Alexis Lossouarn

Pierre-Yves Renard

Cyrille Sabot
email: cyrille.sabot@univ-rouen.fr

Tailored bioorthogonal and bioconjugate chemistry: a source of inspiration for developing kinetic target-guided synthesis strategies

Kinetic target-guided synthesis (KGTS) is a promising tool for the discovery of biologically active compounds. It relies on the identification of potent ligands that are covalently assembled by the biological targets themselves from a pool of reagents. Significant effort is devoted to develop new KTGS strategies, however, only a handful of biocompatible reactions are available, which may be insufficient to meet the specificities (stability, dynamics, active site topology, etc…) of a wide range of biological targets with therapeutic potential. This Review proposes a retrospective analysis of existing KTGS ligation tools, in terms of their kinetics and analogy with other biocompatible reactions, and provides new clues to expand the KTGS toolkit. By way of examples, a non-exhaustive selection of such chemical ligation tools belonging to different classes of reactions as promising candidate reactions for KTGS are suggested.

INTRODUCTION

The search for ligands that bind relevant biological targets through non-covalent interactions with high affinity and specificity is a key feature in the fields of molecular imaging, [START_REF] Chen | Design and development of molecular imaging probes[END_REF] proteomics, 2 molecular diagnostics, and therapeutics. [START_REF] Kalra | Simple Methods and Rational Design for Enhancing Aptamer Sensitivity and Specificity[END_REF] Beyond the high-throughput screening paradigm that uses drug-like molecules for drug discovery, fragment-based drug discovery (FBDD) techniques offer the advantage of a better coverage of the chemical space by identifying high-affinity ligands from small-size and low-affinity fragments that bind target's adjacent binding-sites. [START_REF] Erlanson | Twenty years on: the impact of fragments on drug discovery[END_REF] Importantly, the affinity (i.e. binding affinity, KD) of optimally linked fragments can be dramatically enhanced, up to multiple orders of magnitude as compared to the affinity of parent fragments. Such a super-additivity or "linking effect", is ascribed to the fact that the sum of free energy of binding G (defined by G = -RT ln KD) for two parent fragments includes two unfavourable rigid-body entropy barriers (translational and rotational) upon their complexation to the biological target, while the assembled ligand includes only one unfavourable term. [START_REF] Nazare | Fragment deconstruction of small, potent factor Xa inhibitors: exploring the superadditivity energetics of fragment linking in protein-ligand complexes[END_REF][START_REF] Murray | The consequences of translational and rotational entropy lost by small molecules on binding to proteins[END_REF] Already 40 molecules discovered by FBDD are in various stages of clinical trials in 2020, three of which have been approved by the FDA. The recently introduced kinetic target-guided synthesis (KTGS) strategy brings the biological target to the forefront of the FBDD process. In fact, the biological target is not only used to determine the potency of ligands though biological assays, but also to chemically assemble them through the formation of covalent and irreversible bonds between complementary reactive fragments (Figure 1A). Indeed, fragments that bind the biological target will benefit from their spatial proximity to react together preferentially and form the corresponding multi-site ligand. This methodology has been applied with success for the discovery and the design of numerous ligands and with different classes of biological targets including proteins, and nucleic acids. The term "fragment" will be used throughout the review to refer to chemical precursors for KTGS, although it should be stressed that this term was originally associated with low affinity small molecules that obey a "Rule of Three" (in particular molecular weight <300; number of hydrogen bond donors ≤3; number of hydrogen bond acceptors ≤3; and ClogP ≤3). [START_REF] Congreve | A 'Rule of Three' for fragment-based lead discovery?[END_REF] This was recently clarified by Deprez-Poulain et al. 8 Historically, the covalent bond formation relies on the use of 1,3-dipolar cycloaddition of azides with alkynes leading to the formation of 1,4-and/or 1,5-disubstituted 1,2,3-triazoles. This strategy, also termed "in situ click chemistry", reported by Sharpless and coll. in 2002, has enabled the synthesis of the most potent non-covalent acetylcholinesterase (AChE) inhibitor. [START_REF] Lewis | Click Chemistry In Situ: Acetylcholinesterase as a Reaction Vessel for the Selective Assembly of a Femtomolar Inhibitor from an Array of Building Blocks[END_REF] Beyond drug discovery, this approach has led to impressive discoveries such as the identification of unknown protein conformations, which were not predicted from reported ligand-protein crystallographic complexes, [START_REF] Bourne | Freeze-frame inhibitor captures acetylcholinesterase in a unique conformation[END_REF] or the identification of previously hidden binding pockets of enzymes. [START_REF] Willand | Exploring drug target flexibility using in situ click chemistry: application to a mycobacterial transcriptional regulator[END_REF] KTGS has also been leveraged for biomedical imaging, in particular for positron emission tomography (PET) tracer discovery, [START_REF] Mocharla | From in situ to in vivo: an in situ click-chemistry-derived carbonic anhydrase II imaging agent for positron emission tomography[END_REF] as well as for identifying selective fluorescent probes for a specific topology of G-quadruplex nucleic acids. [START_REF] Hu | A new application of click chemistry in situ: development of fluorescent probe for specific G-quadruplex topology[END_REF] Since then, other chemical ligation strategies were successfully applied, 8, 14-17 such as the recently reported Ugi four-component reaction, [START_REF] Mancini | Protein-Templated Hit Identification via an Ugi Four-Component Reaction[END_REF] Mannich ligation reaction, [START_REF] Wong | The transcription factor STAT5 catalyzes Mannich ligation reactions yielding inhibitors of leukemic cell proliferation[END_REF] or aldol condensation. [START_REF] Fu | Target-guided screening of fragments (TGSOF) in the discovery of inhibitors against EV-A71 3C protease[END_REF] In fact, continuous extension of the repertoire of chemical ligations is needed in particular to face the stability constraints of biological targets, as well as the wide diversity of their active site topologies. Indeed, long in vitro incubation times (from days to weeks) in particular for reactions with high energy barriers, can induce the aggregation of proteins, or impact their folding through the lack of important intermolecular interactions, which are ubiquitous in the cellular environment. Such deleterious events would lead to the premature loss of templating ability of unstable proteins, and thus generate false negative data. It is thus necessary to know, or determine in advance, their stability under the incubation conditions (incubation time, pH and temperature) used in KTGS experiments. In this context, faster ligation tools, characterized by lower energies of activation (Ea-ut) would be highly valuable, keeping in mind that a key for the success of KTGS strategy relies on a noticeable difference in kinetics between templated and untemplated reaction, which results from a marked difference in their activation energy Ea-t and Ea-ut, respectively (Figure 1B). It is important to note that the success of KTGS is also dependent on the ability of the two complementary reactive fragments to bind simultaneously the biological target, and conserve an optimal binding mode upon their ligation, in order to favor the aforementioned "linking effect" or super-additivity of fragment binding energies. [START_REF] Borsi | Entropic contribution to the linking coefficient in fragment based drug design: a case study[END_REF] Finally, the nature of the linker itself, termed as "linkage effect", can significantly contribute to the overall affinity of the ligand through establishment of additional stabilizing H-bonding, π-π stacking, and dipole-dipole interactions, or through its intrinsic physicochemical properties (e.g. lipophilicity). [START_REF] Nazare | Fragment deconstruction of small, potent factor Xa inhibitors: exploring the superadditivity energetics of fragment linking in protein-ligand complexes[END_REF][START_REF] Wong | The transcription factor STAT5 catalyzes Mannich ligation reactions yielding inhibitors of leukemic cell proliferation[END_REF] The triazole ring system has proven to be such a notable example of linker that actively participates in binding to biological targets, which may explained why this in situ click reaction is involved in most of the successful KTGS described to date. [START_REF] Bourne | Freeze-frame inhibitor captures acetylcholinesterase in a unique conformation[END_REF] In this Review, we have drawn a parallel between reported KTGS ligations and their counterpart transformations belonging to the bioorthogonal or bioconjugation toolkit (Figure 2A). However, these biocompatible reactions which have been optimized in order to exhibit fast rate constants (low activation energy Ea), cannot be used in KTGS without appropriate adjustments of their reactivity in order to get slower untemplated reactions kinetics (higher activation energy Ea-ut, Figure 2B)., and thus sufficient differences between template and untemplated reactions. This retrospective analysis may guide the development of new ligation reactions amenable for future KTGS applications, and some of these emerging reactions that might be suitable for this concept are proposed in the second part of this Review. Bioorthogonal chemistry and bioconjugation share some features such as fast rate constants, high chemoselectivity and the ability to proceed under physiological conditions. However, these are two distinct classes of reactions. Bioorthogonal reactions are defined as chemical reactions that can proceed in biological media without interfering with natural chemical functions, while bioconjugation reactions involve chemical functions (generally electrophilic systems) that react with naturally occurring functional groups (such as amines or sulfhydryl moieties). [START_REF] Wall | One-pot thiol-amine bioconjugation to maleimides: simultaneous stabilisation and dual functionalisation[END_REF][START_REF] Fredy | Diverted Natural Lossen-type Rearrangement for Bioconjugation through in Situ Myrosinase-Triggered Isothiocyanate Synthesis[END_REF] With the advent of bioorthogonal [START_REF] Cañeque | Visualizing biologically active small molecules in cells using click chemistry[END_REF] and bioconjugate chemistry, [START_REF] Koniev | Developments and recent advancements in the field of endogenous amino acid selective bond forming reactions for bioconjugation[END_REF] in recent years, an increasing number of experimental [START_REF] Lossouarn | Maleimide-based metal-free ligation with dienes: a comparative study[END_REF][START_REF] Decuypere | Sydnone-alkyne cycloaddition: applications in synthesis and bioconjugation[END_REF][START_REF] Petri | Comparative reactivity analysis of small-molecule thiol surrogates[END_REF] and theoretical studies, [START_REF] Tang | Accurate prediction of rate constants of Diels-Alder reactions and application to design of Diels-Alder ligation[END_REF][START_REF] Ravasco | Predictive Multivariate Models for Bioorthogonal Inverse-Electron Demand Diels-Alder Reactions[END_REF] computational analysis [START_REF] Liu | Bioorthogonal Cycloadditions: Computational Analysis with the Distortion/Interaction Model and Predictions of Reactivities[END_REF] on rate constants of such chemical reactions have been made available to the scientific community. Accordingly, in the light of reported ligation reactions for biorthogonal ligation or bioconjugation, a non-exhaustive list of new reactions amenable to KTGS strategies will be proposed (Figure 2A), as well as clues to tune the kinetics of these reaction to increase their potential to meet the requirements for use in KTGS strategies.

