Peristalsis

N Gorbushin

L Truskinovsky

Peristalsis by pulses of activity

Peristalsis by actively generated waves of muscle contraction is one of the most fundamental ways of producing motion in living systems. We show that peristalsis can be modeled by a train of rectangular-shaped solitary waves of localized activity propagating through otherwise passive matter. Our analysis is based on the FPU-type discrete model accounting for active stresses and we reveal the existence in this problem of a critical regime which we argue to be physiologically advantageous.

Peristalsis is a series of actively generated wave-like muscle contractions and relaxations which propagate along the body of an organism. Smooth muscle tissues develop such contractions to produce a peristaltic wave in the digestive tract [1,2]. Crawling by peristalsis enables animals like snails, earthworms, slugs, and terrestrial planarians, to advance in narrow spaces [3][4][5][6][7][8], moreover, based on geometrical symmetries only, peristaltic waves were shown to be an optimal motility strategy in such systems [9].

In this Letter we develop a prototypical model of a peristalsis in a segmented limbless organism. We assume that it crawls along a flat surface by extending its forward end and then bringing up its rear end. To achieve this goal the organism generates a solitary wave which travels from the front to the rear. The space-time distribution of activity in such living systems is known to be highly adaptive [10] and the mechanism of this adaptability has recently become a subject of great interest in robotics [11,12].

Peristaltic waves are also of general physical interest as elementary nonlinear excitations of active matter. Propagating active pulses reminiscent of peristaltic waves are ubiquitous in nature from shimmering in honeybees [13] to Mexican waves on stadiums [14]. Comparable phenomena in the form of propagating activity bands are also observed in flocking colonies of swarming robots and other similar systems [15,16]. Some of these behaviors can be quantified using models of excitable media [17] or models involving some kind of globally synchronized CPG (central pattern generator) [18]. However, such models have been questioned in other cases clearly dominated by mechanical sensory feedback and neuromechanical proprioception [19]. Given the distinctly mechanical functionality of peristalsis, we forgo the reaction-diffusion framework [20] and neglect the possible role of CPG, and assume instead that physical forces not only drive the associated localized waves of activity but also secure the signaling pathways controlling, for instance, the necessary internal delays.

As a toy model, capturing only the main effects, we consider a mass-spring chain capable of generating active stresses. It is implied that behind such activity is an endogenous machinery of the type involved in muscle tetanization and we assume that the associated energy flow through the system can be adequately represented by a non-constitutive component of stress. We show that the ensuing, apparently purely mechanical, model can generate directional peristaltic locomotion without relying on externally coordinated actuators or digital controllers. Our intentionally minimalistic approach emulates (and can be extended towards) more comprehensive continuum theories of active media with internally generated active stresses known for both fluids [21,22] and solids [23][24][25][26]. While these models directly account for energy consumption and energy dissipation, our approximate model neglects both.

Passive solitary waves have been long employed in the actuator-driven soft robotics imitating peristalsis [27][28][29]. Instead, here we rely on self-driven active solitary waves and show that peristalsis can be modeled by a train of such waves propagating through otherwise passive matter. Rather remarkably, our analysis reveals the existence of a critical motility regime in such systems where active pulses assume realistic rectangular shape with variable width. This ensures broad repertoire of responses and we argue that so-interpreted criticality may be a characteristic feature of the physiological peristalsis.

Bodies of annelid animals are usually divided into a series of metameres, the segments that are fundamentally similar in muscular structure and functionality [7,30]. To model such organisms we first neglect friction [31][32][33], and represent them schematically as a chain of springs connected in series. The dynamics of such system is described by the FPU equations [32]

ρa 2 ∂ 2 ε n ∂t 2 = σ(ε n+1 ) + σ(ε n-1 ) -2σ(ε n ), (1)
where ε n (t) = (u n+1 (t)u n (t))/a is the strain in a spring whose ends undergo displacements u n (t)and a is the equilibrium length of a spring. The inertial term, allowing the system to overcome the discreteness-induced trapping, proved to be important in ultra-soft robotics [29,30,34,35]. In physiological setting the apparent mass density ρ can be viewed as a parameter introducing an activity-related time delay in the response of stretch receptors [START_REF] Badoual | Proceedings of the National Academy of Sciences[END_REF][START_REF] Serra-Picamal | [END_REF]. We assume that the constitutive relation for the stress σ has two branches: passive and active, see Fig. 1(a). For simplicity, the soft elastic response along the passive branch is considered linear σ = Eε, with elastic modulus E. To describe the active branch (analog of muscle tetanus), we write σ = σ a + Eε, where σ a > 0 is a constant active stress; in more detailed models it can be a variable with sigmoidal response taking a value zero in the passive phase [19]. We further assume that switching from passive to active response takes place when the 'unjamming' threshold strain ε c is reached [START_REF] Serra-Picamal | [END_REF]. We neglect the possibility of hysteresis and assume that unloading below the threshold ε c brings the system back into the passive state, see Fig. 1(a). To nondimensionalize the system (1) we normalize length by the system size L, time by L/c, where c = E/ρ, and stress by E.

