Javier Pizarro-Cerdá

Dror Shlomo Chorev

Benjamin Geiger

Pascale Cossart
email: pascale.cossart@pasteur.fr

Javier Pizarro-Cerda
email: javier.pizarro-cerda@pasteur.fr

The diverse family of Arp2/3 complexes

The Arp2/3 complex has so far been considered to be a single seven-subunit protein complex required for actin nucleation and actin filament polymerization in diverse critical cellular functions including phagocytosis, vesicular trafficking, lamellipodia extension and cytokinesis. The Arp2/3 complex is also exploited by bacterial pathogens and viruses during cellular infectious processes. Three recent studies suggest that some subunits of the complex are dispensable in specific cellular contexts, pointing to the existence of alternative Arp2/3 complexes containing other components such as vinculin or α-actinin, as well as different isoforms or phosphorylation variants of canonical Arp2/3 subunits. This diversity should be considered when assigning specific Arp2/3 assemblies to different actin-dependent cellular processes.

Introduction

The actin cytoskeleton is one of the main components of eukaryotic cells, not only providing the molecular basis for cellular morphogenesis and migration, but also participating dynamically in mechanical resistance to deformation, uptake of extracellular material, intracellular vesicular transport, cytokinesis and cell adhesion. The actin cytoskeleton also participates in the organization of complex cellular structures such as filopodia, lamellipodia and podosomes [START_REF] Fletcher | Cell mechanics and the cytoskeleton[END_REF][START_REF] Pollard | Actin, a central player in cell shape and movement[END_REF].

Polymerization of actin monomers into actin filaments requires the activity of cellular actin nucleators. The Arp2/3 complex, the first nucleator identified in eukaryotic cells, plays a central role in many cellular processes and is highly conserved from trypanosomes to the fission yeast and humans [START_REF] Berriman | The genome of the African trypanosome Trypanosoma brucei[END_REF][START_REF] Goley | The ARP2/3 complex: an actin nucleator comes of age[END_REF][START_REF] Krause | Steering cell migration: lamellipodium dynamics and the regulation of directional persistence[END_REF]. Other nucleators include formins, Spire, Cordon-bleu (COBL) and Leiomodins [START_REF] Campellone | A nucleator arms race: cellular control of actin assembly[END_REF].

The Arp2/3 complex is composed of seven subunits [START_REF] Pollard | Structure and function of the Arp2/3 complex[END_REF] and it has been traditionally considered as a single entity, associated with the vast majority of cellular processes in which its function is required and has been studied. Three recent studies [START_REF] Chorev | Regulation of focal adhesion formation by a vinculin-Arp2/3 hybrid complex[END_REF][START_REF] Kühbacher | Genome-Wide siRNA Screen Identifies Complementary Signaling Pathways Involved in ListeriaInfection and Reveals Different Actin Nucleation Mechanisms during ListeriaCell Invasion and Actin Comet Tail Formation[END_REF][10] in the same mammalian cell system reveal that diverse Arp2/3 complexes may regulate different cellular and pathogen-associated functions, raising the interesting possibility that Arp2/3 complex compositions may have been overlooked, paving the way for the identification of novel complexes associated to different actin polymerization-mediated processes.

Discovery and functions of the classical Arp2/3 complex

The Arp2/3 complex was first isolated from Acanthamoeba castellanii during a search for ligands of the actin-binding protein profilin [11]. It contained seven proteins: the actin-related proteins Arp2 (44-kD) and Arp3 (47-kD) considered as "unconventional actins", together with a 40-kD protein similar to a WD40 ß-propeller protein from Dictyostelium discoideum, and four additional proteins of 35-, 19-, 18-and 13-kD [11].

Subsequently, the Arp2/3 complex was also identified and associated with actin-rich structures in the fission yeast Schizosaccharomyces pombe [START_REF] Balasubramanian | Fission yeast Sop2p: a novel and evolutionarily conserved protein that interacts with Arp3p and modulates profilin function[END_REF] and in the budding yeast Saccharomyces cerevisiae [START_REF] Winter | The complex containing actin-related proteins Arp2 and Arp3 is required for the motility and integrity of yeast actin patches[END_REF]. In human cells, the Arp2/3 complex consists of Arp2 and Arp3, together with the Arp complex subunits ARPC1, ARPC2, ARPC3, ARPC4 and ARPC5 [14]. While consensus exists concerning the Arp2 and Arp3 nomenclature, different names have been used in the literature concerning the other Arp2/3 complex subunits: a nomenclature proposal across species is presented in Table 1.

The function of the Arp2/3 complex was shown for the first time to be critical in triggering actin polymerization when it was isolated from a subcellular fraction of human platelets that sustained actin assembly by the bacterial pathogen Listeria monocytogenes [START_REF] Welch | Actin polymerization is induced by Arp2/3 protein complex at the surface of Listeria monocytogenes[END_REF]. The L. monocytogenes surface protein ActA activates the Arp2/3 complex to initiate actin polymerization, and was the first actin nucleation promoting factor (NPF) to be identified [START_REF] Welch | Interaction of human Arp2/3 complex and the Listeria monocytogenes ActA protein in actin filament nucleation[END_REF][START_REF] Kocks | L. monocytogenes-induced actin assembly requires the actA gene product, a surface protein[END_REF]. Several mammalian NPFs were subsequently identified, including WASP [START_REF] Yarar | The Wiskott-Aldrich syndrome protein directs actin-based motility by stimulating actin nucleation with the Arp2/3 complex[END_REF], N-WASP [START_REF] Rohatgi | The interaction between N-WASP and the Arp2/3 complex links Cdc42-dependent signals to actin assembly[END_REF], Scar/WAVE [START_REF] Machesky | Scar, a WASp-related protein, activates nucleation of actin filaments by the Arp2/3 complex[END_REF], and cortactin [START_REF] Weed | Cortactin localization to sites of actin assembly in lamellipodia requires interactions with F-actin and the Arp2/3 complex[END_REF] (see Text Box 1). The purified A. castellanii Arp2/3 complex was shown to nucleate the formation of actin filaments at 70° from other filaments [22]. A series of elegant microscopy and biochemistry investigations then definitively established the key role of Arp2/3 in actin polymerization and the formation of branched structures [START_REF] Svitkina | Arp2/3 complex and actin depolymerizing factor/cofilin in dendritic organization and treadmilling of actin filament array in lamellipodia[END_REF][START_REF] Blanchoin | Direct observation of dendritic actin filament networks nucleated by Arp2/3 complex and WASP/Scar proteins[END_REF] (see Text Box 2).

As mentioned above, the function of the Arp2/3 complex is subverted by bacterial pathogens at different stages of their infectious processes [START_REF] Cossart | Actin-based motility of pathogens: the Arp2/3 complex is a central player[END_REF]. The Gram-positive pathogen L. monocytogenes uses Arp2/3 not only to mediate intra-and inter cellular movements but also to trigger cellular invasion [START_REF] Pizarro-Cerdá | Entry of Listeria monocytogenes in mammalian epithelial cells: an updated view[END_REF][27][START_REF] Sousa | Src, cortactin and Arp2/3 complex are required for Ecadherin-mediated internalization of Listeria into cells[END_REF]. The Gram-negative pathogen Shigella flexneri also requires Arp2/3 function for actin-based motility [29] and for bacterial internalization within host cells [START_REF] Bougnères | Cortactin and Crk cooperate to trigger actin polymerization during Shigella invasion of epithelial cells[END_REF]. Interestingly S. flexneri does not express an ActA-like protein but instead recruits on its surface, via the protein IcsA/VirG, the NPF N-WASP which in turn activates Arp2/3 to mediate actin-based motility [29,[START_REF] Suzuki | Neural Wiskott-Aldrich syndrome protein is implicated in the actin-based motility of Shigella flexneri[END_REF]. The Gram-negative bacteria Rickettsia parkerii and R. conorii activate Arp2/3 during early stages of bacterial intracellular motility via a protein called RickA [START_REF] Jeng | A Rickettsia WASP-like protein activates the Arp2/3 complex and mediates actin-based motility[END_REF][START_REF] Gouin | The RickA protein of Rickettsia conorii activates the Arp2/3 complex[END_REF][START_REF] Reed | Rickettsia Actin-Based MotilityOccurs in Distinct PhasesMediated by Different Actin Nucleators[END_REF]. Moreover, R. parkerii requires Arp2/3 activity to invade diverse host cells [START_REF] Reed | Rickettsia parkeri invasion of diverse host cells involves an Arp2/3 complex, WAVE complex and Rho-family GTPase-dependent pathway[END_REF]. Vaccinia virus is able to move at the surface of cells on actin-based structures [START_REF] Cudmore | Actin-based motility of vaccinia virus[END_REF], which requires the function of the Arp2/3 complex [START_REF] Frischknecht | Actin-based motility of vaccinia virus mimics receptor tyrosine kinase signalling[END_REF]. Other bacteria including Mycobacteria [START_REF] Stamm | Mycobacterium marinumEscapes from Phagosomes and Is Propelled by Actin-based Motility[END_REF] and Burkholderia thailandensis also move via an actin-based motility requiring Arp2/3 functions [START_REF] Sitthidet | Actin-Based Motility of Burkholderia thailandensis Requires a Central Acidic Domain of BimA That Recruits and Activates the Cellular Arp2/3 Complex[END_REF][START_REF] Gouin | Actin-based motility of intracellular pathogens[END_REF][41][START_REF] Gouin | Intracellular Bacteria Find the Right Motion[END_REF].