RETROSPECTIVE ANALYSIS OF LIGATION REACTIONS SUCCESSFULLY USED IN KTGS STRATEGIES AND THEIR CORRELATION WITH BIOORTHOGONAL AND BIOCONJUGATE LIGATION REACTIONS

A retrospective analysis of reactions developed for KTGS reveals that they all had their counterpart either in bioorthogonal chemistry or in the bioconjugate chemistry toolbox (Table 1). It should be stressed that all 2 nd order rate constants given in Table 1 and 2 have been determined in the absence of the biological target mostly under aqueous-based conditions. These two sets of reactions are particularly attractive due to their high chemoselectivity and biocompatibility, as well as their ability to proceed under mild aqueous conditions. [START_REF] Boutureira | Advances in chemical protein modification[END_REF] However, KTGS relies on the use of reactions with dramatically slower reaction rates (up to 10 6 fold), with values generally in the range of ~10 -2 to 10 -4 M -1 s -1 . In order to reach this requirement, kinetics of bioorthogonal reactions or bioconjugation are altered either by the absence of catalysts (such as the Huisgen reaction), or by using akin less reactive chemical moieties, yielding less effective ligation processes, yet acceptable for KTGS applications through the proximity effect. KTGS related to bioorthogonal chemistry. It is worth noting that reactions in this section belong to the bioorthogonal toolbox and as such, are considered as universal KTGS reactions, as contrasted to bioconjugation tools that may induce side-reactions with biological targets. First, the copper(I)-catalyzed 1,3-dipolar cycloaddition reaction of azides with alkynes (CuAAC) to yield corresponding 1,4disubstituted (anti) 1,2,3-triazole products was developed independently by Meldal and Sharpless in 2002, as an easy-to-use ligation tool (Scheme 1). [START_REF] Tornoe | Peptidotriazoles on solid phase: [1,2,3]-triazoles by regiospecific copper(i)-catalyzed 1,3-dipolar cycloadditions of terminal alkynes to azides[END_REF][START_REF] Rostovtsev | A Stepwise Huisgen Cycloaddition Process: Copper(I)-Catalyzed Regioselective "Ligation" of Azides and Terminal Alkynes[END_REF][START_REF] Kolb | Click Chemistry: Diverse Chemical Function from a Few Good Reactions[END_REF] Of note, 1,5-disubstituted (syn) analog scaffolds can regioselectively be formed in the presence of ruthenium (RuAAC), [START_REF] Zhang | Ruthenium-catalyzed cycloaddition of alkynes and organic azides[END_REF] or nickel (NiAAC) [START_REF] Kim | Nickel-Catalyzed Azide-Alkyne Cycloaddition To Access 1,5-Disubstituted 1,2,3-Triazoles in Air and Water[END_REF] catalyst. Accordingly, both regiosiomers are selectively synthetically accessible in high yields.

Scheme 1. Alkyne-azide 1,3-dipolar cycloaddition reactions.

In contrast, this reaction is slowed-down by several orders of magnitude in the absence of copper catalyst, [START_REF] Mock | Cycloaddition induced by cucurbituril. A case of Pauling principle catalysis[END_REF][START_REF] Mock | Catalysis by cucurbituril. The significance of bound-substrate destabilization for induced triazole formation[END_REF][START_REF] Presolski | Tailored ligand acceleration of the Cu-catalyzed azide-alkyne cycloaddition reaction: practical and mechanistic implications[END_REF] making it suitable for KTGS. This strategy was first introduced for KTGS by Sharpless and Finn in 2002 through the discovery of an acetylcholinesterase (AChE) noncovalent inhibitor having a dissociation constant in the femtomolar range. [START_REF] Lewis | Click Chemistry In Situ: Acetylcholinesterase as a Reaction Vessel for the Selective Assembly of a Femtomolar Inhibitor from an Array of Building Blocks[END_REF] In this case, the AChE generated predominantly the syn isomer in the sequestered region within the AChE gorge, while, our team observed a reversal of regioselectivity moving the reaction site toward the upper part of the gorge. [START_REF] Oueis | Reaction site-driven regioselective synthesis of AChE inhibitors[END_REF] In the former example, the anti isomer not formed in the enzyme, was less active by more than one order of magnitude. [START_REF] Manetsch | In situ click chemistry: enzyme inhibitors made to their own specifications[END_REF] Importantly, triazole ring systems are also considered as amide bioisosteres, and as such, they may actively contribute to proteins binding. [START_REF] Bourne | Freeze-frame inhibitor captures acetylcholinesterase in a unique conformation[END_REF][START_REF] Bonandi | The 1,2,3-triazole ring as a bioisostere in medicinal chemistry[END_REF] A few years later, the development of the SPAAC (stain-promoted azide-alkyne cycloaddition), a catalyst-free version of this transformation involving more reactive cycloalkynes was reported by Bertozzi for uncatalyzed covalent modification of biomolecules in living systems (entry 2). [START_REF] Agard | A Strain-Promoted [3 + 2] Azide-Alkyne Cycloaddition for Covalent Modification of Biomolecules in Living Systems[END_REF] In fact, while both copper and copper-free strategies lead to the formation of triazole-containing products from bioorthogonal azides and alkynes, strained alkynes exhibit favorable second order reaction rate constants (up to 1 M -1 s -1 ) [START_REF] Debets | Bioconjugation with strained alkenes and alkynes[END_REF] for bioorthogonal labelling, without the need for a cytotoxic copper catalyst. Very recently, a slower strain-promoted reaction, with a rate constant of 10 -3 M -1 s -1 , [START_REF] Ni | Heteroatom-embedded medium-sized cycloalkynes: concise synthesis, structural analysis, and reactions[END_REF] based on the use of a heteroatom-embedded cycloalkyne, 4,8-diazacyclononyne, proved amenable for KTGS.

This approach enabled the templated synthesis of a recombinant 14-3-3ζ protein inhibitor. [START_REF] Masuda | Copper-Free Huisgen Cycloaddition for the 14-3-3-Templated Synthesis of Fusicoccin-Peptide Conjugates[END_REF] Of note, the KTGS carried out with the terminal alkyne analogue of the 4,8-diazacyclononyne derivative, proved unsuccessful. This comparative study illustrates the importance of extending the repertoire of ligation tools by tailoring the reaction rate constant.

The sulfo-click chemistry involves thio acids and sulfonyl azides yielding N-acyl sulfonamide linkers. It was first used as chemoselective ligation tool for peptide synthesis. [START_REF] Merkx | Resin-Bound Sulfonyl Azides: Efficient Loading and Activation Strategy for the Preparation of the N-Acyl Sulfonamide Linker[END_REF][START_REF] Merkx | Highly efficient coupling of betasubstituted aminoethane sulfonyl azides with thio acids, toward a new chemical ligation reaction[END_REF][START_REF] Rijkers | Sulfo-click' for ligation as well as for site-specific conjugation with peptides, fluorophores, and metal chelators[END_REF] Shortly after, this amidation reaction was leveraged by Manetsch in KTGS for the in situ assembly of Bcl-XL inhibitors (entry 3). [START_REF] Hu | Bcl-XL-Templated Assembly of Its Own Protein-Protein Interaction Modulator from Fragments Decorated with Thio Acids and Sulfonyl Azides[END_REF] With a reaction rate constant recently determined of 10 -1 M -1 s -1 , 52 the sulfo-click reaction represents a rare example of KTGS reactions whose bioorthogonal counterpart was used without additional tuning of their reactivity. It is believed that this reaction is at the borderline in terms of kinetics to be used both in bioorthogonal chemistry (lower limit) and in KTGS applications (upper limit).

KTGS related to bioconjugate chemistry. Aside from bioorthogonal reactions, a greater number of KTGS reactions have their counterpart in bioconjugate chemistry. First, iodoacetamide is a well-known alkylating agent to block all free cysteine residues of proteins by reacting irreversibly with their sulfhydryl groups (entry 4). [START_REF] Goddard | DERIVATIVES OF KERATIN[END_REF] While this reaction proceeds with high rate constants (10 M -1 s - 1 ), their chlorinated analogue reacts, in contrast, much more sluggishly (10 -2 10 -1 M -1 s -1 ). [START_REF] Polgar | On the reactivity of the thiol group of thiolsubtilisin[END_REF] The latter derivative was used by Huc in 2001 in the design of inhibitors of bovine carbonic anhydrase (CA II). However, due to the fast alkylation of thiols with alkyl chlorides, this strategy required competitive experiments involving one thiol and two different alkyl chlorides, in order to determine whether the biological target affects or not the relative proportion of the two formed thioethers. [START_REF] Nguyen | Using an Enzyme's Active Site To Template Inhibitors[END_REF] Thio-Michael addition of biothiols with maleimides is one of the most widely used bioconjugation ligation tool, [START_REF] Friedmann | Antimitotic action of maleimide and related substances[END_REF] and prominent applications include the synthesis of marketed antibody-drug conjugates Adcetris (brentuximab vedotin) or Kadcyla (trastuzumab emtansine). [START_REF] Renault | Covalent Modification of Biomolecules through Maleimide-Based Labeling Strategies[END_REF] This reaction is characterized by high reaction speed 58,-59 and forms a thiosuccinimide adduct which has the propensity to undergo retro-Michael addition. However, by fine-tuning of the Michael acceptor, the reaction can be considerably slowed down (up to 10 5 -fold) by using acrylamide partners, [START_REF] Lee | A genetically encoded acrylamide functionality[END_REF] forming an irreversible thioamide linker (entry 5). As a KTGS counterpart, an enzymatic hydrolysis/Michael addition cascade led to the formation of thioamide binders of m-AChE with IC50 values in the low and sub-nanomolar range. [START_REF] Oueis | First enzymatic hydrolysis/thio-Michael addition cascade route to synthesis of AChE inhibitors[END_REF] This thio-Michael strategy was further illustrated by the discovery of bivalent kinase inhibitors by Soellner. [START_REF] Kwarcinski | Discovery of Bivalent Kinase Inhibitors via Enzyme-Templated Fragment Elaboration[END_REF] Interestingly, acrylamide moiety which was recently genetically encoded in Escherichia coli, has found to be stable in cells, despite the ubiquitous presence of free thiols. [START_REF] Lee | A genetically encoded acrylamide functionality[END_REF][START_REF] Lee | Phospha-Michael Addition as a New Click Reaction for Protein Functionalization[END_REF] This important result opens new opportunities for the use of acrylamide moieties for KTGS in cells.

Finally, modification of proteins through the amine side chain of lysine residues is a popular method due to their high abundance on protein surfaces. [START_REF] Anderson | The Use of Esters of N-Hydroxysuccinimide in Peptide Synthesis[END_REF] In this context, N-hydroxysuccinimide (NHS) esters are routinely used to form the corresponding amide bonds.