σ σa V S 1 S 2 ε ε+ ε- V 2 * * V 2 * 1 1 (a)
It will be convenient to first deal with a continuum approximation of the discrete problem. To this end we introduce the continuous strain field ε(x, t), where ε(nh, t) = ε n (t) and assume that h = a/L 1; we will also use the convenient rescaling ũ = u/ε c , ε = ε/ε c and σ = σ/ε c . If we now Padéapproximate the nonlocal operator in the right hand side of (1) and leave only the lowest order terms, we obtain [38][39][40] 1 -

h 2 12 ∂ 2 ∂x 2 ∂ 2 ε ∂t 2 = ∂ 2 σ ∂x 2 . ( 2
)
To generate a solitary wave solution of (2) we impose a traveling wave anzats ε(x, t) = ε(η) where η = (x -V t)/h and V is the dimensionless velocity of the pulse.

If we center the active pulse performing local extension at η = 0 and denote its width by 2d we can write the associated stress distribution in the form σ(η) = ε(η) + σa rect(η/(2d)), where rect(x) = H(x + 1/2) -H(x -1/2) and H(x) is the Heaviside function. We require that ε(η) → 0 as η → ±∞ and impose at η = ±d the matching conditions ε = dε/dη = 0 and set ε(±d) = 1. Then, integrating (2) twice and applying the boundary/matching conditions, we obtain the equation

V 2 -1 - V 2 12 d 2 dη 2 ε = σa rect η 2d , (3)
where the right hand side implicitly depends on ε. Similar solitary wave solution, describing local contraction , can be obtained if we set σ(η) = ε(η) + σa (1 -rect(η/(2d))) and require that ε → λ when η → ±∞ where λ = σa /(V 2 -1).

The extension pulses exist in the range 1 < V < V * where V * ≡ σa /2 + 1, so they are supersonic which does not mean that they are fast given that the underlying elastic medium is almost an acoustic vacuum [START_REF] Nesterenko | Dynamics of heterogeneous materials[END_REF]. The contraction pulses exist in the range V * < V < V * * where V * * ≡ √ σa + 1 > V * . The phase portrait of the system (3) at 1 < V < V * is shown in Fig. 1(b). Two non-degenerate saddle points at ε± lie on the same Rayleigh line V 2 = (σ(ε)σ(ε + ))/(ε -ε+ ), see Fig. 1(a). Solitary waves describing the extension pulses, correspond to homoclinic trajectories starting and ending at ε+ . Periodic trains of such pulses correspond to closed trajectories encircling the degenerate center at ε = 1. As V → V * homoclinic trajectories become heteroclinic and the solitary waves turn into kinks; at V = V * we have S 1 = S 2 in Fig. 1(a) and therefore the associated macroscopic discontinuity is dissipation free [START_REF] Truskinovskii | [END_REF]. The structure of these solutions is similar to the one in flocking models [15,43] modulo the fact that here we omit the explicit description of inflow and outflow of energy. Note that the contraction pulses, which exist in the complimentary range of parameters V * < V < V * * , correspond to homoclinic trajectories starting and ending at ε-.

In view of the piecewise linear nature of our model, the homoclinic solution of (3) can be written explicitly

ε(η) =            e -(η-d)/z , η > d, λ + (1 -λ) cosh(η/z) cosh(d/z) , -d < η < d, e (η+d)/z , η < -d, (4)
where z = V / 12(V 2 -1) and tanh(d/z) = -(1λ) -1 .

Defining the amplitude of the pulse as A = max ε(η)min ε(η), we find for extension pulses that A = λ + (1λ)/ cosh(d/z). The corresponding explicit solution for contraction pulses is presented in [44]. The functions d(V ) and A(V ) for both types of pulses are shown in Fig. 2(a,b). The two families are separated by the critical value of parameter V = V * where the solitary waves take the form of infinitely separated kinks. At this point the parameter d, playing the role of the correlation length, diverges even though the pulse amplitude remains finite (taking the value A = 2). In the limits V → 1 and V → V * * we obtain sonic waves in passive and active states, respectively; note that the passive limit is singular, see [44] for more details. The typical functions ε(η) for different values of V are shown in the insets in Fig. 2. We emphasize that only the near-critical pulses have a physiologically realistic rectangular form.

Using the relation ε = -V dũ/dη, we can obtain the amplitude of the displacement increment culminating the passing of a pulse, for instance, in the case of a stretching pulse ∆ũ = 2dλ. As a rough description of a peristaltic wave train, represented by a succession of N such pulses, we can write ∆ũ = 2dN λ.