In S. pombe and S. cerevisiae, deletions of genes encoding each of the subunits of the Arp2/3 complex cause severe growth defects or lethality [43,44], suggesting a major role for all subunits in vivo. In particular, Arp2/3 had been shown to be important for the formation and function of cortical actin patches where clahtrin-mediated endocytosis takes place [START_REF] Winter | The complex containing actin-related proteins Arp2 and Arp3 is required for the motility and integrity of yeast actin patches[END_REF]. Mammalian Arp2/3 complex was localized to regions of lamellipodial protrusion [14,[START_REF] Machesky | Mammalian actin-related protein 2/3 complex localizes to regions of lamellipodial protrusion and is composed of evolutionarily conserved proteins[END_REF] and together with cofilin and other actin-binding proteins, was shown to control the organization and tread-milling of actin filaments in lamellipodia [START_REF] Svitkina | Arp2/3 complex and actin depolymerizing factor/cofilin in dendritic organization and treadmilling of actin filament array in lamellipodia[END_REF]. The Arp2/3 complex has been associated to other cellular functions requiring actin polymerization including phagocytosis [START_REF] May | Involvement of the Arp2/3 complex in phagocytosis mediated by FcgammaR or CR3[END_REF], trafficking within and from the Golgi apparatus [START_REF] Luna | Regulation of protein transport from the Golgi complex to the endoplasmic reticulum by CDC42 and N-WASP[END_REF] as well as formation of focal adhesions [START_REF] Demali | Recruitment of the Arp2/3 complex to vinculin: coupling membrane protrusion to matrix adhesion[END_REF]. The critical role of Arp2/3 in humans is highlighted by the Wiskott-Aldrich syndrome (WAS), a recessive X-linked genetic disorder characterized by mutations in the WAS protein (WASP), which is characterized by defects in the actin-rich immunological synapse between T cells and antigen presenting cells, leading to severe defects in immunological responses [49,[START_REF] Kirchhausen | Disease mechanism: unravelling Wiskott-Aldrich syndrome[END_REF]. Initial detailed analysis of the contribution of each Arp2/3 complex subunit to actin polymerization, using L. monocytogenes ActA as a NPF in a baculovirus expression system in insect cells, indicated that only Arp2 and Arp3 are directly involved in actin polymerization, the role of the other subunits being less clear [START_REF] Gournier | Reconstitution of human Arp2/3 complex reveals critical roles of individual subunits in complex structure and activity[END_REF]. More recent structural evidence [START_REF] Rouiller | The structural basis of actin filament branching by the Arp2/3 complex[END_REF] confirms an initial prediction that ARPC2 and ARPC4 provide the main surface for interaction of the complex with the mother actin filament [START_REF] Gournier | Reconstitution of human Arp2/3 complex reveals critical roles of individual subunits in complex structure and activity[END_REF]. ARPC3 is proposed to form a bridge between Arp3 and the mother actin filament [START_REF] Rouiller | The structural basis of actin filament branching by the Arp2/3 complex[END_REF] but complexes lacking ARPC3 display minor functional defects [44,[START_REF] Gournier | Reconstitution of human Arp2/3 complex reveals critical roles of individual subunits in complex structure and activity[END_REF]. While ARPC1 is supposed to make only minor contacts with the mother actin filament [START_REF] Rouiller | The structural basis of actin filament branching by the Arp2/3 complex[END_REF], complexes lacking this subunit are far less effective in actin nucleation, suggesting additional roles for ARPC1 including binding of NPFs [START_REF] Kelly | Actin binding to the central domain of WASP/Scar proteins plays a critical role in the activation of the Arp2/3 complex[END_REF]. ARPC5 was proposed to tether Arp2 to the rest of the complex [START_REF] Rouiller | The structural basis of actin filament branching by the Arp2/3 complex[END_REF].

Several reports also suggest a functional role played by phosphorylation of different subunits of the Arp2/3 complex. ARPC1 phosphorylation by p21-activated kinase (Pak1) was reported to be crucial for mammalian cell motility [START_REF] Vadlamudi | p41-Arc subunit of human Arp2/3 complex is a p21activated kinase-1-interacting substrate[END_REF]. It has been suggested that Arp2 phosphorylation is required and critical for Arp2/3 complex binding to the pointed end of actin filaments and actin nucleation in cultured Drosophila cells [START_REF] Leclaire | Phosphorylation of the Arp2/3 complex is necessary to nucleate actin filaments[END_REF], but mutation of the phosphorylated residues had only subtle effects on motility in Dictyostelium [START_REF] Choi | Phosphorylation of Actin-related Protein 2 (Arp2) Is Required for Normal Development and cAMP Chemotaxis in Dictyostelium[END_REF]. As shown recently, phosphorylation of Arp3 by the Legionella pneumophila kinase LegK2 inhibits actin polymerization at the surface of bacterial-containing phagosomes [57].

Several subunits of the Arp2/3 complex (i.e. Arp3, ARPC1 and ARPC5) display more than one isoform [14], but the functional significance of these variants had not been investigated in detail. While the major subunit Arp3 is detected in all tissues, a gene encoding the isoform ARP3β was detected predominantly in brain neuronal cells and was proposed to play a role in the development and/or maintenance of nerve cells [START_REF] Jay | ARP3beta, the gene encoding a new human actin-related protein, is alternatively spliced and predominantly expressed in brain neuronal cells[END_REF]. Two variants of ARPC1 presenting 70% homology had been known for long [START_REF] Balasubramanian | Fission yeast Sop2p: a novel and evolutionarily conserved protein that interacts with Arp3p and modulates profilin function[END_REF][START_REF] Machesky | Mammalian actin-related protein 2/3 complex localizes to regions of lamellipodial protrusion and is composed of evolutionarily conserved proteins[END_REF] and a mutation in the gene ARPC1A was shown to impact cell migration and invasion in pancreatic cancer [START_REF] Laurila | Characterization of the 7q21-q22 amplicon identifies ARPC1A, a subunit of the Arp2/3 complex, as a regulator of cell migration and invasion in pancreatic cancer[END_REF]. ARPC5 was also found to display a second isoform, named ARPC5B, which exhibited a regular expression in many tissues but with the highest levels in the brain, while the original ARPC5A was found highly enriched in the spleen and thymus [START_REF] Millard | Identification and characterisation of a novel human isoform of Arp2/3 complex subunit p16-ARC/ARPC5[END_REF].

Diversity of Arp2/3 complexes

Focal adhesions. Association of the Arp2/3 complex to focal adhesions in human skin cells had previously been shown to require interactions with vinculin [START_REF] Demali | Recruitment of the Arp2/3 complex to vinculin: coupling membrane protrusion to matrix adhesion[END_REF]. A recent native mass spectrometry analysis of proteins extracted from the dense plaques (focal adhesion homologous structures) of chicken smooth muscle revealed surprisingly that Arp2/3 complexes present in these structures, as inferred from mass spectrometry results, are actually "hybrid complexes", consisting of a core composed of Arp2, Arp3 and ARPC2, together with α-actinin and vinculin, or Arp2, Arp3, ARPC2, ARPC3 and vinculin [START_REF] Chorev | Regulation of focal adhesion formation by a vinculin-Arp2/3 hybrid complex[END_REF]. This study therefore supported, for the first time, the notion that alternative Arp2/3 complexes that do not consist of the seven classical subunits are involved in specific cellular processes. Notably, these alternative complexes contain vinculin that can mediate the recruitment of the complex to focal adhesions and compete with ARPC1B in HeLa cells; knock-down of ARPC1B has therefore a positive effect on focal adhesion and stress fiber formation, as the equilibrium is shifted towards Arp2/3-vinculin hybrid complexes formation [START_REF] Chorev | Regulation of focal adhesion formation by a vinculin-Arp2/3 hybrid complex[END_REF].

Listeria monocytogenes infection. Specific roles for ARPC1A and ARPC1B were recently identified in human genome-wide RNA interference (RNAi) screens investigating HeLa cell infection by L. monocytogenes [START_REF] Kühbacher | Genome-Wide siRNA Screen Identifies Complementary Signaling Pathways Involved in ListeriaInfection and Reveals Different Actin Nucleation Mechanisms during ListeriaCell Invasion and Actin Comet Tail Formation[END_REF]. Knock-down of ARPC1B but not of ARPC1A significantly diminished bacterial entry, highlighting a critical contribution for ARPC1B function in this context. Moreover, it was observed that ARPC4 and ARPC5 subunits do not contribute to L. monocytogenes cellular invasion. The contribution of the different Arp2/3 subunits to bacterial actin-tail formation was also studied, identifying a major contribution of Arp2, Arp3, ARPC1A, ARPC2, ARPC3 and ARPC4 in L. monocytogenes actin-based motility, but no role for ARPC1B nor ARPC5 was found [START_REF] Kühbacher | Genome-Wide siRNA Screen Identifies Complementary Signaling Pathways Involved in ListeriaInfection and Reveals Different Actin Nucleation Mechanisms during ListeriaCell Invasion and Actin Comet Tail Formation[END_REF]. These results therefore show not only that ARPC5 is dispensable for both bacterial entry and actin-tail formation but also that ARPC1 isoforms contribute to different cellular processes during L. monocytogenes infection, at least in HeLa cells. ARPC4 was found dispensable for bacterial entry, but taking into account the central place of this subunit in Arp2/3 complex function according to previous functional and structural results [START_REF] Gournier | Reconstitution of human Arp2/3 complex reveals critical roles of individual subunits in complex structure and activity[END_REF][START_REF] Rouiller | The structural basis of actin filament branching by the Arp2/3 complex[END_REF], it is possible that residual ARPC4 upon RNAi treatment suffices for partial complex function.

Vaccinia virus mobility.