As this chemical ligation may be too fast to be implemented in KTGS (10 -1 10 2 M -1 s -1 , entry 6), [START_REF] Cline | Kinetics and mechanisms of the aminolysis of N-hydroxysuccinimide esters in aqueous buffers[END_REF] protein-templated amidation reactions were carried out by Rademann with poorer leaving groups, displaying a lower electron withdrawing potency, namely phenol and trifluoroethanol, in order to suppress non-templated background reaction. Importantly, this approach enabled the identification of a superadditive inhibitor of factor Xa in the nanomolar range from fragments with millimolar affinities. [START_REF] Jaegle | Protein-Templated Formation of an Inhibitor of the Blood Coagulation Factor Xa through a Background-Free Amidation Reaction[END_REF] This was ascribed to the "linking effect" i.e., to the minimization of the entropic penalty upon binding of fragments. Besides, this ligation also holds great promise since amide bonds are present in a large proportion of small-molecules therapeutics, partly due to their ability to be engaged in hydrogenbonding interactions with biological targets.

CLUES FOR EXPANDING THE REPERTOIRE OF KTGS REACTIONS

The latter bibliographic analysis, which shows systematic correspondences between KTGS and bioorthogonal or bioconjugate chemistry, should open a new avenue for discovering a series of suitable reactions for KTGS. Accordingly, studies which report chemical modifications of existing biocompatible reactions that lead to a significant decrease of their reaction rate, should be thoroughly considered for KTGS. By way of examples, a selection of such chemical ligations belonging to different classes of reactions (cycloaddition, nucleophilic substitution, addition, etc…) have been identified from the literature. These reactions, proposed as eligible reactions for KTGS, are reported in Table 2. KTGS inspired from bioorthogonal and chemoselective ligation. Diels-Alder reactions possess many attractive features such as high chemoselectivities and biocompatibilites with several applications in bioorthogonal chemistry [START_REF] Oliveira | Inverse electron demand Diels-Alder reactions in chemical biology[END_REF][START_REF] Li | A bioorthogonal ligation enabled by click cycloaddition of oquinolinone quinone methide and vinyl thioether[END_REF] and chemoselective ligations. [START_REF] Lossouarn | Maleimide-based metal-free ligation with dienes: a comparative study[END_REF][START_REF] Renault | Covalent Modification of Biomolecules through Maleimide-Based Labeling Strategies[END_REF][START_REF] Montgomery | Versatile Peptide Macrocyclization with Diels-Alder Cycloadditions[END_REF][START_REF] Jouanno | Kondrat'eva ligation: Diels-Alder-based irreversible reaction for bioconjugation[END_REF] Besides, biological systems such as ribozymes, catalytic antibodies, and more recently natural enzymes were proved to be able to promote Diels-Alder reactions. [START_REF] Jeon | Natural [4 + 2]-Cyclases[END_REF] In particular, the inverse electron-demand 1,2,4,5-tetrazine-based reaction is a well-established bioorthogonal ligation reaction which proceeds with high second order kinetic rate constants (k ~ 110 6 M -1 s -1 ) in the presence of strained alkynes/alkenes. [START_REF] Darko | Conformationally Strained trans-Cyclooctene with Improved Stability and Excellent Reactivity in Tetrazine Ligation[END_REF][START_REF] Blackman | Tetrazine ligation: fast bioconjugation based on inverseelectron-demand Diels-Alder reactivity[END_REF] Although such extremely fast reactions are not suitable for KTGS, dramatic decreases in rate constants were observed with linear alkenes (k ~ 10 -3 M -1 s -1 ), thus reaching the chemical reactivity window of KTGS reactions (Table 2, entry 1). [START_REF] Lee | Genetically encoded unstrained olefins for live cell labeling with tetrazine dyes[END_REF] Similarly, tetrazines have found to react with "mini-tag" cyclopropanes very efficiently, illustrated by several biological applications (entry 2). [START_REF] Patterson | Functionalized cyclopropenes as bioorthogonal chemical reporters[END_REF][START_REF] Yang | Synthesis and reactivity comparisons of 1-methyl-3-substituted cyclopropene mini-tags for tetrazine bioorthogonal reactions[END_REF] In contrast, less electron-deficient 1,2,4-triazines displayed theoretical rate constants with cyclopropanes that meet KTGS kinetics requirements (k ~ 10 -5 M -1 s -1 ). [START_REF] Kamber | 1,2,4-Triazines Are Versatile Bioorthogonal Reagents[END_REF] It is worth noting that during the finalization of this article, Disney elegantly showed that tetrazine ligation was amenable to KTGS by using sterically congested cyclopropenes, which markedly slowed down the tetrazinebased chemical ligation process. [START_REF] Angelbello | A Toxic RNA Templates the Synthesis of Its Own Fluorogenic Inhibitor by Using a Bio-orthogonal Tetrazine Ligation in Cells and Tissues[END_REF] This study showed that r(CUG) exp , a RNA repeat expansion that causes myotonic dystrophy type 1 (DM1), was able to template its own inhibitor with significant rate enhancement in comparison to the azide-alkyne strategy. [START_REF] Rzuczek | Precise small-molecule recognition of a toxic CUG RNA repeat expansion[END_REF] Besides, the maleimide-based Diels-Alder reaction with cyclopentadienes, [START_REF] Sun | Carbohydrate and protein immobilization onto solid surfaces by sequential Diels-Alder and azide-alkyne cycloadditions[END_REF] is characterized by high reaction rates in aqueous systems (~ 1010 2 M -1 s -1 ). [START_REF] Lossouarn | Maleimide-based metal-free ligation with dienes: a comparative study[END_REF][START_REF] St Amant | Tuning the Diels-Alder Reaction for Bioconjugation to Maleimide Drug-Linkers[END_REF] This chemoselective ligation was recently used for the production of robust antibodydrug conjugates. [START_REF] St Amant | A Diene-Containing Noncanonical Amino Acid Enables Dual Functionality in Proteins: Rapid Diels-Alder Reaction with Maleimide or Proximity-Based Dimerization[END_REF] In this context, recent studies from our group have shown that switching to linear diene such as trans-trans-2,4-hexadiene, the maleimide-based cycloaddition proceeded with a second-order kinetic rate constant in the range amenable for KTGS, namely 10 -3 M -1 s -1 . [START_REF] Lossouarn | Maleimide-based metal-free ligation with dienes: a comparative study[END_REF] This experimental value shows that maleimide/hexadiene pair would be suitable for KTGS applications, in particular with biological targets that do not contain free cysteine residues, the latter being likely to react with maleimides.

KTGS inspired from bioconjugate chemistry. As shown previously, templated alkylation of thiols with alkyl chlorides requires competition assays since thiols readily react with such electrophiles in the absence of the protein target. [START_REF] Nguyen | Using an Enzyme's Active Site To Template Inhibitors[END_REF] Alternatively, fluoroacetamides which have shown to react very smoothly with thiols (10 -3 M -1 s -1 , entry 4), [START_REF] Polgár | On the Reactivity of the Thiol Group of Thiolsubtilisin[END_REF] would represent an effective backgroundfree alternative that could be explored.

Besides, 2-vinyl/alkynyl pyridinium systems were recently identified as new bioconjugation tools by Bernardes for the fast and selective modification of cysteine-tagged proteins, with reaction rates in the range of 110 M -1 s -1 (entry 5). [START_REF] Matos | Quaternization of Vinyl/Alkynyl Pyridine Enables Ultrafast Cysteine-Selective Protein Modification and Charge Modulation[END_REF] This study also revealed that the addition of thiols was dramatically slower with 2-vinyl/alkynyl pyridine analogs, with reaction rates of 10 -4 10 -3 M -1 s -1 that meet the requirements of KTGS.

In addition, in the context of the development of bioconjugation techniques, ethynylphosphonamidates have very recently shown to undergo a cysteine-selective reaction with proteins or antibodies (entry 6). [START_REF] Kasper | Cysteine-Selective Phosphonamidate Electrophiles for Modular Protein Bioconjugations[END_REF] On the other hands, vinyl analogues, vinylphosphonamidates, proved far less reactive with thiol groups. [START_REF] Gao | Synthesis of a phosphonate-linked aminoglycoside-coenzyme a bisubstrate and use in mechanistic studies of an enzyme involved in aminoglycoside resistance[END_REF] Accordingly, as illustrated with the KTGS developed Michael addition of thiols with acrylamides, such an additional reactive moiety could be investigated for KTGS experiments.

Finally, sulfonyl chlorides, introduced first by Weber in 1952, [START_REF] Weber | Polarization of the fluorescence of macromolecules. II. Fluorescent conjugates of ovalbumin and bovine serum albumin[END_REF] have shown to react readily with amino groups present in proteins, such as for example the fluorogenic reagent dansyl chloride, [START_REF] Gray | Dansyl chloride procedure[END_REF] widely used for the labelling of biomolecules (entry 7). In contrast, sulfonyl fluoride analogues are more stable in particular towards hydrolysis, and react much more slowly with amines, [START_REF] Gahtory | Quantitative and Orthogonal Formation and Reactivity of SuFEx Platforms[END_REF] which makes them well suitable for use as warheads for activity-based protein profiling. Importantly, the S-F bond may be activated by hydrogenbonding between the leaving fluoride ion and proximal residues in the active site. [START_REF] Jones | Structure-based design and analysis of SuFEx chemical probes[END_REF] [START_REF] Moree | Peptides containing the sulfonamide transition-state isostere: synthesis and structure of N-acetyl-tauryl-L-proline methylamide[END_REF][START_REF] Radkiewicz | A Theoretical Investigation of Phosphonamidates and Sulfonamides as Protease Transition State Isosteres[END_REF] As a result, KTGS based on phosphonamide or sulfonamide moieties are expected to provide a substantial "linkage effect" on the overall affinity of protease templated ligands.

Finally, as a proof of concept, these reactions could first be tested on biological targets that have already proved successful for KTGS, such as G-quadruplex, lysine demethylase 5C (KDM5C), or Factor Xa, by targeting close analogs of ligands which were already assembled by these biological targets (Scheme 2). As examples, the tetrazine ligation and thiol-yne addition reaction could be investigated with G-quadruplex [START_REF] Panda | Target guided synthesis using DNA nano-templates for selectively assembling a G-quadruplex binding c-MYC inhibitor[END_REF] and KDM5C [START_REF] Miyake | Metalloprotein-Catalyzed Click Reaction for In Situ Generation of a Potent Inhibitor[END_REF] respectively, to furnish analogues of triazole-based ligands recently highlighted by in situ click chemistry (Scheme 2A and B). On the other hand, SuFEx chemistry could be investigated in place of the amidation reaction, which was successfully investigated with Factor Xa (Scheme 2C). 14 Scheme 2. Potential ligands and targets for proof-of-concept experiments, in regard with the ligands already unveiled through successful KTGS strategies.