To construct the actual periodic solution we can use the representation σ(η) = ε(η) + σa j rect(η -jD/(2d p )), where it has been assumed that each pulse has a half width d p and the whole active lattice has the period 2D. The matching conditions are now ε = dε/dη = 0 and ε(jD

± d p ) = 1.
While the whole solution can be again written explicitly, see [44], here we only present the transcendental equation for the correlation length d p which is now regularized by the fixed system size D:

(1 -λ) tanh(d p /z) = tanh((d p -D)/z).
The obtained family of solutions is parameterized by V and incorporates both stretching (active) and contraction (passive) pulses. To distinguish between the two it is convenient to redefine the half-width as d = min (d p , Dd p ) and the amplitude as A = |ε(0)ε(D)|. The resulting functions d(V ) and A(V ) are shown in Fig. 3. The obtained periodic solutions can be used to model peristaltic locomotion. Suppose that the organism generates a periodic train of stretching pulses that propagate rearward with a velocity V so that when a pulse reaches the tail, another one is initiated at its head with a fixed delay controlled by the parameter D. Observations show that when such pulse moves towards the tail, the latter fattens and gets anchored due to local increase of friction. With such an anchor present, the locomotion naturally occurs into the direction opposite to the direction of the pulse. We mimic such motility pattern in Fig. 4 (left), where the pulse is taken from the range V < V * .

Suppose that the motion of the animal is of stick-slip-type with each advance corresponding to passing of a single pulse producing the forward displacement of the head ∆ũ = 2d p λ. Since the next pulse arrives after the time 2D/V , the mean translational velocity of the system is v = ∆ũV /(2D). Note that the simultaneous propagation of multiple peristaltic pulses along the animal body is also a possibility which we do not consider here. The branch of contraction pulses corresponding to V > V * also generates a motility pattern shown in Fig. 4 (right), In this case the 'fattening' and the resulting anchoring takes place around the pulse. Such regimes, driven by passive pulses in otherwise active medium, are not realistic due to the necessity to maintain active state throughout the whole body of the organism.

1 2 t 0 1 2 ũ ∆ũ ∆ũ V V 1
The behavior of the function v(V ) for both kinds of motility (V ≶ V * ) is shown in Fig. 4 (center). We emphasize that around the critical point V = V * , the macroscopic velocity v behaves singularly: it varies over a broad range around a single value of the control parameter. The corresponding individual pulses take the form of elongated rectangles which is one of the most characteristic features of peristaltic waves. Since the width of these rectangular pulses, and therefore the resulting motility velocity, can vary significantly, being positioned near such critical point, can help an organism to adapt its responses. The implied anomalous sensitivity to controls can facilitate optimal behavior in complex physiological conditions and would then be highly functional.

Given that the biological systems exhibiting peristalsis are often segmented, the question arises whether our oversimplified (quasi) continuum model (2) adequately represents the dynamics of its discrete prototype (1). To answer this question we now briefly consider the traveling wave solutions of the original discrete system.

We maintain the same normalization and use again the ansatz εn (t) = ε(η), where η = (nh -V t)/h. The discrete strain field ε(η) satisfies the equation

V 2 d 2 ε dη 2 = σ(η + 1) + σ(η -1) -2σ(η), (5)
with σ(η) = ε(η) + σa H(-η). Solitary wave can be built directly from kinks which we therefore consider first. Suppose that a switching point where ε(0) = 1 is placed at η = 0. Application of the Fourier transform to (5) allows one to obtain an explicit solution:

εk (η) = ε+ - σa 2π ∞ -∞ ω 2 (k) (0 + ik)L(k) e -ikη dk, (6)
where L(k) = ω 2 (k) -(kV ) 2 , ω 2 (k) = 4 sin 2 (k/2) and ε+ = ε(∞). Computing the integral in (6) we obtain the representation of this solution in the form of infinite series

εk (η) =          ε+ + σa k∈Z - ω 2 (k) kL (k) e -ikη , η > 0, ε--σa k∈Z + ω 2 (k) kL (k) e -ikη , η > 0, (7)
where Z ± = {k : L(k) = 0, ±Im k > 0} and ε-= ε+ + λ. This solution exists only for the critical value of velocity V = V * , see [44] for details, which suggests that admissible kinks in this model must necessarily satisfy the dynamic Maxwell condition. To construct solutions representing stretching pulses we must solve (5) with the stress given by the same rect function as in the continuum case and use the same matching condition ε(±d) = 1. The linearity of the system at fixed d hints that the solitary wave solution can be obtained as a linear combination of two kinks ε(η) = εk (ηd)εk (η + d). The nonlinear relation d = d(V ) can be then found from the matching condition εk (0)εk (2d) = 1. By inverting the equation (e -ik -1)û(k) = ε(k) in the Fourier space we can also find the displacement field u(η), see [44], and compute the total displacement which is again ∆ũ = 2dλ. The typical dependencies d(V ), and A(V ) for stretching pulses are illustrated in Fig. 5, where they are compared with the corresponding results of the continuum theory with h = 1; the comparison confirms the overall adequacy of the continuum approximation despite the choice of a finite value for our 'small' parameter.