In a recent study of actin polymerization by Vaccinia virus, specific roles for ARPC1B and ARPC5B have been found [10]. Indeed, it has been observed that Arp2/3 complexes containing ARPC1B and ARPC5B (named ARPC5L in this work) are significantly more efficient at promoting actin assembly than those containing ARPC1A and ARPC5A. Actin networks induced by complexes containing the subunits ARPC1B and ARPC5B were found more stable since in the presence of these specific subunits, cortactin stabilizes the Arp2/3 complexes against coronin-mediated disassembly [10].

Overall, these three reports indicate that only Arp2, Arp3 are directly partaking in the actin nucleating activity of the complex, together defining an actin nucleation core module, whereas the other subunits serve alternative roles such as determining the efficiency of actin nucleation, localization of the complex, as well as serving as an autoinhibitory mechanism [START_REF] Chorev | Regulation of focal adhesion formation by a vinculin-Arp2/3 hybrid complex[END_REF][START_REF] Gournier | Reconstitution of human Arp2/3 complex reveals critical roles of individual subunits in complex structure and activity[END_REF][START_REF] Rouiller | The structural basis of actin filament branching by the Arp2/3 complex[END_REF]. Together, the other subunits thus define the regulatory module of the Arp2/3 complex. In the case of the L. monocytogenes model, it is interesting to mention that vinculin inactivation by RNAi did not perturb bacterial cellular invasion nor actin-based motility [START_REF] Kühbacher | Genome-Wide siRNA Screen Identifies Complementary Signaling Pathways Involved in ListeriaInfection and Reveals Different Actin Nucleation Mechanisms during ListeriaCell Invasion and Actin Comet Tail Formation[END_REF], raising the possibility that other cellular molecule(s) not yet identified may participate to the localization/modulation of the Arp2/3 complex during L. monocytogenes infection-related processes.

Concluding Remarks

While the Arp2/3 complex has been classically considered as a single molecular entity for 20 years since its discovery, an emerging possibility from recent research suggests that multiple versions of the Arp2/3 complex may co-exist in cells (Figure 1 presents a summary of currently described complexes and their mode of regulation). Indeed, the subunits ARPC1, ARPC3, ARPC4 and ARPC5 can be replaced by vinculin and α-actinin in focal adhesions [START_REF] Chorev | Regulation of focal adhesion formation by a vinculin-Arp2/3 hybrid complex[END_REF].

In the case of the L. monocytogenes system, even if dispensable, ARPC5 can be detected at both bacterial entry sites and actin comet tails by fluorescence microscopy, indicating that Arp2/3 complexes containing this subunit, while not required, may still be recruited during both processes [START_REF] Kühbacher | Genome-Wide siRNA Screen Identifies Complementary Signaling Pathways Involved in ListeriaInfection and Reveals Different Actin Nucleation Mechanisms during ListeriaCell Invasion and Actin Comet Tail Formation[END_REF]. It is possible that Arp2/3 complexes of different composition have overlapping functions during L. monocytogenes infection, but current data suggests that the precise composition of different Arp2/3 complexes plays a role in fine-tuning actin rearrangements in both instances. This hypothesis is supported by the observation that ARPC4 can be found predominantly early during bacterial actin comet tail formation and that knock down of ARPC4 affects initial actin polymerization at the bacterial surface rather than actin tail elongation indicating that different Arp2/3 complexes may be required in a sequential manner. The fine tuning of Arp2/3 complex actin polymerization activity depending on the subunit composition is also supported by results on the Vaccinia virus system [10].

Overall, the reports discussed herein point to the possibility that Arp2/3 is a natural modular nano-machine, capable of regulating its activity via replacement of its subunits.

FigureFigure 1 :

 1 Figure

TableTable 1 :

 1 Arp2/3 complex nomenclature

Acknowledgments

This work was supported by the Institut Pasteur, the Institut National de la Santé et de la Recherche Médicale (INSERM Unité 604), the Institut National de la Recherche Agronomique (INRA Unité Sous Contrat 2020), the Institut Pasteur "Programmes Transversaux de Recherche" (PTR 460 and PTR521 to JPC), L'Agence Nationale de la Recherché (ANR-15-CE15-0017 StopBugEntry to JPC), Fondation Le Roch Les Mousquetaires, European Research Council Advanced Grant (670823 BacCellEpi to PC) and under grant agreement n o 294852-SynAd (to BG). P.C. is an International Senior Research Scholar of the Howard Hughes Medical Institute. B.G. holds the Erwin Neter Chair in Tumor and Cell Biology. The authors declare no conflict of interest.


François Biraben

Lucile Julien

François Nez
email: francois.nez@lkb.upmc.fr

T W Hänsch

Tilman Esslinger

Nathalie Picqué

Thomas Udem

Thirty years of hydrogen spectroscopy in Paris

several years, we also moved to the 1S-3S transition, a transition from the ground state, which is nowadays still under study in our group.

We show in this paper how Paris works in atomic hydrogen have been stimulated by the ones of Ted Hänsch's groups, first in Stanford and then in Garching, and how they developed in a context where an healthy competition turned often into collaboration and joint efforts for a common goal.

The beginning of the Paris story

At the beginning, we (FB and LJ) were then only two people on the hydrogen project in Paris, with the scientific support of Bernard Cagnac who founded a few years before in our laboratory a group devoted to Doppler-free two-photon spectroscopy. He himself had pointed out for the first time the advantage of this spectroscopy for the study of the 1S-2S transition in hydrogen [4] but without undertaking such an experiment since Ted Hänsch seized this subject very quickly. It is why we decided to study other hydrogen lines.

One of us (F.B.) had already a very good expertise in high-resolution spectroscopy, having studied two-photon transitions in sodium [5] and rare gas atoms [6,7]. The other one (L.J.) had previously measured atomic structures and Lamb shifts in excited hydrogen states by an anticrossing method [8].

Our target was to excite the n ≥ 8 states because the wavelength range needed 730-778 nm was easily obtained with our homemade dye laser.

In a first step, we had to build a 2S metastable atomic beam, which was obtained by the following method: molecular hydrogen is dissociated by a RF discharge in a

Introduction

Hydrogen spectroscopy in Paris is a long story which began in 1983. One of us (FB) decided to use the tunable cw monomode laser he had himself developed [1] to excite two-photon transitions in atomic hydrogen. At this time, Ted Hänsch's work on hydrogen-the simplest and the most fascinating of the atoms-were worldwide known: in particular, the first observation of the 2S Lamb shift in an optical spectrum recorded by saturated absorption [2] and the one of the 1S-2S two-photon transitions using a pulsed laser [3].

The Paris idea was then to choose other transitions than the 1S-2S one, actually transitions starting from the 2S metastable state, to excite Rydberg S or D states. After pyrex tube; an effusive 1S beam flows into a first vacuum chamber through a Teflon nozzle; 2S state is excited by electronic bombardment which bends the atomic beam and then makes it collinear with the two laser beams used for the two-photon excitation. The grid of the electron gun creates an equipotential volume around the metastable beam to shield it from quenching electric fields. The interaction between atoms and laser beams takes place in a second vacuum chamber delimited by two holes 96 cm apart from each other. The atoms remaining in the 2S state are detected in a third chamber where two electrodes quench the 2S state and the resulting Lyman-α fluorescence is detected. The two-photon transition signal is recorded through the decrease of the number of metastable atoms at the end of the beam. The schematized geometry of the excitation is shown in Fig. 1.

To enhance the two-photon excitation probability, the whole metastable beam was placed inside a Fabry-Perot cavity, the length of which was locked on the laser frequency by monitoring the reflected polarization, using the famous method proposed by Hänsch and Couillaud [9].

The search for n = 8 signal was not straightforward. Our laser frequency was scanned by locking it to a pressure swept Fabry-Perot cavity and known by comparison with a reference cavity whose length was not perfectly determined. Moreover, the Rydberg constant was actually known with a poorer exactness than believed, so that we did not look for our resonance at the correct frequency. After several months of efforts and changing the curvature of the enhancing cavity mirrors to optimize the excitation probability along the beam, we finally obtained signals (see Fig. 2) in both hydrogen and deuterium and were able to perform a preliminary measurement of the n = 8 isotopic shift [10]. The way to absolute measurements was now open for us.

Wavelength measurements and first determinations of the Rydberg constant

The largest signal amplitude was the 2S 1/2 (F = 1) → 8D 5/2 one, with a 10% decrease of the metastable beam intensity, and an experimental width of 1.4 MHz (relative linewidth 1.8 × 10 -9 ), which was at that time the narrowest one obtained in hydrogen. This width has to be compared to the 550 kHz natural width of the 8D states. The main broadening and shift effects were the light Rapidly, we observed not only the 2S-8D transitions in H and D at 778 nm but also the 2S-10D ones in H at 760 nm, and we undertook to determine the Rydberg constant from the measurement of the three 2S -nD 5/2 transition wavelengths.

The most recent determinations of the Rydberg constant were previously deduced either from the one-photon 2S-3P and 2P-3S Balmer-α transitions [11][12][13], or from the 1S-2S two-photon transition, both methods in which Ted Hänsch took a pioneer role. The first one has the disadvantage of being limited to about 10 -9 by the natural width of the P level involved. The second one was much more promising, but needs the development of a tunable laser source at 243 nm. Although the 1S-2S signal had been already observed with a relative linewidth of 5 × 10 -9 using cw excitation [14], the first wavelength measurement had been performed with a pulsed laser and a much broader signal [15].

Compared to the 1S-2S transition, the 2S-nD ones (n ≥ 8 or 10) were limited by a much larger natural width but were competitive at that time for the determination of the Rydberg constant, since the 2S Lamb shift was precisely measured by RF spectroscopy [16,17].