CONCLUSIONS

A bibliographic analysis showed that reactions successfully used in KTGS have strong connections with preexisting fast bioorthogonal or bioconjugate chemistry reactions. However, rate constants of KTGS reactions require to be dramatically slower, generally in the range 10 -2 -10 -4 M -1 s -1 . It comes out that reaction rate of fast biocompatible reactions can be significantly tuned in order to meet kinetics requirements of KTGS transformations. In particular, this can be achieved by playing on different factors such as removing the catalyst, switching strained unsaturated systems by linear alkene or alkyne functions, or using less electrophilic/nucleophilic analogous chemical moieties. Several reliable electrophilicity/nucleophilicity scales are now available, and supporting theoretical studies can now be routinely implemented to supplement reported studies. However, it should be kept in mind that, in contrast to bioorthogonal and bioconjugate chemistry, the size of reactive functions matters in KTGS and should be also taken into consideration. In fact, bulky chemical functions may drastically disturb the binding mode of the parent fragments, and therefore negatively impact the affinity of resulting ligands. Furthermore, as many as possible, effective chemical processes should also be associated to new KTGS strategies.

They will be useful to provide readily obtained authentic standards readily obtained in order to assay the success of the KTGS process, as well as to upscale the synthesis of ligands unveiled by successful KTGS experiments, in order to further study, and tune their biological activity.

Table of Contents graphic

Figure 1 .

 1 Figure 1. Principle of kinetic target-guided synthesis (KTGS) with different classes of biological targets (A); and energy of activation of template & untemplated reactions (B).

Figure 2 .

 2 Figure 2. Bioorthogonal and bioconjugate reactions as a source of inspiration for KTGS ligations (A); Impact of chemical modifications of bioconjugation/bioorthogonal tools on their activation energy (Ea) in order to meet energy requirements (Ea-ut) of KTGS (B).

(

  The loop symbol is used with naturally occurring chemical functions, while the star and sphere symbols are used with synthetic functions. [b] Unless specified, rate constants were determined under aqueous-based conditions. [c] determined in aqueous formic acid at 40 °C. [d] Determined in CD3CN at 25 °C.

(

  leaving group ability of the sulfonyl derivative a [a] The loop symbol is used with naturally occurring chemical functions, while the star and sphere symbols are used with synthetic functions. [b] Rate constants were determined under aqueous-based conditions.

Table 1 . Retrospective analysis of KTGS reactions and their related bioorthogonal or bioconjugation reactions. a

 1

	Entry	Reaction	Reaction used in KTGS
			(untemplated reaction 2 nd order rate constant)

b

Related bioorthogonal or bioconjugate reaction

Table 2 . Perspective of KTGS reactions a

 2

	Entry	Reaction	Plausible reaction for KTGS	Related bioorthogonal or bioconjugate reaction
			(untemplated reaction 2 nd order rate
			constant) b

  Sulfur(IV)fluoride exchange (SuFEx) of sulfonyl fluorides has emerged as a new generation of click chemistry. Importantly, for these latter two examples, both phosphonamidates and sulfonamides are protease transition-state isosteres, as they are non-hydrolyzable functional groups capable of mimicking the tetrahedral transition-state intermediate of enzyme-based peptide bond hydrolysis.

Funding

The authors thank Agence National pour la Recherche (ANR-16-CE07-0004, "Click_and_Fish") for Ph.D fellowship to A.L. This work was also partially supported by the Centre National de la Recherche Scientifique (CNRS), INSA Rouen, Rouen University, and the Labex SynOrg (ANR-11-LABX-0029).

AUTHOR INFORMATION Corresponding Author

Cyrille Sabot  CNRS, UNIROUEN, INSA Rouen, COBRA (UMR 6014), Normandie Univ, 76000 Rouen, France; orcid.org/0000-0002-5014-7217; Email: cyrille.sabot@univ-rouen.fr

Authors

Alexis Lossouarn  CNRS, UNIROUEN, INSA Rouen, COBRA (UMR 6014), Normandie Univ, 76000 Rouen, France; Pierre-Yves Renard  CNRS, UNIROUEN, INSA Rouen, COBRA (UMR 6014), Normandie Univ, 76000 Rouen, France; orcid.org/0000-0001-9094-9778

Notes

The authors declare no competing financial interest.


Paul Rouquette

Claude Amra

Myriam Zerrad

Michel Lequime

Optimization of micro-cavities for huge enhancement patterns in freespace and modal space

Keywords: Micro-cavity, enhancement patterns, free-space, modal space

We show how to design huge enhancement of emission patterns in multilayer planar micro-cavities. Optimization is carried out arbitrarily for both the free-space pattern of plane waves and for the modal pattern of guided modes. The energy carried by these patterns is compared for various stacks, and their control is discussed.

INTRODUCTION

Optical sources within micro-cavities have been extensively studied for at least 2 decades [START_REF] Rigneault | Modal analysis of spontaneous emission in a planar micro-cavity[END_REF]- [START_REF] Riel | Tuning the emission characteristics of top-emitting organic light-emitting devices by means of a dielectric capping layer: An experimental and theoretical study[END_REF]. These days it is well known that over-coating these cavities with multilayer optics allows their emission patterns in free-space to be modified. While sources within single cavities radiate a quasi-lambertian pattern, more elaborate structures (such as Fabry-Perot filters) enable the patterns in angular or spectral regions to be confined. Applications concern lighting and biophotonics, micro-sources and single photon sources, antennas, optical sensors, etc... The theoretical modelling has allowed to optimize the efficiency of the radiated pattern in free-space with remarkable results. The Fabry Perrot structure is, in most of the cases, used. However, the energy transferred to guided modes has been less studied because of the complexity of the problem.

In recent work we introduced the zero-admittance technique [START_REF] Amra | Energy density engineering via zero admittance domains in all-dielectric stratified materials[END_REF] for designing huge field enhancement in specific multilayer optics (ZAL coatings). This problem differs from that of micro-cavities since the optical coating is free of sources, and is illuminated from a high-index medium to allow total internal reflection (TIR) to occur. The field enhancement can be arbitrarily controlled (amplitude, wavelength, incidence and polarization, depth localization) with the design technique and increases with the layer number; its amplitude is bounded by intrinsic (absorption, roughness, thermal and non-linear effects, damage threshold) and extrinsic bandwidth phenomena such as divergence and spectral width, design accuracy and transverse uniformity) conditions. Here enhancement means that the electromagnetic stationary field in the coating is much greater (by several decades) than the incident field.

The objective of this paper is to show that similar techniques are available for designing huge enhancement of emission patterns in micro-cavities. Specific layers are introduced to over-coat given cavities and are first shown to create asymptotic poles in the reflection coefficient of the multilayer structure in the free-space (plane waves) frequency range. The result is a huge enhancement of the free-space pattern, by several decades. In a second step the technique is extended to the modal (guided waves) window, where another technique allows exact poles to be constructed and to attain huge enhancement of arbitrary modes. We compare the energy carried in free space to that transferred to the guided modes, and address the question of controlling this ratio. Results are discussed for a series of stacks. This technique could be of great interest in the fabrication of ultra-sensitive optical sensor and differs from other works that rely mostly on numerical optimization [START_REF] Munzert | Multilayer coatings for Bloch surface wave optical biosensors[END_REF], [START_REF] Michelotti | Design rules for combined label-free and fluorescence Bloch surface wave biosensors[END_REF].

RECALLS ON MICRO-CAVITIES

The aim of this section is to recall how the emission patterns in micro-cavities can be obtain. We work in a single wavelength (or harmonic) regime with a temporal dependence exp(-𝑗𝜔𝑡), 𝜔 = 2𝜋𝑐/𝜆being the temporal pulsation, and 𝜆 the illumination wavelength. All media are assumed to be linear, isotropic, and non-magnetic. The end media (substrate and superstrate) are assumed to be non-absorbing. The planar geometry of the all-dielectric stack is shown in Figure 1. Such a stack can be designed either as an optical filter or coating (when illuminated at infinity from the surrounding media), or a micro-cavity (when the sources are located within the stack) or a waveguide (case of integrated optics). In Figure 1 all thicknesses have the same order of magnitude as the wavelength. At each interface (i) of the coating, an electric of magnetic current can be found and stands for the optical sources.

Figure 1: geometry of a planar multilayer stack, with n0 and nS the refractive indices of the superstrate and the substrate Since we are interested in the emission pattern of optical sources in micro-cavities, it is useful to remind how the energy balance can be analysed in those structures. The energy balance can be written [START_REF] Amra | Electromagnetic power provided by sources within multilayer optics: free-space and modal patterns[END_REF], [START_REF] Amra | Mutual coherence and conical pattern of sources optimally excited within multilayer optics[END_REF], [START_REF] Amra | Electromagnetic Optics of Thin-Film Coatings: Light Scattering, Giant Field Enhancement, and Planar Microcavities[END_REF] as:

𝐹 = Φ + 𝐴 ( 1
)
where 𝐹 is the total power provided by the sources, Φ the Poynting flux through a closed surface surrounding the cavity, and A the absorption within the cavity. We assume the absence of interaction between fields and currents (weak regime), and that the cavity supports a unique electric current 𝐽 𝑖 ⃗ ⃗ (𝑟 ) located at one interface (i). Under these conditions, in a single wavelength regime each term of relation (1) follows [START_REF] Amra | Electromagnetic power provided by sources within multilayer optics: free-space and modal patterns[END_REF], [START_REF] Amra | Mutual coherence and conical pattern of sources optimally excited within multilayer optics[END_REF], [START_REF] Amra | Electromagnetic Optics of Thin-Film Coatings: Light Scattering, Giant Field Enhancement, and Planar Microcavities[END_REF]:

𝑑𝐹 𝑑𝜑 = ∫ 𝑓(𝜎 )𝑑𝜎 +∞ 0 𝑑Φ 𝑑𝜑 = ∫ 𝑔(𝜎 )𝑑𝜎 +∞ 0 𝐴 = 𝜔 2 ∫ ℑ{𝜖(𝑧)}|𝐸(𝜎 , 𝑧)| 2 𝑑𝑧 𝑧 𝑝 0 ( 2
)
where 𝜎 = 𝜎(𝑐𝑜𝑠𝜑, 𝑠𝑖𝑛𝜑)is the spatial pulsation in polar direction 𝜑 in the range [0; 2𝜋], 𝑟 = (𝑥, 𝑦) the spatial transverse coordinates and 𝑧 an altitude in a direction perpendicular to the multilayer. Note that all energy quantities are given per unit of current surface area. The stationary electromagnetic field in the volume of the stack is denoted 𝐸(𝑧, 𝜎). Also, recall that all quantities depend on ω, whose notation is omitted for the sake of simplicity. The temporal Fourier transform of permittivity 𝜀(𝑡, 𝑧) is denoted 𝜀(𝑧).