We tested the numerical stability of the obtained discrete pulses by solving a range of initial value problems. An example of a stable propagation of a rectangular stretching pulse is shown in Fig. 6(a) for the chain with 1000 springs. We used initial conditions ũn (0) = 0, u1 (0) = v 0 and un (0) = 0, n > 1 and assumed free ends. The time evolution of the displacement field, shown in Fig. 6(b), illustrates the creation of the main driver of peristaltic motility, the finite displacement behind the propagating pulse.

To conclude, we developed a model of a dynamic passiveto-active transformation taking place in the front of a steadily moving pulse with the corresponding reverse transformation taking place in its rear. The resulting solitary wave solutions for the active medium were extended as periodic trains and used to model the peristaltic mode of self propulsion. We found that at the critical value of parameter the model generates singular regimes with diverging effective correlation length and argued that such criticality may be functional. Our results can be used for biomimetic reproduction of wormlike motion in applications ranging from endoscopic diagnostics [45] to pipeline inspection [START_REF] Yamashita | IEEE International Conference on Robotics and Biomimetics[END_REF]. While in the existing robotic systems activity is usually imitated by globally synchronized distributed actuators [9,20,35] our focus on local mechanical feedback in muscle-type soft deformable materials opens new avenues in the modeling of peristalsis, see also [START_REF] Pehlevan | [END_REF][48][START_REF] Desimone | The Mathematics of Mechanobiology[END_REF]. The proposed prototypical model can serve only as a proof of concept and future work allowing one to make quantitative predictions, should incorporate energy supply and dissipation and account for realistic 3D geometry. The value of the parameter d in the corresponding interval of velocities V * < V ≤ V * * can be found from the equation εk (0) + εk (2d) = 1.

Using the same idea we can find the solution describing the train of stretching pulses. In this case we need to use the ansatz σ(η) = ε(η) + j σa rect(η j /(2d p )). The strain and displacement fields are now represented via infinite sums ε(η) = j εk (η jd)εk (η j + d p ) and ũ(η) = j ũk (η jd)ũk (η j + d p ). The equation for finding d p ( see the main text) is now j εk (-2jD)εk (2d p -2jD) = 1.

(S22)

In the discrete case we can also define the parameter d = min (d p , D pd p ) observing that d p = D/2 at V = V * .

Figure 1 .

 1 Figure 1. (a) Stress-strain relation with passive (blue) and active (red) branches. Kink, connecting ε± at the critical value of velocity V = V * when S2 = S1, is dissipation free. (b) Phase portrait of the continuum system (3) at 1 < V < V * . Stretching pulse corresponds to the homoclinic trajectory starting and ending at ε+; contraction pulse, which exists at V * < V < V * * , is a homoclinic trajectory starting and ending at ε-(not shown).

Figure 2 .

 2 Figure 2. Parametric dependence of the structure of active pulses at σa = 100: (a) the half-width d = d(V ); (b) the amplitude A = A(V ); insets show the strain profiles at: V = 5 (point P ), V = V * ± δ, where δ = 10 -7 (points Q and R) and V = 8 (point S). Vertical asymptotes mark the location of the critical point V = V * .

Figure 3 .

 3 Figure 3. Trains of active pulses at σa = 100: (a) half-width of a single pulse d(V ); (b) its amplitude A(V ). Vertical asymptotes correspond to V = V * ; insets illustrate the typical profiles ε(η).

Figure 4 .

 4 Figure 4. Schematic presentation of peristalsis by trains of active pulses at σa = 100 and D = 3. Motility by stretching pulses is shown on the left, by contraction pulses -on the right. The central plot shows dependence of the average velocity v on V ; vertical asymptote corresponds to the critical point at V = V * .

Figure 5 .

 5 Figure 5. Comparison of the predictions of discrete (red) and continuum (black) models at σa = 100 : (a) half-width, (b) amplitude. The red markers show the discrete pulses obtained numerically by solving a one-parametric set of initial value problems. The inserts compare discrete and continuum distributions of strains (a) and displacements (b) at the near critical speed V = V * -10 -7 .

Figure 6 .

 6 Figure 6. Results of numerical experiments with discrete chain at σa = 100 and v0 = 200 (see the text): (a) strains, (b) displacements. The generated stretching pulse correspond to V ≈ V * , d ≈ 1.88 and A ≈ 1.998. Insets show the comparison of the numerical solution in the co-moving frame (markers) with the corresponding analytical solution (solid line).

Figure 1 :

 1 Figure 1: Stable propagation of a pulse at V = 6 and σa = 100. The inset shows the superimposed analytical solution.

The authors thank G. Mishuris, P. Recho, A. Vainchtein for helpful discussions and acknowledge the support of the French Agence Nationale de la Recherche under the grant ANR-17-CE08-0047-02.