We measured our transition wavelengths [18] by interferometric comparison with an iodine-stabilized heliumneon laser at 633 nm. The wavelength of this laser was known at 2 × 10 -10 since it had been itself compared to one of the "Institut National de Métrologie." We deduced our first value of the Rydberg constant with an uncertainty of 5.2 × 10 -10 , but this value was in slight disagreement with the other ones published. As often in such a situation, such a discrepancy is uncomfortable for a new team in the field, but is a strong encouragement to pursue it.

During the following years, we studied also the 2S-8S transition and performed a detailed study of the line profiles taking into account for each transition all shifting and broadening effects for various values of the excitation power. Calculated profiles were then fitted to experimental signals in order to determine the line positions corrected from these effects. Examples of such fits are shown in Fig. 3 for the 2S 1/2 -8S 1/2 and 2S 1/2 -8D 5/2 transitions. Here the quenching field at the end of the atomic beam is modulated and the 2S signal demodulated by a lock-in amplifier. A good agreement is observed for both transitions, even if their line profiles have different asymmetries.

During this period, we also worked to reduce systematic effects and were able to extend our method to higher n levels. For that purpose, our interaction region was drastically shielded from stray electric fields by painting the walls of the vacuum chamber with Aquadag, a graphite liquid mixture: we were then able to record 2S-20D two-photon signals and to estimate an upper limit for the electric field experienced by the atoms (≤2 mV/cm), and for the resulting Stark effect on our signals [19]. Another improvement was to measure the velocity distribution of our metastable atomic beam, by probing the 2S-3P Balmer-α one-photon transition with a collinear 656 nm laser beam [20].

In 1989, our result for the Rydberg constant with an uncertainty of 1.7 × 10 -10 was mainly limited by the standard He-Ne laser [21]. During the same time, a good agreement of the results obtained on other transitions in hydrogen was found [22][23][24], especially after elimination of chirping effects in pulsed amplifiers, previously used for the 1S-2S excitation.

However, in these measurements, as in others, our laser wavelength was measured by interferometric comparison to the one of the standard lasers using the so-called virtual mirrors method, and the resulting uncertainty arose both from the knowledge of the reference laser frequency and from the comparison method itself. The further reduction of the uncertainty needed to abandon wavelength measurement for direct frequency comparisons.

Let us also point out here that in 1990 our homemade excitation dye laser was turned into a Ti:sapphire laser (TiSa), perfectly suitable for our 730-778 nm range and much more convenient in terms of stability, efficiency and easiness of running.

Direct comparison of two optical frequencies

As explained in the following, absolute optical frequency measurements were a quite complicated issue in the early 1990s. On the other hand, the determination of the Rydberg constant from the study of an optical transition was limited by the knowledge of the Lamb shifts of the levels involved. Since in atomic hydrogen the ratio between transition frequencies is very close to simple rational fractions, both Ted's group in Garching and our Paris group decided to complete their experimental setup in order to directly compare transition frequencies with a ratio close to 4:1: the 1S-2S and 2S-4S/4D transitions in Garching, the 1S-3S and 2S-6S/6D ones in Paris.

In Garching, a new 2S metastable beam was then developed with a Ti:sapphire laser at 972 nm for the atom excitation. In a similar way as in Paris, residual 2S atoms were detected at the end of the interaction region. The numerical calculation of the 2S-4S/4D signal profile was an opportunity for a fruitful collaboration between our two groups [25].

In Paris, the experiment was not easy to achieve, but during one year we benefited from the precious help of Derek Stacey, who had an expertise in the UV domain since he had studied the 1S-2S transition in the Oxford group [27]. A 1S beam was build and we had to develop a new laser source at 205 nm. For that purpose we chose to perform two successive frequency doubling stages of the Ti-sapphire radiation at 820 nm used for the 2S-6S/6D excitation. The first doubling in an enhancement external cavity was quite efficient, and we obtained more than 400 mW at 410 nm with our LBO crystal [28]. However, the second doubling step in a BBO crystal was far more challenging since 205 nm is the shortest wavelength which can be generated by SHG in such a crystal and because of photochemical reactions on the crystal faces and photorefractive effects in its bulk. We finally got only about 1 mW in a quasi-continuous mode where the enhancement cavity length is modulated [29].

Compared to the 1S-2S transition, the 1S-3S one is much broader and the signal much smaller, so that it was not straightforward for us to see the signal. It was finally observed in 1995. On the other hand the 2S-6S/6D transitions are much narrower than the 2S-4S/4D ones and, following the Garching and Yale groups [25,26], we were able to determine the 1S Lamb shift with an uncertainty of 46 kHz [30], using the experimental value of the 2S one. The results obtained by the three groups were in good agreement with each other and, with comparable uncertainties, they notably improved the knowledge of the 1S Lamb shift. However, they were limited by the precision of the 2S Lamb shift, measured in the RF domain. To go further, it was necessary to perform the absolute frequency measurement of two different hydrogen optical frequencies.

Absolute optical frequency measurements in Paris

The first absolute frequency measurement of an hydrogen transition was the one of the 2S-8D transitions [31], performed in our group during the PhD thesis of the third of us (F.N., also called F2). At that time, such a measurement needed to have a good frequency reference in the laboratory and a convenient frequency chain to compare the frequency to be measured to the one of the reference.

Frequency references

Our wavelength measurements had been previously performed by interferometric comparison with an I 2 -stabilized He-Ne laser at 633 nm. This reference laser had been itself calibrated by comparison with the Bureau International des Poids et Mesures (BIPM) standard lasers through an intermediate laser, so that its frequency was known with an uncertainty of 1.6 × 10 -10 . In order to implement direct frequency measurements in hydrogen, we decided to look for new references around 778 nm, the wavelength of the 2S-8S/nD two-photon transitions. After a tentative work with the hyperfine components of the IBr molecule located in this frequency range [32], we finally chose the ones of the 5S-5D two-photon transitions in Rb [33]. Laser diodes could easily be locked to these transitions, giving a new optical frequency standard at 778 nm. Its frequency was measured, thanks to a collaboration with the Laboratoire Primaire du Temps et des Fréquences (LPTF, presently LNE-SYRTE) and the BIPM, with an uncertainty of 2 kHz [34]. This frequency standard with high metrological features [35] is still used in our hydrogen setup.

At the same time, after several years of effort and thanks to the tenacity of B. Cagnac, a 3-km-long optical fiber has been placed underground between our laboratory and the LPTF (LNE-SYRTE). This opened up the possibility to transfer optical frequencies between our two laboratories with an accuracy of a few Hz [36] and then to connect our experiment to the primary frequency standard, the Cs atomic clock.

Frequency chains

The first frequency chain built in Paris to measure the 2S-8S/8D transitions of hydrogen used actually two frequency standards: the I 2 -stabilized He-Ne laser at 473 THz (633 nm) and the CH 4 -stabilized He-Ne laser at 88 THz (3.39 μm). It took advantage of the quasi-coincidence of the frequency difference between these two frequency standards with the one of our transitions (2 photons at 385 THz). Both standard frequencies were precisely known, thanks to recent calibrations done at the LPTF, and the 89 GHz frequency gap was measured in our laboratory firstly using a Fabry-Perot cavity [37], secondly using a Schottky diode which means that we realized for the first time a direct link to the Cs clock without any interferometry [31]. The Rydberg constant was deduced with an uncertainty of 2.2 × 10 -11 .

In the following, this result could be improved to 9 × 10 -12 thanks to the Rb frequency standard and the optical fiber discussed above, and the measurement was extended to deuterium, giving a new determination of the 2S Lamb shift in this atom [38].

As the direct frequency comparison with our Rb standard was only possible for the 2S-8S/8D transitions, we were obliged to build-up again a specific frequency chain to measure other transitions in hydrogen. As a new target, we chose the n = 12 levels which are very sensitive to stray electric fields (the quadratic Stark effect varies as n 7 ) and then give complementary information to possible systematic effects on the n = 8 transitions. This choice was moti- vated by the possibility to compare the transition frequency at 799 THz (2 TiSa photons at 750 nm to excite the n = 12 levels in H) to twice the one of the Rb standard at 385 THz, using a standard OsO 4 -stabilized CO 2 laser at 29 THz (≈10 µm) to measure the gap. One has indeed:

In this new chain, we used an optical frequency divider to reduce by a factor 2 the frequency difference between our TiSa laser and the Rb standard, as suggested for the first time by Hänsch et al. [39].

The experiment was simultaneously carried out in our laboratory and in the LPTF, thanks to our link by a double optical fiber. An auxiliary 809 nm radiation (370.5 THz) delivered by a laser diode in our laboratory was sent to the LPTF to be compared to the frequency difference between a laser diode at 750 nm (385 THz) and the CO 2 /OsO 4 laser. A radiation at 750 nm was also send to our laboratory to be compared to our TiSa laser, whose frequency was mixed (1) with the one of the auxiliary laser diodes at 809 nm. The frequency sum obtained was compared to twice the one of the Rb standard. The two following equations were then realized, respectively, in LPTF and in LKB:

The four 2S-12D transitions in hydrogen and deuterium were measured and a careful analysis of the line profiles was carried out, including various systematic effects, in particular the Stark effect. Using this result and the ones obtained by other groups, especially the Garching one on the 1S-2S transition, the Rydberg constant could be deduced with an uncertainty lower than 8 × 10 -12 [40,41].