In relation [START_REF] Rigneault | Spontaneous emission in multilayer microcavities: modal theory extended with Fourier-Green analysis for dissipative structures[END_REF] the 𝑓 and 𝑔 functions are spectral densities of power and flux respectively, per unit spatial pulsation and polar angle. The power density is given as [START_REF] Amra | Electromagnetic power provided by sources within multilayer optics: free-space and modal patterns[END_REF], [START_REF] Amra | Mutual coherence and conical pattern of sources optimally excited within multilayer optics[END_REF], [START_REF] Amra | Electromagnetic Optics of Thin-Film Coatings: Light Scattering, Giant Field Enhancement, and Planar Microcavities[END_REF]:

𝑓(𝜎 ) = 𝑑𝐹 𝑑𝜎𝑑𝜑 = 2𝜋 2 ℜ { 1 ΔY i (𝜎) } |𝐽 ̂𝑖(𝜎 )| 2 ( 3
)
where 𝐽 ̂𝑖(𝜎) is the spatial Fourier transform of 𝐽 𝑖 ⃗ ⃗ (𝑟 ) and the quantity Δ𝑌 𝑖 = 𝑌 𝑖 ′ -𝑌 𝑖 is the difference in the complex admittances from each side of surface (i), that is, in layers (i) and (i-1) respectively. Note that the admittances are discontinuous at surface i where the current is present. Relation (3) shows how the cavity coefficient

1 ΔY i (𝜎)
drives the power provided by the current, which can be inhibited or enhanced (at constant current). It should be stressed that all terms in relation ( 1) are cavity-related (they depend on the cavity coefficient), so that the optical balance is always satisfied [START_REF] Amra | Electromagnetic Optics of Thin-Film Coatings: Light Scattering, Giant Field Enhancement, and Planar Microcavities[END_REF].

The flux density emerging into the end media is given by:

𝑔(𝜎 ) = 𝑔 -(𝜎 ) + 𝑔 + (𝜎 ) (4)
Where 𝑔 + -is relative to the flux by reflection (𝑔 -) or transmission (𝑔 + ), that is:

𝑔 -(𝜎 ) = dΦ - dσd𝜑 = 2𝜋 2 𝜎ℜ{𝑛 ̃0}|𝐸 0 -(𝜎 )| 2 and𝑔 + (𝜎 ) = dΦ + dσd𝜑 = 2𝜋 2 𝜎ℜ{𝑛 ̃𝑆}|𝐸 𝑆 -(𝜎 )| 2 (5)
In relations [START_REF] Benisty | Impact of Planar Microcavity Effects on Light Extraction-Part I: Basic Concepts and Analytical Trends[END_REF], 𝑛 ̃ denotes the effective index in the superstrate (𝑛 ̃0) or in the substrate (𝑛 ̃𝑠).

Throughout all these formulae, light polarization is considered by the effective indices and admittances. Finally, one can check [START_REF] Amra | Electromagnetic Optics of Thin-Film Coatings: Light Scattering, Giant Field Enhancement, and Planar Microcavities[END_REF] for a lossless cavity that the provided power is equal to the sum of the emerging fluxes.

In relation [START_REF] Benisty | Impact of Planar Microcavity Effects on Light Extraction-Part I: Basic Concepts and Analytical Trends[END_REF] the flux is given as a function of fields, while in (3) the power was given as a function of the current. We will also need to express the power as a function of the fields [START_REF] Amra | Electromagnetic Optics of Thin-Film Coatings: Light Scattering, Giant Field Enhancement, and Planar Microcavities[END_REF] which can be written in the absence of magnetic currents:

𝑓(𝜎 ) = 𝑓 -(𝜎 ) + 𝑓 + (𝜎 ) = 2𝜋 2 𝜎[ℜ{𝑌 𝑖 ′ }|𝐸 𝑖 ′ (𝜎)| 2 -ℜ{𝑌 𝑖 }|𝐸 𝑖 (𝜎)| 2 ] = 2𝜋 2 𝜎ℜ{Δ𝑌 𝑖 }|𝐸 𝑖 (𝜎)| 2 (6)
As an illustration and because of future need for normalization in the next sections, the emission pattern of 2 single cavities is plotted at a wavelength of 633nm in Figure 2. The first cavity (on the left) is a single very thin (1nm) high-index layer between an air superstrate and an air substrate, with the current located at half distance from each interface. We observe a quasi-lambertian pattern in both reflection and transmission, which will be used for normalization of the patterns from other cavities. The second cavity (on the right) is a 𝑄𝑊 = 𝐿(𝐻𝐿) 𝑞 stack matched at wavelength 𝜆 0 0 = 633𝑛𝑚for normal incidence 𝜃 0 0 = 0°. The current is located at the top surface of the mirror. This second cavity pattern is normalized at each angle by the previous one, which highlights the cavity effect (moderate enhancement in reflection by a factor of 4, and strong reduction in transmission). TE polarization is used for all patterns. Notice that huge enhancement is not expected here and will be discussed in the next section.

ENHANCEMENT PATTERNS IN FREE SPACE WITH ZAL COATINGS

Analytical discussion

Field enhancement still remains one of the key objectives of nanophotonics [START_REF] Bahaa | Fundamentals of Photonics[END_REF], [START_REF] Novotny | Principles of Nano-Optics[END_REF], regardless of the geometry (planar, radial, concentric) of the component. Enhancement means that the stationary field in the stack is much greater than the input (incident) collimated (plane wave) field in the superstrate. In the case of planar multilayers, we recently highlighted an analytical design technique for creating huge (several decades) field enhancement with zero-admittance layers [START_REF] Amra | Electromagnetic Optics of Thin-Film Coatings: Light Scattering, Giant Field Enhancement, and Planar Microcavities[END_REF], [START_REF] Amra | Energy density engineering via zero admittance domains in all-dielectric stratified materials[END_REF] a technique that is relevant for threshold reduction of micro-sources and high-sensitivity optical sensing. The enhancement can be controlled in amplitude and for arbitrary illumination conditions (incidence, wavelength and polarization), provided the stack is under total internal reflection (TIR). Hence the superstrate is a high-index material with respect to the substrate material (𝑛 0 > 𝑛 𝑠 ) and oblique incidence (𝜃 0 ) in the superstrate is required beyond the refractionlimit angle, that is, in terms of spatial pulsation: 𝜎 0 = 𝑘 0 sin 𝜃 0 > 𝑘 𝑆 with𝑘 0,𝑆 = 2𝜋𝑛 0,𝑆 /𝜆andσ 0 istheincidentspatialpulsation [START_REF] Benisty | Method of source terms for dipole emission modification in modes of arbitrary planar structures[END_REF] The TIR regime is a key feature since it allows the complex admittance to be cancelled at a particular altitude 𝑧 𝑐 of the stack (which is the origin of huge enhancement in quarter-wave stacks under TIR), while in free space this zero admittance value must be approached asymptotically (as in Fabry-Perot filters). So, if one considers a quarter wave stack with a ZAL layer over coating the stack we obtain, at a specific spatial pulsation 𝜎 𝑐 > 𝑘 𝑆 and at the top interface of the ZAL layer (𝑧 = 𝑧 𝑝-1 ):

𝑌 ′ (𝜎 𝑐 , 𝑧 = 𝑧 𝑝-1 ) = 0 ( 8
)
We are now interested in the optimization of light emitted by the micro-cavity in the absence of illumination. As per relation (3), maximum enhancement in the cavity would be provided at spatial frequencies 𝜎 𝑚 which satisfy the condition:

Δ𝑌 𝑖 (𝜎 𝑚 ) = Δ𝑌 𝑖 ′ (𝜎 𝑚 , 𝑧 = 𝑧 𝑖 ) = 0 (9)

Note that relations [START_REF] Björk | Modification of spontaneous emission rate in planar dielectric microcavity structures[END_REF][START_REF] Delbeke | High-efficiency semiconductor resonant-cavity light-emitting diodes: a review[END_REF] are respectively related to TIR (𝑘 𝑠 < 𝜎 𝑐 < 𝑘 0 ) and modal (max(𝑘 0 , 𝑘 𝑠 ) < 𝜎 𝑚 < max(𝑘 𝑖 )) frequency ranges which do not intersect [START_REF] Amra | Mutual coherence and conical pattern of sources optimally excited within multilayer optics[END_REF], [START_REF] Amra | Electromagnetic Optics of Thin-Film Coatings: Light Scattering, Giant Field Enhancement, and Planar Microcavities[END_REF] so that they do not seem related a priori. This result is illustrated in Figure 3, where the abscissa is the normalized spatial frequency (𝜈 * = 𝜆𝜎/2𝜋) analogous to 𝑛𝑠𝑖𝑛𝜃. Since we are concerned with an optimization of the reflected pattern only, we start with relation ( 6) which can be further developed as [START_REF] Amra | Electromagnetic Optics of Thin-Film Coatings: Light Scattering, Giant Field Enhancement, and Planar Microcavities[END_REF]:

𝑓 -(𝜎 ) = -2𝜋 2 𝜎ℜ{𝑌 𝑖 } |𝐽 ̂𝑖(𝜎 )| 2 |Δ𝑌 𝑖 | 2 ( 10
)
We assume the current to be located at the top interface (p-1) of the ZAL layer. With relation ( 8), we obtain:

Δ𝑌(𝜎 𝑐 , 𝑧 𝑝-1 ) = -𝑌(𝜎 𝑐 , 𝑧 𝑝-1 ) = -𝑌 𝑝-1 (𝜎 𝑐 ) (11)
So that:

𝑓 -(𝜎 𝑐 ⃗⃗⃗ ) = 2𝜋 2 𝜎 𝑐 ℜ{𝑌 𝑝-1 (𝜎 𝑐 )} |𝐽 ̂𝑖(𝜎 𝑐 )| 2 |Y p-1 (𝜎 𝑐 )| 2 (12)
and the question is whether this last quantity can be maximized at 𝜎 𝑐 . For that we use the transfer matrix from interface 0 to p-1 to find an expression of Y p-1 [START_REF] Amra | Electromagnetic Optics of Thin-Film Coatings: Light Scattering, Giant Field Enhancement, and Planar Microcavities[END_REF]:

𝑌 𝑝-1 (𝜎 𝑐 ) = 𝑛 ̃𝐿 2 𝑛 ̃0 ( 1 𝛽 2𝑞 ) with𝛽 = 𝑛 ̃𝐻/𝑛 ̃𝐿 ( 13
)
We can then conclude that the single wavelength cavity pattern at the ZAL frequency 𝜎 𝑐 < 𝑘 0 is enhanced in the form:

𝑓 -(𝜎 𝑐 ⃗⃗⃗ ) = 2𝜋 2 𝜎 𝑐 𝛽 2𝑞 𝑛 ̃0 2 𝑛 ̃𝐿 |𝐽 ̂𝑖(𝜎 𝑐 )| 2 (14)
The key result here is the proportionality to the term 𝛽 2𝑞 of the effective index ratio, which creates a huge enhancement pattern of the micro-cavity.