Supplementary material for the paper: "Peristalsis by pulses of activity" N.Gorbushin 1 and L. Truskinovsky The travelling waves in this model solve the differential equation:

where ε+ is the integration constant defined by the condition that ε(η) → ε+ when η → ∞. In the case of stretching pulses, we look for solutions with V > 1. The chosen ansatz for σ(η) distinguishes between a kink, a solitary wave and a train of pulses which are heteroclinic, homoclinic and periodic solutions, respectively. In what follows we adopt the normalization and the notations from the main text. Kinks are characterized by a single transition event at η = 0 and the corresponding equation reads

The integration of this equation gives

where λ = σa /(V 2 -1) and z = V / 12(V 2 -1) and ε-= ε+ + λ. Imposing the matching condition ε(0) = 1 we obtain the closure condition ε± = 1 ∓ λ/2 which is equivalent to the requirement that V = V * and that the two areas in Fig. 1 of the main text are equal: S 1 = S 2 . Integration of the equation εk (η) = dũ k (η)/dη allows one to reconstruct the displacement field

Solitary wave solutions satisfy the condition ε± = 0. The governing equation takes the form

The active stress σa is now applied on the finite interval 2d and we should integrate the above equation under the assumption that ε(±d) = 1.

Solution for this problem exist in the interval 1 < V ≤ V * . The strain and displacement fields can be written down explicitly

To obtain the field ũ(η) we again set the trivial integration constant to 0 by assuming that ũ(η) → 0 when η → ∞. To close the system we must impose the continuity of the derivative dε/dη at η = ±d. We can then compute the function and obtain other important relations:

The model describes not only the stretching but also the contraction pulses. To obtain the latter we need to solve the equation

The boundary conditions must be now chosen in the form ε → λ when η → ±∞.

Solution for this problem exist in the interval V * < V ≤ V * * . It can be again written explicitly

The parameters here are

This solution degenerates in two limiting cases. First, at V = 1 the integration of (S5) gives

Imposing the continuity of derivatives at η = ±d we obtain d = 0 and A = 0 and, therefore, ε(η) ≡ 1 and σ(η) ≡ 1. In the other singular limit V = V * * we have λ = 1 and therefore from (S10) we find that ε = 1 for |η| > d. The continuity of strain derivatives at η = ±d again gives d = 0 and A = 0, hence ε(η) ≡ 1 but now σ(η) ≡ 1 + σa .

To obtain the trains of active pulses we need to solve the equation

where η j = η -2jD. The integration of this equation gives with the matching conditions mentioned in the main text gives

e dp/z 1 + e 2(dp-D)/z e ηj /z + e -(ηj +2D)/z , -D < η j < -d p .

(S13)

The value of the parameter d p can be found as a positive real root of the transcendental equation

which is equivalent to the requirement of continuity for the first derivative of strain at η j = ±d p . By integrating (S13), while respecting the continuity of displacements, we obtain

(j -1)∆ũ + ze dp/z 1 + e 2(dp-D)/z e ηj /ze -(ηj +2D)/z , -D < η j < -d p .

(S15)

DISCRETE PROBLEM

The discrete problem reduces to a solution of the advance-delay differential equation

In view of the partial linearity of the problem, it will be enough to construct the kink-type solution while individual pulses and trains of pulses can be obtained as linear combinations of kinks with appropriately adjusted continuity conditions. As above, the kink solution can be obtained if use in (S16) the ansatz σ(η) = ε(η) + σa H(-η). We can then apply the Fourier transform εk (k) = ∞ -∞ εk (η)e ikη dk and rewrite the problem as an algebraic one

where the kernel is

), is the dispersion relation and (0+ik) -1 = lim α→0+ (α+ik) -1 stands for the Fourier transform of the Heaviside function .

The solution of the original problem can be presented in the form of the inverse Fourier transform

where the contour integrals can be computed by the residue method. The kernel function L(k) has a double zero at k = 0. The rest of the roots are simple and complex, located in both half-planes. They can be organized in the sets

Then the explicit series solution of the discrete problem can be written in the form:

Here ε+ is a homogeneous solution of the problem with the boundary conditions ε(η) → ε± at η → ±∞ and where ε-= ε+ +λ.

To apply the matching condition at η = 0 we need to consider an infinitely large circle in the complex plane. The contour integration in this case gives the relation

If we also recall that L(-k) = L(k) and L(k) = L(k) we obtain k∈Z -ω 2 (k)/(kL (k)) = k∈Z + ω 2 (k)/(kL (k)) and, hence, ε(0) = ε+λ/2 = ε-+ λ/2. By applying the matching condition ε± = 1 ∓ λ we obtain again the equal area construction S 1 = S 2 , illustrated in Fig. 1(a) of the main text. The obtained condition means that the entropy production on the corresponding jump discontinuity in the coarse grained continuum problem would be equal to zero. We can also construct the discrete displacement by inverting the relation (e -ik -1)û k (k) = εk (k). Following the same scheme as above we obtain

e -ik(η-1/2) , η < 1/2.