Frequency combs

As in Paris, the study of hydrogen transitions in Ted Hänsch's group, first in Stanford and then in Garching, needed during the same period successively the design of an auxiliary standard laser [42], the development of a frequency chain with a frequency divider [43], the use of a transportable calibrated standard [44] and finally a direct link to a metrology institute through an optical fiber [45]. All these means were used to measure the 1S-2S frequency with an increasing and unbeatable precision.

Moreover, during the same time occurred the so-called frequency comb revolution, the main father of which is Ted Hänsch, rewarded by the well-deserved Nobel Prize jointly awarded to John Hall and Ted Hänsch in 2005. The life in our laboratories was totally changed. No more need of complicated frequency chains depending on the transition to be measured: a fs laser coupled to a photonic fiber was the unique instrument needed to measure optical frequencies, allowing one to compare them directly to the Cs frequency standard. What we dreamed about was realized.

Of course, the 1S-2S transition was the first one in hydrogen to be measured in this new way [46], and other improvements followed [47,48]. For us, it was time to leave 2S-nS/nD and focus on 1S-3S.

From 2S-nS/nD to 1S-3S spectroscopy

At the turn of the century, it seemed to us that it was no more possible for us to push further our precision on the transitions starting from the 2S metastable state. Let us point out the advantages and the limitations of these transitions.

They are in a frequency range easy to reach with a TiSa laser and have small natural widths of a few hundreds of kHz. The stray electric field was reduced at best and estimated by recording the transitions toward n = 20 levels, and the velocity distribution measured (see above). However, the number of metastable atoms in our atomic beam was small 2 × 10 6 at/s being a limiting factor for our signal-to-noise ratio. We then used a high laser power to excite the transitions: up to 150 W in each direction of propagation inside the enhancement cavity. Due to the conjugated effect of saturation of the 2S depopulation and light shifts, the line profiles are asymmetric (see Fig. 3) but well understood by numerical simulations. Extrapolation to null power of the center of the line is performed using a procedure detailed in [41].

The features of the 1S-3S transition are quite different. The number of 1S atoms is about eight orders of magnitude larger than the metastable one. The natural width of 1 MHz is similar, but the excitation wavelength which lies in the UV range (205 nm) is responsible for a better quality factor for the line. The laser power is only a few milliwatts and the light shift much smaller. Moreover, the residual Stark shift effect is negligible.

However, since we have no tunable laser source to probe the velocity distribution of the 1S beam through a one-photon transition, we have no simple way to determine the second-order Doppler effect which shifts the line by -v 2 /2c 2 , almost 150 kHz. This is why we have proposed and implemented an original method allowing to measure this shift [49] in our thermal atomic beam. The basic idea is to apply a transverse magnetic field to the atoms, inducing a quadratic motional Stark effect varying also as v 2 and able to partially compensate the second-order Doppler effect near a 3S-3P anticrossing [50].

With this method and using a frequency comb, we measured for the first time the absolute frequency of the 1S-3S transition [51], with an uncertainty of 13 kHz (4.5 × 10 -12 ). After the measurement of the 1S-2S transi- tion in Garching, our result was in 2010 the second most precise one of the optical transitions in hydrogen. This experiment is still being improved. We have in particular developed a new laser source at 205 nm, by frequency mixing of two radiations at 266 and 894 nm, respectively generated by frequency doubling of a Verdi laser (Coherent Inc.) at 532 nm and by our TiSa laser. By this method, we obtain up to 15 mW of cw operation at 205 nm [52].

In 2013, the 1S-3S signal was observed with a signal-to noise ratio up to 170 after 3.5 h of integration, and with a width of about 1.5 MHz, to be compared to the 1 MHz natural width of the line. The absolute frequencies of the two radiations at 532 nm and 894 nm were measured with an optical frequency comb referenced to the Cs clock of the LNE-SYRTE laboratory. The statistical uncertainty on the deduced transition frequency has been significantly improved to reach a value of 2.2 kHz [53]. However, a complete analysis of systematic shifts of the line, in particular collisional shifts, was still needed to determine the 1S-3S transition frequency.

This problem is presently under study. Our experiment is in progress and a recent signal is recorded in Fig. 4.

Hydrogen spectroscopy and the Rydberg constant

When we began the study of 2S-nD transitions during the 1980s, our first goal was the determination of the Rydberg constant. It was simply deduced from our transition frequencies, using the experimental value of the 2S Lamb shift. Such a method was limited to an uncertainty of 1.2 × 10 -11 due to the 2S Lamb shift measurement.

To go further in precision, two different optical transitions were needed, which were found in the 1S-2S and 2S-8D transitions. As the quantity L(1S) -8 L(2S) had been precisely calculated [54][55][56], the comparison of the two optical frequency intervals allows the determination of the Rydberg constant with an uncertainty of 8.6 × 10 -12 . The advantage of this method is that it needs neither the 2S Lamb shift nor the proton radius and is applicable both to hydrogen and deuterium.

Finally, the most precise value of the Rydberg constant is given by the least squares adjustment of the fundamental constants performed by the Committee on Data for Science and Technology (CODATA). It takes into account data concerning all the fundamental constants, but is limited to spectroscopic data on hydrogen (e-p) only and e-p diffusion experiments. The uncertainty deduced for the Rydberg constant by the last adjustment is 5.5 × 10 -12 [57] resulting mostly from the uncertainties obtained by the Paris and the Garching groups on the 2S-nS/nD and 1S-2S transitions.

In the same manner as for the Rydberg constant, hydrogen 1S and 2S Lamb shifts can be deduced from the same measurements, using the scaling law quoted above. The value of the 2S-2P interval obtained is more precise than the one given by direct microwave spectroscopy. The 1S Lamb shift with an uncertainty of 24 kHz only would give a test of the QED two-loop corrections in QED if the proton charge radius r p was perfectly known. On the other hand, the QED calculations are a way to determine r p , which is complementary with the e-p scattering experiments.

The hydrogen atom, a still hot topic

At the end of the 1990s, it was then clear that the proton radius became a limiting factor to test theoretical predictions of QED on the hydrogen atom. It is why we joined in 1998 the international collaboration, now called CREMA, constituted to determine r p from 2S-2P spectroscopy of muonic hydrogen at the Paul Scherrer Institute (PSI) in Switzerland. The Paris and the Garching groups were active together in this new adventure, which is not closed at the present time.

The idea was to measure the 2S-2P Lamb shift interval in muonic hydrogen, an atom made of a proton and a negative muon (µ -). As the muon is 207 times more massive than the electron, the correction due to the finite size of the nucleus is largely enhanced. Indeed, the sensitivity of the frequency of the 2S-2P transition to r p in such a spectroscopy at the 10 -6 level is enough to get a determi- nation of r p 10 times better than the best determination of r p from the electronic hydrogen spectroscopy and quantum electrodynamics.

The principle of the experiment can be summarized in three steps: production of muonic hydrogen (µ-p) atoms in the metastable 2S state, excitation of the 2S-2P transition with a laser at 6 µm and then detection of the 2P-1S fluo- rescence photon at 2 keV. Even if it looks simple, each step was challenging. The source of metastable atoms has been realized thanks to the expertise of F. Kottmann, D. Taqqu and R. Pohl. Members of the Paris group took part in the development of the laser chain which produced the 6 µm laser pulse in collaboration with A. Antognini. After three unsuccessful beam times (2002,2003,2007), in 2009, we finally observed two transitions of muonic hydrogen and three of muonic deuterium [58][59][60], but they were not at the frequency predicted by the theory. This was the beginning of the so-called proton size puzzle which has triggered and stimulated a lot of activities in our community Fig. 4 1S-3S signal recorded on July 6, 2016. The excitation laser at 205 nm is swept through an acousto-optic modulator. The integration time is here 4.5 h and the signal linewidth 1.36 MHz in atomic frequency units. The signal background is mainly due to stray fluorescence induced by the UV excitation light but also outside of it, both theoretical and experimental ones.

On the side of experiments in hydrogen atom, a new RF measurement of the 2S-2P hydrogen interval is currently in progress in the group of Hessels [START_REF] Hessels | Frontiers in Optics 2015. OSA Technical Digest[END_REF] while the Paris and Garching groups pursue their efforts to improve their optical measurements: in the one-photon 2S-4P transition [START_REF] Beyer | [END_REF] (cw laser in Garching) and in the two-photon 1S-3S transition [53,63] (pulsed laser in Garching and cw laser in Paris). At the level of precision needed to clarify the proton puzzle, all stray shifting effects have to be carefully studied, including quantum cross-damping interference effects [64][65][66].

New results are looked forward in the near future, not only on the above hydrogen experiments, but also on hydrogenic systems [67][68][69][START_REF] Pohl | Proceedings of LEAP 2016[END_REF].

Conclusion

This review of more than thirty years of hydrogen spectroscopy in Paris and also in Ted Hänsch's groups in Stanford and Garching shows how the study of this simple atom has stimulated the development of laser technology and new frequency measurement methods in parallel with QED calculations.

Thanks to the development of frequency combs, ultrastable cavities and laser references, the optical part of an hydrogen experiment is almost no more an issue, except for the lack of a powerful cw 121 nm laser to optically cool atomic hydrogen, even if 1S-2P spectroscopy has been already performed with a low power source in Ted Hänsch's group [START_REF] Eikema | [END_REF]. The last experimental frontier in hydrogen spectroscopy is certainly the atomic source: hydrogen spectroscopy is still performed on atomic beams.