It should be stressed that huge enhancement occurred at the ZAL frequency 𝜎 𝑐 though this frequency does not belong to the modal frequency, in contrast with the poles 𝜎 𝑚 . The reason is that 𝜎 𝑐 is not an exact, but an asymptotic zero of the admittance difference:

lim 𝑞→∞ Δ𝑌 𝑝-1 (𝜎 𝑐 ) = 0 (15)

Numerical results

In Figure 4 we have plotted the angular pattern of a multilayer stack with a ZAL layer. The design of the coating is Glass/L(HL) q /ZAL/Air. The stack was formerly designed for a free-space illumination for TE polarization, at wavelength 𝜆 0 0 = 633𝑛𝑚 and incidence 𝜃 0 0 = 45°. Here there is no free-space illumination, but a current located at the top surface of the ZAL layer (interface p-1=13 with q = 6). The emission pattern is plotted for TE polarization in the incidence plane (𝜑 = 0°) at wavelength 𝜆 0 = 𝜆 0 0 . In order to quantify the gain of this cavity, the pattern is normalized at each normal angle 𝜃 to the pattern value (at the same angle) of the thin layer cavity to the left of Figure 2.

Following [START_REF] Michelotti | Design rules for combined label-free and fluorescence Bloch surface wave biosensors[END_REF] the cavity is expected to enhance the source emission at the same 𝜆 0 0 and 𝜃 0 0 . This is indeed what we observe in Figure 4 since there is an intense peak of four decades enhancement at 45°. Note also the narrow angular width ∆𝜃 0 of the peak, where most of the energy is concentrated; the percentage of energy in the peak is 14% for ∆𝜃 0 = 0.01° and 36% for ∆𝜃 0 = 0.1°. These values must be multiplied by a factor of 2 because of the symmetrical peak. Contrary to its integral, the peak magnitude is not bounded due to the fact that the width and amplitude of the peaks are respectively proportional and inversely proportional to the imaginary indices of the thin film materials. Other peaks are present but cannot be seen in the left-hand figure because their amplitude is much lower by several decades. They can be seen on the right-hand figure where I0 is plotted for the reflected pattern and IS for the transmitted pattern.

ENHANCEMENT OF GUIDED WAVE IN MICRO-CAVITIES

Until this point, we were limited to free-space optics and never considered the energy carried by guided modes within the cavity in directions parallel to the interfaces. In this section we address the question of modal efficiency, that is, the ability to reach a maximum power carried by these modes without radiative losses. In other words, starting from a given source in the cavity, the question is how to optimize the power provided to the modal window, or to a specific mode in this window, and to quantify this power in comparison with the free-space power in the surrounding media.

Working with the ZAL coating and the ZAL frequency

An initial idea was to keep the same ZAL coating but modify the superstrate so that the zero-admittance frequency 𝜎 𝑐 is transferred to the modal space. Hence the glass superstrate is replaced by air, which turns the modal frequency range into: max(𝑘 𝑆 , 𝑘 𝑆 ) = 𝑘 𝑆 < 𝜎 𝑐 < max(𝑘 𝑖 ) = 𝑘 𝐻 [START_REF] Amra | Energy density engineering via zero admittance domains in all-dielectric stratified materials[END_REF] where k 𝑠 is for the air medium. The surface current is still located at the top surface (p-1) of the ZAL layer. The power density 𝑓(𝜎, 𝜑 = 0) of the cavity is shown in Figure 5 for TE polarization. Eight modes can be seen in the modal window, with the TE0 mode at the highest frequency, while the zero-admittance frequency 𝜎 𝑐 is close to the TE6 mode (with the highest amplitude). Actually, the TE6 frequency tends to the ZAL frequency when the layer number of the QW stack increases, that is:

lim 𝑞→∞ 𝜎 6 = 𝜎 𝑚 (17)
As expected, we observe an intense peak in the vicinity of the ZAL frequency 𝜎 𝑐 ≈ 𝜎 6 .

Figure 5: Power density versus normalized frequency, plotted for the ZAL cavity where the superstrate is replaced by air Recall here that in contrast with the free-space situation where the poles are complex, there are 8 exact poles in the modal window at which the reflection function diverges. For that reason we used small imaginary indices (10 -6 ) for thin film dielectric materials in order to remove these singularities.

Results can be found in Table 1. In this table we give the modal energy carried by each mode and for different current localization after normalization by the energy provided in the free-space window. This allows the efficiency of the process to be quantified. We first observe that, when the current is localized on the p-1 interface (#13), the total modal power (last column) is greater by a factor of 4 than the free-space power, and that among the modes the greatest energy is carried by the TE6 mode (close to the ZAL frequency). Results vary with the depth localization of current. The greatest modal efficiency is reached at the 9 th interface, with maximum power transferred to the TE4 mode. 1: Energy provided to each mode, after normalization by the energy provided in the free-space window. ZAL cavity with air superstrate and substrate. TE polarization. Each line is for a particular current localization (see text)

Modes

Designing an exact pole with a MZAL coating

In sub-section 3.1 we worked with the ZAL coating and observed that the zero-admittance (ZAL) frequency 𝜎 𝑐 was an asymptotic pole (see relation [START_REF] Amra | Electromagnetic Optics of Thin-Film Coatings: Light Scattering, Giant Field Enhancement, and Planar Microcavities[END_REF]). The best optimization would be to design an arbitrary exact pole (as opposed to asymptotic pole). While this was not possible in free space where exact poles are necessarily complex [START_REF] Deppe | Spontaneous emission from a dipole in a semiconductor microcavity[END_REF], the modal window offers the opportunity to construct an exact pole. To this end, we consider a QW stack alone and, in the absence of absorption, look for real pole 𝜎 𝑚 that satisfies:

Δ𝑌 𝑝 (𝜎 𝑚 ) = 0 ⇔ 𝑌 𝑝 (𝜎 𝑚 ) = 𝑛 ̃𝑆(𝜎 𝑚 ) (18)
We coat the QW stack with a with a single layer of refractive index n and we seek the thickness layer that satisfies the condition 𝑌 𝑝+1 = 𝑛 ̃𝑆, where subscript (p+1) is used to localize the new last interface of the stack.

Relation (20) gives the thickness of the additional layer once its refractive index is chosen. By analogy with the ZAL layer, such a layer will be denoted MZAL to refer to the modal window to which it applies. Note that while the ZAL layer cancels the admittance in the free-space window, the MZAL layer cancels an admittance difference in the modal window.

Note that a MZAL could also be designed at the top interface. For that, following the same method, we coat the stack with a single layer of refractive index n and we seek the thickness that satisfies the condition 𝑌 -1 ′ = -𝑛 ̃0, where the subscript (-1) is used to localize the new top interface of the stack. We obtain: The MZAL layer allows an exact pole to be constructed whatever the coating design. Hence a complex stack is not required, and for this reason we address the case of a single layer over-coated with a MZAL layer on the superstrate side. The design of the coating is Air/MZAL H/Air and the question is how to quantify its performance with respect to the previous coatings.

The MZAL is a high-index layer still calculated for a pole at 45° and 633nm wavelength. Results are plotted in Figure 6 for 3 different current localizations. Two poles appear in the modal window for TE polarization. The modal efficiency is given in Table 2. A maximum value is obtained at the top surface of the single layer (#1), with maximum energy transferred to the TE0 mode that we constructed. We also notice that the modal efficiency is much greater (19) than that (4) of the QW stack, which highlights the results of this sub-section. We have checked that such efficiency can be increased (up to 25fold) if the high index material (Ta2O5) is replaced by a higher index one (Nb2O5, n = 2.3216 at 633nm) as is currently used in thin films. Note, however, that though these single stacks provide high efficiency in the modal window on account of their exact poles, this efficiency vanishes in free space since their behaviour is not asymptotic.

Figure 2 :

 2 Figure 2: Angular patterns at 633nm for a single very thin layer cavity (left side, SI units) and for a quarter-wave stack (right side). The second pattern is normalized with respect to the first (see text). TE polarization.

Figure 3 :

 3 Figure 3: Free-space and modal frequency windows. The abscissa is the normalized spatial frequency

Figure 4 :

 4 Figure 4: TE normalized angular pattern at 633nm of the ZAL micro-cavity in polar (left figure) and cartesian (right figure) coordinates (see text).

Figure 6 :

 6 Figure 6 : power density for a single layer coated with a MZAL layer (see text)

  The admittances 𝑌 𝑝+1 and 𝑌 𝑝 are related through: Due to the lossless regime of the guided waves, the 2 emerging effective indices (𝑛 ̃0𝑎𝑛𝑑𝑛 ̃𝑠) and the 2 admittances are pure imaginary numbers (𝑌 = 𝑗𝑌 " )[START_REF] Amra | Electromagnetic power provided by sources within multilayer optics: free-space and modal patterns[END_REF],[START_REF] Amra | Mutual coherence and conical pattern of sources optimally excited within multilayer optics[END_REF],[START_REF] Amra | Electromagnetic Optics of Thin-Film Coatings: Light Scattering, Giant Field Enhancement, and Planar Microcavities[END_REF] while the effective indices in the layer media are real (k 𝑠 < 𝜎 𝑚 < k 𝐿 < k 𝐻 ). Seeking a solution for 𝑌 𝑝+1 (𝜎 𝑚 ) = 𝑛 ̃𝑆(𝜎 𝑚 ), we obtain:

	𝑌 𝑝+1 =	𝑌 𝑝 cos 𝛿 + 𝑗𝑛 ̃sin 𝛿 cos 𝛿 + 𝑗 𝑛 ̃sin 𝛿 𝑌 𝑝	with𝛿 = √𝑘 2 -𝜎 2 𝑒	(19)
		tan 𝛿 = 𝑛 ̃𝑛 ̃𝑆 ′′ -𝑌 𝑝 ′′ 𝑛 ̃2 + 𝑛 ̃𝑆 ′′ 𝑌 𝑝 ′′

Table 2 :

 2 Energy provided to each mode, after normalization by the energy provided in the free-space window. Case of a single layer coated with a MZAL layer. TE polarization. Each line is for a particular surface localization (see text)

