to the closeness to the potential limits of the TEA. Future work should also concentrate on the FMM simulation of the smoothed structure to take into account the effects of fabrication errors on the diffraction efficiency.

In summary, scalar models that make some allowance for the "thickness" of the DOE structure, for example, by modeling fabrication defects as a multi-phase level DOE rather than a binary phase DOE, appear to be a useful intermediate modeltechnique enabling the use of computationally expensive full vector models to be postponed. In the example studied here, the limit of the TEA model seems to be when DOE feature sizes are about twice the illuminating wavelength, but further investigation is required on this point, particularly on the relative importance of the structure's width and height.

  OCIS codes: (090.1970) Diffractive optics; (220.0220) Optical design and fabrication; (220.3740) Lithography; (050.6624) Subwavelength structures; (220.4241) Nanostructure fabrication.

Fig. 1 .

 1 Fig. 1. (a) Binary Fourier DOE, where the dashed line is the reflection symmetry axis. (b) Target image.

Fig. 2 .

 2 Fig. 2. (a) AFM image of the DOE. (b) Etching profile of the DOE.

Fig. 3 .

 3 Fig. 3. Average Hermitian symmetry at different DOE feature sizes.

Fig. 4 .

 4 Fig. 4. Average reflection symmetry at different DOE feature sizes.

Fig. 5 .

 5 Fig. 5. Different models for the etching profile: (a) flawless model, (b) trapezoid model, and (c) rounding model.

Table 1 .

 1 Experimental Diffraction Efficiency ηm;n % Obtained in Diffraction Order m;n for the Fabricated DOE

				n
	m	-2	-1	0	1	2
	2	1.74	1.81	2.86	3.59	1.98
	1	1.77	2.50	2.86	4.20	3.07
	0	2.84	2.90	3.98	2.95	2.69
	-1	3.72	4.43	3.02	1.70	2.90
	-2	2.05	3.09	2.90	2.80	1.09

Table 2

 2

		. Simulated Diffraction Efficiency ηm;n % of
	the DOE Using Our Scalar Propagator
				n
	m	-2	-1	0	1	2
	2	1.81	3.14	3.63	4.36	3.17
	1	3.14	2.45	3.18	4.26	4.36
	0	3.63	3.18	0.36	3.18	3.63
	-1	4.36	4.26	3.18	2.45	3.14
	-2	3.17	4.36	3.63	3.14	1.81

Table 3 .

 3 Simulated Diffraction Efficiency ηm;n % of the DOE Using FMM for Linearly Polarized Illumination at 45°n

	m	-2	-1	0	1	2
	2	1.70	2.13	3.08	3.85	2.18
	1	2.13	2.70	3.04	4.10	3.29
	0	3.08	3.04	1.89	3.10	3.17
	-1	3.85	4.10	3.10	1.75	2.91
	-2	2.18	3.29	3.17	2.91	1.00

Acknowledgment. The authors would like to thank Prof. Pierre-Emmanuel Durand for the AFM measurements.