The other frontier to overcome is in the perturbative QED calculations. Nowadays, the 1S-2S transition frequency is measured with an uncertainty of 10 Hz that is 250 times smaller than the theoretical uncertainty! From the beginning of quantum mechanics, the hydrogen atom is a source of new discoveries in physics and can be considered as "the Rosetta stone of modern Physics" as pointed out in a paper of Schawlow et al. [72]. Coming back to the Paris adventure, two of us who were at the beginning of this story are now retired, although still active in research. We did not imagine that after more than thirty years, the hydrogen atom would be always so fascinating and rich of secrets still to discover.

Fig. 1

 1 Fig. 1 Geometry of the two-photon excitation for the observation of 2S-nS/nD signals. After interaction with the two counter-propagating laser beams, the remaining 2S atoms are detected through electric quenching and Lyman alpha fluorescence

Fig. 2

 2 Fig. 2 2S-8D signal recorded on November 9, 1984. The two fine structure components of the 8D level are visible, separated by about 57 MHz. The laser frequency is modulated and the 2S signal demodulated by a lock-in amplifier. The red trace is the transmission of a FP cavity used for the calibration, having a low finesse at this wavelength

Fig. 3

 3 Fig.32S 1/2 -8S 1/2 (upper part) and 2S 1/2 -8D 5/2 (lower part) signals recorded in 1996, with fitted calculated profiles superimposed. Vertical scale number of 2S atoms recorded after demodulation (see text) by a lock-in and offset. The signal amplitudes are, respectively, 4.4 and 21.0 % of the 2S population. Horizontal scale frequency of the acoustooptic modulator used to scan the excitation laser. The different asymmetries of the lines arise from the respective weight of saturation and light shift. The saturation is almost negligible for the 8S line and important for the 8D one

( 2 ) 2 ( 3 )

 223 370.5 THz (auxiliaryDL) = 385 THz -29 THz /399.5 THz (Ti-Sa) + 370.5 THz (auxiliaryDL) = 2 × 385 THz

Acknowledgements

We are pleased to acknowledge here all people who worked with us on our "hydrogen experiments" in Paris: Jean-Claude Garreau, Sophie Bourzeix, Béatrice Chatel (De Beauvoir), Gaëtan Hagel, Olivier Arnoult, Sandrine Galtier and Hélène Fleurbaey as Ph.D. students, Mark D. Plimmer, F. de Tomasi and L. Jozefowski as post-docs and our colleagues M. Allegrini, D.N. Stacey and C. Schwob. The authors are indebted to the people of LNE-SYRTE laboratory for the time reference and constant help during all these years.


M Solal

The origin of duo-lateral position sensitive detector distortions

Keywords: PSD, Continuous position sensitive detector, model, simulation, distortion

The origin of duo-lateral position-sensitive detector distortions

M.

I. Introduction

In some applications, when multiplicity (number of incident particles) and counting rate are not critical, it is well suited to use a silicon continuous Position Sensitive Detector (PSD) for locating particles, the active area is up to 60 mm x 60 mm. Continuous PSD are available in standard devices and only four to five electronics channels are needed for the readout of the whole surface of detection and high spatial resolution is achieved when incident particles are at high energy [1,2]. PSD can be associated with an image intensifier tube (IPSD) and a crystal plate for gamma rays detection [1,3] or can be used directly to localize particle beam or heavy nuclei (alpha, carbon, phosphor, etc …). For example, an alpha particle of 5.5 MeV gives a typical spatial resolution of 1 mm (FWHM) for a PSD alone and 300 µm (FWHM) for an IPSD. Since the first PSD development which began in the sixties, spatial non linearity has been largely observed [1..4]. In previous works, algorithms were used to correct spatial aberration [4] but the origin of the distortion has not yet understood; this is particularly true for the duo-lateral detector where the observed cushion distortion is not in accordance with this PSD geometry. In this paper, we will briefly recall the principle of continuous PSD and describe the simulation model we developed. The distortions observed experimentally will be compared to simulation results. Then the origin of the distortion will be analysed and explained. This study will allow choosing optimum PSD parameters for POCI (Per Operative Compact Imager) [1 .. 3, 5].

II. Method.

II.A. Principle

The continuous PSD is a 500 µm thick fully depleted diode of a large area, the distances between the electrodes define the active surface L x L (Fig. 1). For pin-cushion PSD, the front surface is covered by P+ ion implanted resistive layer: the corresponding resistance expressed in Ohm / square is R sq = 10 to 22 kΩ.

In integrated devices, a film resistor of thickness d, length L and the width W has a resistance

W L R W d L R sq = ⋅ = ρ
, where the ratio (L/W) is sometimes called the number of squares. Notice that the resistance R is independent of the sizes of the square. The back side is formed by a metallic cathode (Fig. 1): such PSD type shows strong distortions. In corrected pin-cushion PSD the four electrodes are linked via a low resistive deposited strip (Fig. 1) with resistance typically ~ 2 kΩ [4]. The low resistive strip reduces the classical spatial distortion but causes a strong thermal noise [5]. In duo lateral PSD both sides are ionimplanted resistive sheets R sq = 4 to 10 kΩ/sq; there is a bore implantation for the P+ side and a phosphorus implantation for the N side. The X location is measured on the bore side while the y location is measured on the phosphorus side (Fig. 1

.b).

All PSD (Tetra lateral, pin cushion or duo lateral) are based of the same location principle; each electrode receives a fraction i i of the total induced current due to drift of e-h pairs generated by incident particle. Induced current are integrated in a charge preamplifier.

For duo-lateral detectors, the determination of X and Y incident particle positions is based on the

following relationships: Q e1 = Q t .(L/2+X)/L, Q e2 = Q t .(L/2-X)/L, Q h1 = Q t .(L/2+Y)/L, Q h2 = Q t .(L/2-Y)/L,
where the origin is at the centre of PSD, Q

t = Q e1 + Q e2 = Q h1 + Q h2 is the total charge collected, L is the PSD dimension. For pin-cushion detectors, Q e1 is replaced by Q ec + Q ed , Q e2 by Q ea + Q eb , Q h1 by Q ea + Q ec and Q h2 by Q eb + Q ed .
The position measurement is then given by:

        - - ⋅ ⋅ =         h2 h1 e2 e1 t Q Q Q Q Q 1 2 L Y X , (1)
Q t = Q ea +Q eb +Q ec +Q ed = Q e1 +Q e2 = Q h1 +Q h2
is proportional to the total energy deposited on the PSD. For pin-cushion PSD it is better (smaller noise and ballistic deficit) to measure Q t from the backside.

II. B. Spatial resolution

Spatial FWHM resolution in both planes is given by

2 1 2 2 2 2 . ) 2 L ( 1 L 355 . 2 ) (             + ⋅ = i E dQ Q U U δ , ( 1.2)
where U = X or Y and i dQ is the Equivalent Noise Charge (ENC) on a position electrode [5].

The main contribution to ENC is the Johnson noise (2kT/R det ) due to the low value of the resistive layer.

Other noise contributions can be quadraticly added, for a CR-RC2 filter the equivalent noise charge is

      ⋅ + ⋅ +         ⋅ + ⋅ + = C A qI R C e R kT ENC f leak s π τ τ τ τ π .
The prefactor normalizes the noise to the signal gain, the second term combines the serial noise of the preamplifier e s and the detector impedance (capacitive and resistive part), and the last term is the contribution of excess noise [5,6].

Formula of spatial resolution (1.2) is perfectly verified by experiments [2]. If the charge is not completely collected (ballistic deficit mode) due to a too short integration time constant, the formula (1.2) should be replaced by

( ) ( ) ( ) ( )         + + + + + + ⋅ ⋅       ⋅ ⋅ =         ∑ = 2 2 2 2 2 2 355 . 2 ) ( ) ( C A D B B A D C D A i i Q Q Q Q Q Q Q Q Q ENC L Y X δ δ
, where i Q is the effective charge collected [5].

II.C. Spatial distortion

Spatial non linearity is often observed on two dimensional detectors. This effect is inherent to the geometry of the PSD and the electronics readout. Moreover, the observed distortion increases when the time constant of the spectroscopy filter decreases. A higher time constant reduces the distortion but increases the parallel noise contribution and hence degrades the spatial resolution [5].

The electrodes configuration is shown in Fig. 1 for a duo-lateral and a pin-cushion detector. To evaluate the spatial linearity of the PSD, a thick lead mask perforated by an array with 5 mm spaced 1 mm holes was put on the front surface of the PSD. A 241 Am source, placed at 20 cm distance from the mask, irradiated the whole detector (fig. 2.a). In the case of pin-cushion detectors, the strong distortion is simply linked to the geometrical configuration of the electrodes (fig. 2.b) [7]. In the case of duo-lateral detectors the weak distortion could not be attributed to the position of the electrodes which are parallel to each other.

In our knowledge, there is no analytical solution allowing to analyse the observed distortion in duo-lateral PSD. The resistive sheet and capacitance are continuously distributed, then, based on the finite element model of R.L. Cowin [6] (numerical computation), M. Solal [5] developed an electronics model simulated with Spice software.

II.D. Charge sensitive preamplifier

The deposited energy induces current which flow to the four input channels via the resistive sheet.

The resistive sheet seen by the preamplifier also requires suitable electronics. As illustration, we present in impedance Z in = 0 Ω, the time integration constant τ = 0 and the sensitivity = 1/C f ). Further the contribution of the input impedance will be taken in a second model.

III. Simulation model

III.A. PSD

In a previous paper [5] the model included 13 x 13 cells. Numerical simulations were performed for the two kinds of pin-cushions PSD. For the 300 µm thick duo-lateral PSD of 42 x 42 mm 2 area, the previous model was completed with two resistive layers R bore and R phosphore , which are linked by the distributed diode capacitance (Fig. 5.a). A particular attention was paid to the edges (Fig.