	Modes	TE0	TE1	total
	Inter 0	2.5355	1.7743	4.3098
	Inter 1	18.7449	0.3490	19.0940
	Inter 2	2.5355	1.7743	4.3098


Houari Bettahar

Cédric Clévy

Nadège Courjal

Philippe Lutz

Force-Position Photo-Robotic Approach for the High-Accurate Micro-Assembly of Photonic Devices

Keywords: Micro-robotics, Nano-positioning, Accuracy, Photo-robotic, Photonics, Fabry-Perot interferences

Recently, micro-assembly of individual photonic elements has been highly attracting to provide new optical functionalities and products with high performances. This approach requires position control in free space and contact detection in constrained space. In both cases, getting accurate measurements remains a key lock. For this sake, the FP (Fabry-Perot) interferences happening during the active alignment of optical elements is exploited. This 1D displacement measurement is combined with multi-DoF (Degrees-of-Freedom) robotic motions and a compliant structure along a photo-robotic approach to provide high accurate estimation of both multi-DoF position and contact forces during the assembly process. This approach has been applied to the assembly of an optical lamella with respect to a fiber. An original contact detection algorithm has been proposed, which relies on correlations between the interference figure and the expected irradiance deduced from the nano-positioner internal sensors. This contact detection enables to estimate the position of the lamella with respect to the fiber along the optical axis and thus control the UV-adhesive thickness required to achieve high optical performances after assembly. Experimental investigations using this new approach shows that a positioning accuracy of 27.6 nm and 2.1 m • (standard deviation values) are reached for positions and orientations respectively. The contact detection algorithm has also been studied experimentally and results shows that the contact position can be detected with a maximum error of 80 nm, which provides a very high interest for many photonic applications.

I. INTRODUCTION

Achieving fast and automated 3D complex tasks of micromanipulation and micro-assembly with very high accuracy is strongly desired in many application fields, such as advanced integrated optics manufacturing (to achieve commercial success with reliable and cost-effective assembly, and packaging technologies) [START_REF] Clévy | Positioning accuracy characterization of assembled microscale components for micro-optical benches[END_REF] [START_REF] Popa | M 3 -deterministic, multiscale, multirobot platform for microsystems packaging: Design and quasistatic precision evaluation[END_REF]. Many emerging industrial on-demand applications made the field of integrated optics particularly challenging because of their always increasing functions and complexity. Indeed, these applications require the integration of various building components coming from different technologies in order to achieve a wide range of microsystem types [START_REF] Das | A multiscale assembly and packaging system for manufacturing of complex micro-nano devices[END_REF] [START_REF] Benson | Assembly of hybrid photonic architectures from nanophotonic constituents[END_REF]. The possibility to achieve hybrid nano-photonic components by assembly provides very original solutions to overcome the monolithical integration difficulties and to achieve 3D multi-functional optical devices fabricated with different technologies [START_REF] Benson | Assembly of hybrid photonic architectures from nanophotonic constituents[END_REF]. Robotic micro-assembly also enables to automate, to optimize the optical performances through closed loop control and to adapt every assembly to the heterogeneities of fabrication processes [START_REF] Böttger | Fully automated hybrid diode laser assembly using high precision active alignment[END_REF].

Positioning and assembly for integrated optics industry is generally done manually based on passive and active positioning for many applications [START_REF] Hauffe | Methods for passive fiber chip coupling of integrated optical devices[END_REF]. In passive positioning [START_REF] Takigawa | Passive alignment and mounting of linbo 3 waveguide chips on si substrates by low-temperature solid-state bonding of au[END_REF], operators may use geometrical devices such as V-grooves, or visual systems such as a microscope or a binocular to provide visual geometrical information which could be used for positioning [START_REF] Ma | Robotic grasping and alignment for small size components assembly based on visual servoing[END_REF] [START_REF] Zheng | Automated visual position detection and adjustment for optical waveguide chips and optical fiber arrays[END_REF]. However, very complex procedures must be established in passive positioning to guarantee optical performances. For this reason, the active approach is mainly used for integrated optics industry and mostly in a manual or teleoperated way [START_REF] Zhang | A novel algorithm for fiber-optic alignment automation[END_REF] [START_REF] Landry | Automated alignment of microstructured optical fibers and conventional singlemode fibers[END_REF]. Micro and nano-positioning systems are used to adjust the position of the optical component in order to obtain the maximum optical irradiance. This manipulation requires dexterity and skills on the part of the operator and a lot of precision and attention which means that it is a tiring task for the operator. Consequently, the performances and quality of the assembled products are strongly operator-dependent. Moreover, optical components are fragile and easily breakable. They typically break at the micro-Newton (µN) force range, a range that cannot be felt by a human (then by the operator) assembling these optical components with tweezers and microscopes. It is also, not reliably measurable with the existing force sensors during assembly. As a result, it is extremely difficult to manipulate effectively and flawlessly optical components for assembly at that scale. This situation decreases overall yields, drives up the assembled optical structure costs. and prevent from scaling up.

Furthermore, closed loop control approaches typically used in robotics opens the possibility to control the relative position between two components in a referenced way. Nevertheless, at the microscale, micro and nanopositioning robots introduce geometrical errors and integrating sensors able to estimate contact forces in the mN range together with relative position is still an open issue [START_REF] Tan | Accuracy quantification and improvement of serial micropositioning robots for in-plane motions[END_REF] [START_REF] Guelpa | Accurate 3d-positioning in a sem through robot calibration[END_REF].

In this scope, a photo-robotic approach was proposed in the previous works [START_REF] Bettahar | Photorobotic positioning for integrated optics[END_REF] [START_REF] Bettahar | Novel strategy for high precision automated robotic positioning based on fabry-perot interferometry principle[END_REF]. It has especially been applied to align an optical lamella with respect to an optical fiber with high accuracy in free space and without any contact between the two components. This approach uses a 1-D interferometric measurement principle based on FP interferences associated to a robot motion in order to achieve identification of the extrinsic geometric parameters. The working principle of this approach was presented and validated experimentally. It demonstrated its interest and high performances for different cases: identification of 2 parameters [START_REF] Bettahar | Photorobotic positioning for integrated optics[END_REF], identification of 4 parameters [START_REF] Bettahar | Novel strategy for high precision automated robotic positioning based on fabry-perot interferometry principle[END_REF] and identification of all extrinsic parameters of the robot [START_REF] Bettahar | Photorobotic extrinsic parameters calibration of 6-dof robot for high positioning accuracy[END_REF].

These works was indeed based on an "position"-based photo-robotic approach and demonstrated the capability to achieve the relative positioning of two optical components.

But, achieving the full assembly process require to consider the contacts between the components notably to control the UV-adhesive thickness required to maintain high optical performances after completing the assembly process. For these purposes, the contribution of the paper especially aims at proposing an algorithm to detect the contact (i.e. the relative distance) between the two optical components. The proposed approach especially synergize both position and force necessary to complete the whole assembly process. This approach is based on the correlation between FP cavity length and the interference figure and thus conduct to an original "force-position"-based photo-robotic approach.

The paper is organized as follows. Section II recalls the FP interferometry working principle. Section III illustrates the correlation between the robotic motions and the FP interference figure and presents the photo-robotic approach. Section IV presents the "force-position" photo-robotic approach for multi-DOF nano-positioning. Section IV investigates the strategy to maximize optical performances. Section V investigates contact detection based on FP interferometry. Section VI concludes the paper.

Optical lamella positioning

Micro-assembly with optimal optical performances Optical lamella placement along the optical axis UV-adhesive

Contact detection

II. FABRY-PEROT INTERFEROMETRY

Interferometric sensing is attracting due to its high resolution, accuracy, wide measuring range and bandwidth. Active alignment induces the FP interferometry phenomenon resulted by the multiple beam interferences that arises when light passes through a cavity bounded by two reflective surfaces as shown in Figure 2. A generic optical system is chosen as a case-study. This generic optical system represents an optical fiber and an optical lamella, as depicted in Figure 2. In our case, the involved surfaces are the fiber extremity (R 1 ) in one hand and the lamella surface (R 2 ) in the other hand.

Based on the FP interferometry principle, when the reflective surface (R 2 ) moves along the optical axis, the FP cavity length (L) changes and modifies the reflected light signal accordingly. Therefore, there is a correlation between the FP cavity length L and the interference figure, defined as follows:

Optical axis

δL = λ 2 • p with p ∈ N (1)
The periodicity of the irradiance signal depends directly on the wavelength of the laser signal (λ = 1560 nm ). p is a natural number, the distance between two resonances (see Figure 2), i.e. the distance variation between two peaks (δL) is a multiple of half wavelength λ of the laser irradiance.

Active positioning and assembly of optical components give the opportunity to measure a highly accurate 1-D relative position along the optical axis with very high resolution based on 1-D FP interference.

III. PHOTO-ROBOTIC POSITIONING APPROACH FOR HIGHLY ACCURATE NANO-POSITIONING

Conventional active alignment techniques is done manually by an operator. The operator moves the optical lamella with respect to the fiber, in order to maximize the optical irradiance, by applying a an non-referenced movements via the nano-positioner. As a results, it leads to local maximum optical irradiance. Moreover, it is time consumption especially for multi-DOF alignment process to reach maximum irradiance, since there is no model relating robotic control variables with the optical irradiance.

For this sake, the proposed photo-robotic approach of our previous paper [START_REF] Bettahar | Novel strategy for high precision automated robotic positioning based on fabry-perot interferometry principle[END_REF] is used, to achieve highly accurate multi-DOF positioning. The photo-robotic approach uses a 6-DOF serial nano-positioning system to hold the optical lamella and align it to a fixed fiber as shown in Figure 3. The positioning system is controlled via the computer using Matlab/Simulink platform. The interfered reflected light is measured using the photo-detector (optical irradiance), where the fibered circulator is used to separate the reflected interfered light from the incident light. The measured reflected light

Reference frame Definition

Rp (Op Xp Yp Zp) is assigned to the positioning system, the origin Op is assigned to its real rotational center of the positioning system, the ---→ OpXp, ---→ OpYp and ---→ OpZp are the expected motion directions of the positioning system.

R l (O l X l Y l Z l )
is assigned to the optical lamella, its origin is assigned to the intersection point between the optical axis and the lamella plane. The axis ---→ O l Z l is perpendicular to the optical lamella plane and it is directed toward the fiber. The axes --→ O l Y l and ---→ O l X l are perpendicular to each other and they are on the lamella plane.