Maryne Duval

Charlie Blons

Sonia Mallet-Ladeira

Damien Delcroix

Lionel Magna

Hélène Olivier-Bourbigou

E Daiann

Sosa Carrizo

Karinne Miqueu
email: kmiqueu@univ-pau.fr

Abderrahmane Amgoune

György Szalóki

Didier Bourissou
email: dbouriss@chimie.ups-tlse.fr

Cu-Catalyzed P-C bond formation/cleavage: straightforward synthesis/ring-expansion of strained cyclic phosphoniums

HAL is

Introduction

Reductive elimination plays a pivotal role in organometallic chemistry. It intervenes in most catalytic cycles of transition-metal mediated cross-couplings as the key product-releasing step. Reductive coupling between L-type ligands (such as phosphines, thioethers, pyridines, Nheterocyclic carbenes…) and X-type groups (such as aryls, alkyls, halogens…) is also well-known (Figure 1). It has recently attracted an upsurge of interest, in particular the coupling of phosphines and aryl groups. [1][2][3][4] It is not only an efficient way to prepare aryl phosphonium salts (Figure 1, path a) 5 but also an undesirable deactivation pathway for catalytic transformations involving phosphine-ligated aryl complexes. Aryl-aryl exchange between transition metals and phosphine ligands is also possible and this M-C/P-C bond metathesis process (Figure 1, path b) has been recently exploited to achieve catalytic functional group exchange between aroyl chlorides and aryl iodides. 6 In all this chemistry, the group 10 metals, Pd and Ni especially, occupy the forefront position due to their easy cycling between the M(0) and M(II) oxidation states. Comparatively, much less is known in the field with the group 11 metals that are inherently reluctant to undergo oxidative addition and therefore to achieve 2-electron redox cycles. Examples of phosphine-aryl coupling at gold have become more numerous but are still rare. [START_REF] He | For C-P coupling reactions catalyzed by gold[END_REF] The formation of aryl phosphoniums has been authenticated as the favourite decomposition pathway of tricoordinate [(R3P)Au(Ar)X] + complexes arising from oxidative addition of aryl halides to gold. [START_REF] Robinson | For an early study in the gas phase[END_REF][START_REF] Kawai | For the fomation of tricoordinate [(R3P)Au(Ar)X] + complexes by halide abstraction using silver salts[END_REF][START_REF] Bender | For aryl-phosphine coupling at copper promoted by supramolecular cages[END_REF][START_REF] Shu | For the detection of aryl-phosphoniums as side-products in gold-catalyzed transformations[END_REF] It has also been observed upon C-H activation of arenes at phosphine-ligated Au(III) complexes [START_REF] Kawai | For the fomation of tricoordinate [(R3P)Au(Ar)X] + complexes by halide abstraction using silver salts[END_REF] and by reacting (N,C)-cyclometallated Au(III) complexes with phosphines. 12 The coupling of aryl groups and L ligands at copper is even scarcer. 13,[START_REF] Beletskaya | For reviews dealing with Cu-catalyzed C-P coupling reactions[END_REF] With phosphines, the only example we are aware of involves a bipyridine-functionalized bromo-arene (Figure 2a). Allen et al. tested other templated substrates, but this was the only one to undergo P-C coupling. 15 A related coupling involving NHC instead of phosphine complexes was recently reported by Fairlamb, Ariarfard and co-workers. 16 Quantitative formation of a 2-arylated benzimidazolium salt was observed upon reaction with PhI (Figure 2b). Based on DFT calculations, the reaction was proposed to involve a Cu(I)/Cu(III) cycle (C-I oxidative addition, NHC-C reductive elimination). As pointed out by Huang et al. in their recent review, 4 all the aryl-phosphine couplings described so far with the coinage metals require a stoichiometric amount of metal complex. 17 In the course of our investigations of coinage metal complexes deriving from the peri-halo naphthyl phosphine such as 1-I, 18,19 we recently discovered an unexpected intramolecular P-C coupling leading to the peri-bridged naphthyl phosphonium salt 2a (Scheme 1). 20 In this transformation, copper behaves very differently from gold which forms a (P,C)-cyclometallated complex. 18 Scheme 1. Reaction of 1-I with CuI: intramolecular phosphine-aryl coupling promoted by copper leading to a peri-bridged naphthyl-phosphonium 2a.

Given the interest but very few P-peri-bridged naphthalenes [21][22][START_REF] Owsianik | A peri-bridged naphthyl phosphonium has been proposed as a transient intermediate in the reaction of 1,8-bis(diphenylphosphino)naphthalene with 1,8-bis(bromomethyl)-naphthalene[END_REF] and the paucity of phosphine-aryl coupling at copper, we embarked in a comprehensive study of this process. This work is reported hereafter. Our aim was in particular to address the following questions: (i) is this transformation general? (ii) is it possible to make it catalytic? and (iii) what is the reactivity of the strained four-membered ring in such phosphoniums? The experimental study has been complemented by a thorough theoretical investigation to gain better understanding of the reaction mechanism and identify the underlying factors.

Results and Discussion

Variation of the substituents at phosphorus

The generality of the reaction was first examined by replacing the two i Pr groups at phosphorus by the diamino moiety ( i Pr)NCH2CH2N( i Pr) that we have previously used in the context of o-carboranyl diphosphine ligands. 24 Upon reaction with CuI, the peri-iodo naphthylphosphine 3 readily formed the corresponding spiro-phosphonium 4, which was characterized by multi-nuclear NMR spectroscopy (Scheme 2). Most characteristic is the symmetrisation of the naphthyl moiety, as apparent in the 1 H and 13 C NMR spectra (Figures S1 andS2). 25 Scheme 2. Reaction of the diamino-substituted peri-iodo naphthyl phosphine 3 with CuI.