III.B. Filters

The first simulation was performed without spectroscopy filter, directly at the charge preamplifier output in order to show PSD intrinsic distortions. In the subsequent simulation we will introduce the filter in order to analyse the observed experimental spatial distortion. The spectroscopy filter used in experiment is the Ortec 571 (Fig. 6) which is implemented with PSpice "Analog Behaviour Model" block. The exact

transfer function T(s) is s k s k s s s s T ⋅ + + ⋅ + ⋅ + × ⋅ + × ⋅ + ⋅ × = 2 2 2 2 1 1 2 1 1 3 1 1 1 84013 , 2 ) ( τ τ τ τ τ ,
where k = 0,8 and τ = 600 ns [8]. IV Results

IV.A Simulated intrinsic PSD response

In this section we will focus on the analysis of the simulated intrinsic non linearities of the duolateral PSD. The modelled PSD (300 µm thick) is considered fully depleted, in experiment the full depletion voltage is 30 Volts. The determination of the centroïd is performed directly at the charge sensitive preamplifier (CSA) output. As described in previous section ( § II.D), this component is assumed perfect.

Consequently the charge collected at the preamplifier is directly proportional to the integral of the current flowing through the contacts which are short circuited. The simulated CSA output waveforms (taken from the bore electrode) are represented in Fig. 7 for different positions 3 mm spaced along X axis. We can notice a very long establishing time (~ 1µsec) even for a particle hitting just near the electrode x1 (X=L/2), however the asymptotic values are strictly linear of X.

The simulated mapping response is obtained by injecting the current source in each nodal mesh (15 x 15 nodes) of equal 3 mm spacing ∆X = ∆Y. The normalized coordinates are noted x = X / (L/2) and y = Y / (L/2). The centroïd (x, y) is calculated for each source position and the corresponding dot is plotted in Fig. 8. The calculated position patterns of the PSD modelled (Fig. 8) calls for two comments: Using this first model, centroïds are calculated with potentials taken at t = 500ns (V x1 (t=500ns) and V x2 (t=500ns)) corresponding to the collection time of the shaping amplifier. The unexpected distortion results from the data processing procedure. Some values of V x2 (t=500ns) observed are small and negative for a hit near the electrode x 1 , then, the centroïd x = 2 1 2 1

x x x x V V V V +
could be greater than 1 (with V x2 <0 instead of 0). Bipolar current pulses on the x2 electrode are expected. The current of opposite sign in two dimensional duo-lateral PSD was measured by E. Laegsgaard for t/(R det C det ) < 0.2 at certain positions [8]. This phenomena was studied by A. J. Tuzzolino who called it "anomalous" current and calculated it in a particular case [10]. The induced current calculation in silicon has been studied for non-grounded electrodes by E. Gatti et al. [11]. Figure 9 shows two responses for different values of y and the same x=1, one of the curves is obtained by simulation for a particle hitting the middle of the electrode and the other one for a particle hitting the electrode edge. In these cases, we expect a short collection time corresponding to 10 ns drift time, this is not the case and we notice two different collection times. We will see in the next paragraph that these different settling times have an effect in the spectroscopy amplifier and explain the distortion. Figure 10 shows the difference in charge collection between bore and phosphorus sides.

The bipolar current pulse and the long collection times are specific to duo-lateral PSD and don't exist in one dimensional PSD neither in pin-cushion or tetra-lateral PSD. The reason for this phenomenon is the both sides are resistive and are capacitively coupled. Such behaviour in pin-cushion PSD is much smaller due to metal plane. Both resistive sheets of duo-lateral PSD are not grounded, that it is explained in figure 11. The drift current is modelled by a current source, with one of its terminal connected close to the preamplifier and the other terminal connected to the resistive sheet. Then, the current flowing into the preamplifier for a particle near the bore electrode is given by ( )

2 1 // . . . 1 1 phos phos c source bore Z Z C j i i ω + × = , (1.a)
and for phosphorus side it is given by ( )

2 1 // . . . 1 1 bore bore c source phos Z Z C j i i ω + × = . ( 1

.b)

Cc is the local coupling capacitor, Zphos1 is the impedance between one of the terminals of i source and the electrode 1 of the phosphorus side. Zphos2 is the impedance between the same terminal of i source and the electrode 2 of the phosphorus side.

Then, the current flowing to the preamplifier can't be instantaneous, the time constant is C c Z with Z = Z 1 // Z 2 which depends on the position, thus explaining the waveforms found in figures 7 to 10 which correspond to the integrated currents. The simulated current for a hit of the phosphorus electrode is given in figure 12, the current collected by the preamplifier connected to the phosphorus electrode is effectively not equal to the drift current. On the opposite electrode, the bipolar current pulse with integral 0 is found, after 500 ns the current turns to positive. IV. B. Response with shaping amplifier

The filter amplifier described in paragraph III.B, is now connected at the output CSA preamplifier.

Figure 13 shows the output voltage response of the filter along x and y axes. We see that the bipolar pulse of the CSA output voltage has an effect onto the response of the spectroscopy amplifier. For positions near the opposite of the collection electrode, the output filter has a bipolar pulse shape. For x or y = -1, the pulses do not stay at the expected value zero. In this case, the maximum sampled value (positive in boron side) leads to calculate a centroïd value smaller than the expected hit position. We see the same effect for the phosphorus side with opposite signal polarity. We can also notice a more important effect on the phosphorus side than on the boron side.

The spatial linearity of the shaping amplifier is given in fig. 14 for incident positions on equidistant lines along the x and y directions. The simulated response is now in good agreement with experimental mapping in all regions. The ovoid points are due to the difference in resistive layers, the phosphorus side is noisier [5]. Spatial distortions are more important along y axis corresponding to the phosphorus channel. This could have been predicted by observing figure 11, the centroïd value for a hit at x=1 is given by

x = 9 . 0 3 . 29 9 . 0 3 . 29 ) 1 ( ) 1 ( ) 1 ( ) 1 ( max max max max + - = - = + = - = - = x v x v x v x v ,
+ - = - = - = - - = + = - y v y v y v y v
, which is lower than y =1.

The difference in shape for phosphorus and bore channel is a consequence of the difference in

IV.C. Improvement

The input impedance of the preamplifier has also been tested. This has been performed by introducing a resistance in parallel with the perfect preamplifier. The results show a degradation of the existing distortion as the input impedance increases. In order to improve the linearity the input impedance of the preamplifier should be as small as possible.

The spatial linearity could be improved by increasing the collection time as discussed in section V.a, or/and by reducing the value of the resistive sheet R detector . This is realized by increasing the depth of ion implant. But these two solutions lead to an increase of the parallel noise (thermal noise) leading to a bad spatial resolution [5].

An alternative solution consists of keeping a value not too small for the resistive sheet, between 8 kΩ to 30 kΩ to avoid a higher thermal current noise 2kT/R detector , and improving the front end electronics.

The duo-lateral PSD needs electronics with a long collection time to avoid ballistic deficit and a short derivative time to reduce parallel noise. The solution adopted for POCI γ camera which is clinically evaluated [2] is the linear pseudo trapezoidal filter [12]. This solution was also used for gamma ray detection in large germanium detector [13] where the problematic is similar. The drift time is related to the distance of carriers travel leading to the ballistic deficits. For the trapezoidal weighting function the width of the constant portion determines the charge collection time, and the width of the two triangular portions is determined by the noise corner time constant [13]. Two-dimensional PSD readout by a trapezoidal filter has been studied by M. Solal, a constant portion width of about four times larger than the triangular width offers the best trade off between spatial resolution and position linearity [5]. But the large width of the constant portion leads to make a linear electronic design with an important number of Salen-key structures [12] and with some difficulties for the practical realisation du to the accuracy of components.

A time variant filter approach is the gated baseline restorer which has been studied and developed by Z. Zojceski [14], the output CSA is integrated during the time where the gate is open, the time derivation is given by the short time of the end of the gate. The gate duration must be adapted to reach the compromise between the ballistic deficit reduction and the noise integration, a pre-filter may be required. L.

Fabris has developed a new pulse processing technique combining ballistic deficit immunity and resilience to parallel noise [START_REF] Lorenzo | IEEE Nucl. Sci. Symp. and Med. Imag. Conference[END_REF], this time variant filter has been tested on a CdZnTe detector where the charge pulses collected have different shapes due to the impurities in the semi conductor.

A new idea based on progress in numerical sampling, is to keep a classical analogue invariant time filter (semi Gaussian or pseudo trapezoidal) and to sample the whole pulse shape. Thereby, any value of the pulse can be held at any time instead of holding the maximum value with a peak detector, classically used to have the maximum signal to noise ratio. We see obviously in figure 13 that the pulse shapes are not linear functions of x or y positions. This consideration could be used for reducing the distortion [START_REF] Selem | LCAM 91406 Orsay[END_REF]. Let us consider a hit located at x = 1, the corresponding pulse value response at time t = 2.0 µsec gives for the centroïd the exact position x = 0 7 . 23

0 7 . 23 ) 1 ( ) 1 ( ) 1 ( ) 1 ( 2 2 2 2 + - = - = + = - = - = = = = = mV mV x v x v x v x v µs t s t µs t s t µ µ = 1.
We can notice at t = 2.0 µsec that the pulses pattern

) ( 0 . 2 i s t x v µ =
improve considerably the linearity along whole the x axis. We also notice figure 13 that the same time t = 2.0 µsec gives also the optimum linearity along the y axis. In this solution the noise is unchanged because the signal is yet filtered by the spectroscopy amplifier but the signal is smaller at certain positions and then the signal to noise ratio is degraded by a factor less than 1.3. This solution is also very versatile, the signal to noise ratio is slightly degraded but the sampling allows choosing the optimum time which had the minimum spatial distortion.