R f (O f X f Y f Z f )
is assigned to the fiber, its origin O f is assigned to the intersection point between the optical axis and the fiber plane. The axis ---→ O f Z f is perpendicular to the optical lamella plane and it is carried by the optical axis and it is directed toward the optical lamella. The axes ---→ O f Y f and ----→ O f X f are perpendicular to each other and they are on the fiber plane. by the photo-detector is also acquired using an acquisition card interfaced with Matlab/Simulink platform. The optical lamella is adjusted using the nano-positioning system in order to search the maximum optical irradiance. As a result, several parameters influencing the interference figure need to control the optical response through controlling the nanopositioning.

Three reference frames are defined, one is assigned to the positioning system, the second is assigned to the optical lamella and the last is assigned to the fiber as shown in Figure 5. The transformation between each two reference frames is achieved by using transformation matrices including the geometric parameters between frames [START_REF] Bettahar | Photorobotic extrinsic parameters calibration of 6-dof robot for high positioning accuracy[END_REF]. The definition of each reference frame is in Table I.

Figure 5 summarizes schematically the whole diagram of the photo-robotic approach. The multi-DOF nano-positioner motion allows to position the optical lamella with respect to the fiber. A 1-D relative distance measurement is achieved using the interferometry figure for each nanopositioner motion. Different optical lamella geometric parameters (β i , θ i , β l , θ l ) can be identified using the 1-D relative distance measurements and the multi-DOF nanopositioner motion. β i is the angle between the plan of the optical lamella and the axis parallel to Y f axis passing by the origin O l , in the (Y f O f Z f ) plan. β l is the angle between the real axis X p and the axis parallel to Y f axis passing by the origin O l in the (Y f O f Z f ) plan (see Figure 4). θ i is the angle between the surface of the optical lamella and the axis parallel to X f axis passing by the origin O l , in the (X f O f Z f ) plan. θ l is the angle between the Y p -axis (real displacement axis) and the axis parallel to X f axis passing through the origin O l , in the (X f O f Z f ) plan. Finally, a photo-robotic model is used to calculate the control angles for the nano-positioner (β p , θ p ) to be applied. The uncertainty distribution of the identified relative misalignment angles β i for each given nano-positioner angle was investigated as well. We obtained rotational measurements with an uncertainty based on standard deviation is 2.1 m • , and the translational measurements with an uncertainty based on standard deviation is 27.6 nm. Fig. 5. The photo-robotic approach experimental implementation.

Fiber

IV. MAXIMIZATION OF OPTICAL IRRADIANCE

In order to understand the irradiance maximization issue, we suppose that, a robotic referenced displacement without relative misalignment angles is applied along Y f or X f axis. The obtained optical irradiance is expected to be constant, but the maximum optical irradiance is not guaranteed. When the optical lamella is translated along the optical axis, toward the fiber without misalignment angles as shown in Figure 6, the corresponding optical irradiance varies increasingly as shown in Figure 7. The closest the optical lamella to the fiber is, the highest oscillation amplitudes (peaks) of the optical irradiance are. If the optical lamella is placed on different FP cavity lengths, from the farthest to the closest (L 1 , L 2 , L 3 ) respectively as shown in Figure 6, the corresponding optical irradiance to the closest FP cavity length L 3 is the lowest compared to those corresponding to the cavity length L 1 , L 2 even if it is the closest to the optical fiber. The optical irradiance which corresponds to the FP cavity length L 2 is higher than the optical irradiance which corresponds to the FP cavity length L 3 even it is farther to the optical fiber than FP cavity length L 3 (see Figure 8). Therefore, the optical irradiance depends on if the induced cavity length corresponds to a maximum peak or not. The correlation between FP cavity length and the interference figure is essential to guarantee the maximum optical irradiance. However, in order to achieve the optimal maximum irradiance, the first closest peak before contact has to be attained. Contact detection is discussed in the next section.

V. CONTACT DETECTION BASED ON FABRY-PEROT INTERFEROMETRY Micro-assembly of optical components requires position control for both, free movements and constrained movements i.e. when the contacts occur. Micro-assembly requires potentially contact detection in constrained space. The correlation between FP cavity length and the interference figure is essential to guarantee the maximum optical irradiance during assembly. In order to achieve the maximum optical irradiance, the first closest peak before contact has to be attained. Therefore, in order to reach maximum optical irradiance, contact detection between the optical lamella and the fiber is necessary, Once the contact is identified, the first maximum peak before contact can be reached.

The contact detection can be achieved by integrating a compliant structure between the lamella and the 6-DOF positioner. In this work we propose to perform contact detection and force estimation based on FP interferometry measure. In order to validate the proposed approach a reference force sensor is integrated to the experimental setup. In this way, the results obtained from force estimation based on FP interferometry measurements are compared to those obtained from the force sensor.

To study the proposed principle, the experimental setup has been modified as schematically shown in Figure 9. The real experimental setup can be seen on Figure 3. Experimental set-up scheme for force estimation and contact detection .

The first step in the validation process is to investigate the repeatability of the force sensor placed at the end-effector of the 6-DOF nano-positioner. For this sake, a forwardbackward motion on the Z-axis is done, where the contact of the optical lamella with the fiber is reached for each forward motion and the displacement continues to estimate the incurred force. Since we are interested in the contact detection, the repeatability of the detected force signal for forward displacement is provided in Figure 10. Translating the optical lamella toward the optical lamella produces simultaneously a periodic sinusoidal irradiance with variable frequency as shown in Figure 11. Before contact detection between the optical lamella and the fiber tip, it represents the case of Figure 13 (a), which produces a signal of the period of 780 nm, which corresponds to half of the used 1560 nm laser wavelength. At z = 6.16 µm, the contact occurs between the optical lamella and the fiber, which represents the case of Figure 13 (b). After contact detection, it represents the case of Figure 13 (c), an applied force by the optical lamella, it conducts to frequency reduction of the measured signal. This frequency reduction is due to the non-flatness of the fiber (fabrication defects), which is not considered in the photo-robotic approach for multi-DOF nanopositioning of the optical lamella with respect to the fiber in section IV.

The difference between this measured signal and the one when no contact exists (red curve named expected irradiance without contact) enables the estimation of the compliant structure compression. Knowing the stiffness of the compliant structure (k = 1.15 mN/µm), it could also be possible to estimate the applied force. The estimated force can be calculated using equation 2.

F e = k(X e -X r ), (2)
The expected displacement X e before contact can be calculated using the same method as presented in section II or it can be taken from the internal sensor of the nanopostioner. After contact, the number of the produced peaks p due to each movement are calculated. Then, the real displacement X r can be calculated using the same formula as presented in section II : X r = p * λ/2.

In order to validate the force estimation after contact based on FP interferometry, it is compared with the force measured by the reference sensor as shown in Figure 12. Results highlight a very good correspondence between both curves. They also show that it is possible to estimate the location of the contact with high precision. These results validate the proposed approach and clearly state the interest for force measurement based on the optical FP interferometry principle.

When the contact occurs, the frequency of the measured irradiance reduces. The difference between this measured irradiance and the one when no contact occurs (red curve named expected irradiance without contact) enables the contact detection as shown in Figure 14. The corresponding flowchart for contact detection using FP interferometry irradiance measurement is shown in Figure 16. By taking the case when there is no contact, the experimental optical irradiance uncertainty of Figure 15 corresponds to a 27.6 nm measurement uncertainty of displacement. When there is a contact the frequency of oscillation of the optical irradiance gets slower. By fixing a periodicity threshold more than the measurement uncertainty, hence, if the optical irradiance periodicity is more than the periodicity threshold, then there is a contact detection. This threshold is fixed to 80 nm.

Yes No

No contact

Expected irradiance with no contact

Experimental irradiance + -

Periodicity error < 𝜺

Contact detection

Fig. 16.

Flowchart for contact detection based on FP interferometry measure.

VI. CONCLUSION

In conventional active alignment, the non-referenced motion of the nano-positioner induces difficulties to analyse the source of this FP cavity variation, which leads to three main drawbacks: time consumption especially for multi-DOF alignment process to reach maximum irradiance, local maximum problem and the alignment task success relies on human expertise.

A "position"-based photo-robotic positioning approach has been used for high accurate multi-DOF positioning of an optical lamella with respect to a fiber in free space. It relies on multi-DOF robot motions associated with 1-D FP interferometry measurements, to estimate poses in a multi-DOF way.

An original contact detection approach has been proposed conducting to an original "force-position"-based photorobotic approach. It relies on correlation between the interference figure and the expected irradiance deduced from the nano-positioner internal sensors. The contact detection enables to estimate the position of the lamella with respect to the fiber along the optical axis which is, for example, usefull to control the UV-adhesive thickness thus achieving high optical performances after completing the assembly process.

Rotational measurements are achieved with an uncertainty based on standard deviation of 2.1 m • , and translational measurements with an uncertainty based on standard deviation of 27.6 nm. Additionally, in constrained space where contact detection is required, the proposed contact detection is based on the correlation between the interference figure and the expected irradiance deduced from the nano-positioner internal sensors. This photo-robotic approach has especially been used to achieve automated multi-DOF assembly of an optical lamella relative to an optical fiber with high accuracy. Experimental results prove the high accurate contact detection with an error of 80 nm as a maximum.

ACKNOWLEDGMENT

These works have been funded by the Franche-Comté region, nano-roboptic project supported by the EUR EIPHI Graduate School (ANR-17-EURE-0002), ANR CEPAGE ANR-16-CE24-0024 and by the French RENATECH network through its FEMTO-ST technological facility.

Fig. 1 .

 1 Fig. 1. The proposed micro-assembly task flowchart.

Fig. 2 .

 2 Fig. 2. Schematic diagram of the case-study optical system (fiber, optical lamella).A translation of the optical lamella along the optical axis produces simultaneously a sinusoidal Reflected irradiance.

Fig. 3 .

 3 Fig. 3. Experimental setup: a 6 DOF nano-positioner used to move a photonic component in front of a fiber.

Fig. 4 .

 4 Fig. 4. Real displacement is along Xp and it is coupled with perpendicularity default angle β i , frames referencing angle β l and δL displacement along the optical axis.

Fig. 6 .Fig. 7 . 4 FP cavity distance L 1 FP cavity distance L 2 FP cavity distance L 3 Fig. 8 .

 6741238 Fig.6. Translation along the optical axis ( Z f -axis) toward the fiber.

Fiber

  Fig. 9.Experimental set-up scheme for force estimation and contact detection .

Fig. 10 .Fig. 11 .Fig. 12 .Fig. 13 .

 10111213 Fig. 10. The corresponding experimental sensed force for 7 forth motion along the Z-axis.

Fig. 14 .

 14 Fig. 14. Contact detection from expected and real FP irradiance periodicity difference.

Fig. 15 .

 15 Fig. 15. Repeatability of the reflected light irradiance during experimentation for 30 translations.

TABLE I THE

 I ASSIGNED REFERENCE FRAMES DEFINITIONS.