Then, we turned our attention to the phenyl-substituted phosphine 5 18 and observed a quite different behaviour (Scheme 3). Under similar conditions (CuI, dichloromethane, 25°C), a white precipitate formed rapidly. The corresponding 31 P NMR signature (broad signal at -12.0 ppm) suggested in this case coordination of the phosphorus atom to copper, without formation of the phosphonium. 26 To shed light into the structure of the obtained product, an X-ray diffraction (XRD) analysis was carried out. It revealed, that 6a is Cu(I) complex with two peri-iodo naphthyl phosphines coordinated to CuI. The copper atom sits in trigonal bipyramidal geometry [P-Cu-P = 130.58(6)°; sum of P-Cu-P and P-Cu-I bond angles = 359.66(5)° and is engaged in weak Cu•••I contacts with the peri-iodine atoms located along the apical axis [3.037(9) and 3.319(10) Å, vs 2.575(8) Å for the equatorial CuI bond; I•••Cu•••I = 167.49(4)° (Table S3)]. [START_REF] Kaddouri | For structurally characterized bis(phosphine)CuX complexes, see for examples[END_REF] Of note, the peri-I•••Cu contacts induce significantly distortions of the naphthyl backbones, as apparent from the I-C•••C-P torsion angles (-20.0(5) and 35.8(5)°). Scheme 3. Reaction of the diphenyl-substituted peri-iodo naphthyl phosphine 5 with CuI and crystallographic structure of the ensuing bis-phosphine CuI complex 6a.

To further analyze the weak Cu•••I contacts observed crystallographically, DFT calculations were carried out on the actual complex 6a using the B3PW91/SDD+f(Cu), SDD(I), 6-31G** level of theory (gas phase). 25 The optimized geometry nicely matches that determined experimentally by XRD (Table S6). In particular, the distances between Cu and the peri-I atoms are well reproduced (3.178 and 3.267 Å). In addition, NBO analysis confirms the presence of secondary I•••Cu interactions, found as weak I(lone pair)→Cu(vacant 4s orbital) donor-acceptor interactions (Table S7) at the secondorder perturbation theory analysis (the corresponding delocalization energies E(2) amount to 12.7 and 10.0 kcal/mol, significantly less than that of the equatorial CuI bond, 77.3 kcal/mol).

Heating complex 6a showed no evolution over time, even under forcing conditions (110°C for 2 days in toluene). Addition of AgSbF6 was then tested to abstract iodide from copper and eventually favour oxidative addition of the C-I bond, as previously observed for the intramolecular activation of a Sn-Sn bond in diphosphine copper complexes. 28 Gratifyingly, the cyclic phosphonium 7a rapidly and cleanly formed under these conditions. The thereby obtained SbF6 salt was characterized by 1 H, 13 C, 31 P NMR spectroscopy, high-resolution mass spectrometry (in both positive and negative modes) as well as elemental analysis. 25 Notably, switching from CuI to Cu(MeCN)4OTf as copper precursor circumvents the use of silver salt and offers a valuable alternative method to prepare phosphonium 7 (Scheme 4). Under these conditions, the diphosphine complex 6b ( 31 P NMR: -7.5 ppm) forms instantaneously and then gradually evolves to the cyclic phosphonium 7b within 4 hours at room temperature. Scheme 4. Reaction of the diphenyl-substituted peri-iodo naphthyl phosphine 5 with Cu(MeCN)4OTf.

From stoichiometric to catalytic transformations

The reaction conditions to access the peri-bridged naphthyl-phosphonium were then optimized using the isopropyl-substituted substrate 1-I. Whatever the copper source employed, CuBr(SMe2), Cu(MeCN)4PF6, Cu(MeCN)4OTf or (Py)CuI, the formation of the phosphonium was instantaneous and quantitative at room temperature according to 31 P NMR spectroscopy. Thus, neither the charge of the Cu precursor nor the ancillary ligands have a significant impact on the reaction. We then sought to decrease the loading in Cu source with the hope to develop a catalytic process. As mentioned above, all known examples of phosphine-aryl coupling at copper are stoichiometric. Pleasingly, complete and clean conversion of 1-I into 2b was observed using 20 mol% for (Py)CuI. The loading can even be decreased to 1 mol% with CuI and Cu(MeCN)4OTf. To further optimize the transformation, different solvents were tested (Table 1). In general, polar solvents were preferred, due to solubility issues. Among those tested (DCM, MeCN, THF, DMF), the best results in terms of reaction rate, cleanness and isolation were obtained with THF and MeCN. In the end, two sets of conditions were established (entries 3 and 4). When using 1 mol% of Cu(MeCN)4OTf in THF, the reaction reached full conversion after 30 minutes at room temperature. The phosphonium 2b precipitated in the course of its formation and was readily isolated (89% yield) by simple filtration. On the other hand, the phosphonium 2b was obtained in only 5 minutes at rt by treating 1-I with 1 mol% CuI in MeCN! Removal of MeCN and washing with THF resulted in a 91% isolated yield. Note that in both cases the phosphonium is obtained as its iodide salt 2b containing ca 1 mol% of CuI2 - counter-anion (the copper content, as determined by ICP-MS, corresponds to ca 1 mol%, in line with the introduced amount of catalyst). 25 Table 1. Screening of the reaction conditions. Catalyst loading is 1 mol% for all entries. [a] Isolated yields in parentheses.