V. Conclusion

The model developed allows prediction of the position non linearity. The results are in good agreement with experimental responses, reproducing all weak distortion in all regions. The model was used to optimize the sheet resistances of the PSD used for the POCI camera. Other parameters could be added.

I.e., in gamma ray detection the IPSD [2] could be simulated by introducing the spatial repartition of the light spot and its time constant coming from the crystal plate. The preamplifier and its input impedance can be simulated at the component level.

The simulation model developed has allowed to a large extent to understand the exact origin of the distortion of the duo-lateral PSD. The long collection time and the bipolar current pulse are due to the resistive sheets at both sides in the duo-lateral PSD. Therefore the pulse shapes at the output of the spectroscopy amplifier depend on the hit location, leading to the spatial distortion. We can conclude that the classical readout by taking the maximum value of the shaping amplifier is the origin of the distortion largely observed, that readout is not well adapted for the duo-lateral PSD. The model presented is now validated and becomes a powerful tool to study alternative pulse processing in order to avoid distortions. PSpice simulations performed in time domain allow also studying any time variant filter processing. One idea is to numerically exploit the whole sampled shape of the pulse which is effectively dependent on the position. Fig. 1. PSD types: (a) in pin-cushion PSD the front side is a resistive sheet; the signals are taken at the four metallised contacts located on the corners and are used to determine the position. The back side is a metallic contact film used to improve the energy resolution. Without resistive strip Rl (standard detector), the spatial resolution is high but the spatial linearity is very bad. With a resistive strip the response is quite linear while the spatial resolution is degraded. (b) In duo-lateral PSD both sides are resistive with contacts located at the four edges on both sides (and orthogonally); this kind of PSD offers both a quite good spatial resolution and a good spatial linearity; but the drawback of this detector is the poor energy resolution (no metallic sheet). Each dot is the calculated position of nodal hit using the simulated amplitudes signals (V x , Vy) at outputs of perfect charge sensitive preamplifiers. Due to the very long collection time, the charge is taken at t = 500 ns which corresponds to the collection time amplifier. We notice a barrel distortion which is the opposite of the expected "pincushion" distortion. Lastly, at edges, maximum calculated coordinates are greater than 1, which is theoretically not possible! Fig. 9. CSA voltage output near (onto) the electrode corresponds to a location x=1 for y =0 and for y = 6/7 (in the middle and at the end of borus side electrode). The collection times are different, V(x=1) depends on y value; the settling time is longer in the middle of the PSD where the distortion is maximum. Because the voltage is taken before the complete settling at t=450ns the distortion shows up. Nevertheless, when a particle hits near the electrode (onto the electrode in this simulation) we expect an instantaneous collection time as it is observed on one dimensional PSD. . Above, the experimental responses of two alpha particles arriving at about 1mm near the electrode in bore side (x ≈ 1, y=0) and near the electrode on phosphorus side (x=0, y ≈ 1). Bellow, the corresponding simulated responses with hits of electrode at (x = 1, y = 0) and at (x=0, y = 1). We notice a long collection time which is much more than the 10 ns drift time in spite of the input impedance of the CSA equal to zero.

Collected currents

-2,E-5 0,E+0

Current in A

x 1 Fig. 11. This schematic explains the long collection time phenomena. All the electrodes are connected to the virtual ground. The hit is located onto the electrode y1 and creates the drift current i source between one terminal and the resistive sheet on bore side. The effective time constant of the collected current i in is given by C c // Z bore1 // Z bore2 . x 1 x 2 y 1 y 2 Fig. 12. Above, the injected current i source on a phosphorus electrode (x=0, y=1). Below, the corresponding collected currents i phos through the virtual ground of the preamplifiers which are connected on both phosphorus electrodes. The simulation confirms that the collection is not equal to the 10 ns drift time current. And the current flowing through the opposite electrode is a bipolar pulse with integral zero, (the current turns to positive after 450 ns). Fig. 13: responses of the shaping (spectroscopy) amplifier for hits located along x axe (left) and y axe (right). We notice at the position x = -1, that the wave is negative at the beginning and then become positive instead of staying at zero. (This is the consequence of the "anomalous" negative current.) Therefore the maximum value classically taken to calculate the centroïd is wrong. This explains the distortion near electrodes, where x calculated is less than the position x expected.

Fig. 3 .

 3 Fig. 3.a. the deficit signal versus the resistive input for a standard charge sensitive amplifier (CSA) and the electronics principle (Fig. 3.b) implemented in POCI camera to suppress the deficit signals. The complete diagram of the designed CSA is given in Fig. 4. It should be stressed that in order to quantify the intrinsic distortion of the PSD, we used a perfect charge sensitive preamplifier in our simulation model (e.g., input

  5.b), and especially in the area around the contact electrodes where the main unexpected distortions are located. More precisely, metal contacts overlap P and N + layers (Fig 5.b). In order to reproduce with sufficient accuracy these weak spatial non lineraties, the number of cells was increased to 15 x 15. The active area L x L is divided into 15 slices of equal width ∆X = ∆Y, to simulate the spatial response a current pulse was injected at the different cell nodes. The whole model is implemented in OrCad Capture CIS 10.3 software and simulated with using OrCad PSpice 10.3 software (distributed by Advanced Logic System Design, http://www.alsdesign.fr).

1 )

 1 Contrary to experimental observations, an unexpected barrel distortion is clearly observed!2) The maximum calculated centroïd at edges is 1.5 in y and 1.2 in x, instead of the theoretically maximum conclusion would be addressed: the intrinsic calculated PSD distortions (based on this first model) are far from the experimental observation.

  charge collection time. The charge collection time is given in first approximation by equations(1), this collection time on the bore side is determined by the resistance of the phosphorus sheet (4 kΩ) and the collection time on the phosphorus side is determined by the resistance of the boron sheet (8 kΩ). The collection time is function of the resistive layer of the opposite side. A short collection time reduces the ballistic deficit, then the distortions on the boron side are less important because the resistance of the phosphorus sheet is lower and vice versa (Fig.14). The position pattern obtained by experiment in figure15confirms this explanation, spatial linearity is improved in both side by R ph = R bo = 4 kΩ and only in y direction by R ph = 8 kΩ and R bo = 4 kΩ. Therefore a distortion on one side could be reduced by reducing the resistance sheet of the other side. In our application POCI (Per Operative Compact Imager) a duo-lateral PSD with resistive layers R ph = R bo = 4 kΩ have been implemented.

Fig. 2 .Fig. 3 .

 23 Fig.2. Typical spatial distortion for a duo-lateral PSD (a) and for a pin-cushion detector (b)[7]. Note that a stronger distortion is observed in the case of pin-cushion detector: this is mainly due to the geometrical configuration of the electrodes[7]. A weak pin cushion distortion (Fig.2.a) is observed near the edges for duo-lateral PSD, which can't be explained by electrode positions.

Fig. 4 :

 4 Fig.4: Main elements of an electronic channel of the front side detector. The electronic channel of the rear side is the same expect for the type of the input and output transistor (feed-back) due to the opposite input polarity signals.

Fig. 6 :

 6 Fig. 6: The real and simulated time step response of the shaping amplifier; time constant τ = 600 ns and k=0.8. The transfer function is implemented by using elementary Laplace blocks from «Analog Behaviour Model» of Pspice library.

Fig. 8 .

 8 Fig. 8. Position pattern reconstruction of the PSD modelled.Each dot is the calculated position of nodal hit using the simulated amplitudes signals (V x , Vy) at outputs of perfect charge sensitive preamplifiers. Due to the very long collection time, the charge is taken at t = 500 ns which corresponds to the collection time amplifier. We notice a barrel distortion which is the opposite of the expected "pincushion" distortion. Lastly, at edges, maximum calculated coordinates are greater than 1, which is theoretically not possible!

Fig. 7 .

 7 Fig. 7. Charge sensitive preamplifier waveforms V x1 (X) along X axis. In this figure the normalized coordinate is noted x = X / (L/2). Far away from the electrode (x ~ -1) the negative signal corresponds to the integration of a negative current. After 1 µs the output voltages aim to the good values. As we can see, V x1 (x=-6/7) turns positive and V x1 (x=-1) goes to the normal value 0. Lastly, we notice a long and anomalous collection time even for a hit on the electrode (x=1)!

  Fig.10. Above, the experimental responses of two alpha particles arriving at about 1mm near the electrode in bore side (x ≈ 1, y=0) and near the electrode on phosphorus side (x=0, y ≈ 1). Bellow, the corresponding simulated responses with hits of electrode at (x = 1, y = 0) and at (x=0, y = 1). We notice a long collection time which is much more than the 10 ns drift time in spite of the input impedance of the CSA equal to zero.

Fig. 14 .Fig. 15 :

 1415 Fig.14. left: simulated mapping response with shaping amplifier, right: corresponding experimental response at right. In both figures we notice a more important distortion (at edges) in phosphorus side (along y axis). The model reproduces also the week distortion along x axis (bore side). We can also notice that the week barrel non linearities at centre are reproduced by the simulation.

  this value is lower than the expected centroïd value x=1. For a

	hit at y=1, the determination of the centroïd value is
	y =	min	(	) 1	min	(	) 1	27	7 .	8 . 1
		min	(	) 1	min	(	) 1	7 . 27	. 1	8

Acknowledgments: The author acknowledges Nguyen Duy of Sté Protec-Feu Nancy for the numerous Pspice simulations and Stephanie Pitre of I.M.N.C. Orsay Paris XI for the exhaustive experimental measurements.