Generalization to peri-bromo and chloro-naphthyl phosphines

The possibility to extend this transformation of peri-iodo naphthyl-phosphines to the other halogens was then investigated. Different copper precursors and solvents were tested, the best results are disclosed in Scheme 5. Under stoichiometric conditions, the peri-bromo phosphine 1-Br reacted a little slower than 1-I (reaction is complete in 30 min instead of < 5 min in DCM at rt). 25 The transformation could nevertheless be efficiently achieved with catalytic amounts of copper. A protocol using 5 mol% of CuBr(SMe2) and heating for 30 minutes at 50°C in MeCN was developed. The corresponding phosphonium salt 2c was thereby isolated in 82% yield after work-up. The perichloro phosphine 1-Cl was much more resilient to react. Heating for 15 days in refluxing DCM with 1 equiv. of Cu(MeCN)4OTf was required. In this case, CuCl gradually precipitated and the phosphonium was obtained as the trifluoromethanesulfonate salt 2d (as deduced from ESI in the negative mode) in 96% spectroscopic yield. Screening different reaction conditions by varying the copper precursor (CuCl), the solvent (MeCN, THF, o-DCB) and the temperature (80°C) did not lead to significant improvement.

Scheme 5.

Optimized conditions for the Cu-mediated transformation of peri-halogenated naphthyl phosphines 1-X into the corresponding peri-bridged naphthyl-phosphoniums 2b-d. [a] Yield based on 31 P NMR spectroscopy.

These results show that copper can activate not only the C-I bond but also C-Br and even the less reactive C-Cl bond of naphthyl-phosphines 1-X. Catalytic transformations using low copper loadings were developed with 1-I and 1-Br, but with 1-Cl, the reaction could not be driven to completion under catalytic conditions.

So far, only two peri-bridged naphthyl-phosphoniums have been reported. 21,22 They were prepared by a completely different route. Peri-bridged naphthyl-phosphines were first synthetized by intramolecular nucleophilic substitution at phosphorus. Then the phosphorus atom was quaternarized with methylating reagents. The copper-catalyzed route, as developed in this work, provides efficient and general access to such compounds. It gives the possibility to explore their reactivity, and our first investigations in this direction are described hereafter.

Ring-expansion of peri-bridged naphthyl-phosphoniums

Peri-bridging imparts significant ring strain (as apparent from the acute bond angles CPC = 77° and PCC = 88°) and phosphoniums such as 2 were thus supposed to be prone to ring-opening reactions.

Our interest for gold complexes and unusual reactivity at gold 29 prompted us to explore the behaviour of phosphoniums 2 towards Au(I) precursors. We were particularly intrigued by the possibility to insert gold into the P-C bond (via oxidative addition or any other pathway). Such a ring-expansion would lead to (P,C)-cyclometallated gold(III) species, complexes we have shown recently to display versatile reactivity 19,30 and to be competent catalysts for the hydroarylation of alkynes. 31 Pleasingly, the phosphonium salt 2b was found to react with gold(I) iodide to give the known (P,C)AuI2 complex 8 (Scheme 6). 19a The reaction is significantly slower (80% spectroscopic yield after 9 d at 50°C) than when starting from the peri-iodo naphthyl phosphine 1-I (100% in 2 h at 50°C). However, with phosphonium 2b, it does not benefit from P-coordination (and chelation assistance), and actually represents a rare example of intermolecular oxidative addition to gold. Performing the reaction in ortho-dichlorobenzene (o-DCB) at 100°C enabled to speed up the reaction (4 d, 82% isolated yield). Then, the phosphonium salt 2e deriving from the peri-bromo naphthyl phosphine 1-Br was reacted with AuBr(SMe2). NMR monitoring indicated complete consumption of the phosphonium 2e after 1 h at 80°C with formation of a distinct intermediate displaying a 31 P NMR resonance signal at 53.5 ppm (Scheme 7). The latter species was unambiguously assigned as the peri-bromo naphthyl phosphine gold(I) complex 9 that was also independently synthesized and fully characterized. 18,25 From 9, the (P,C)-cyclometallated gold(III) complex 10 formed gradually (complete conversion after 6 days at 80°C) by intramolecular oxidative addition of the C-Br bond to gold. From a mechanistic standpoint, the formation of (P,C) cyclometallated gold(III) complexes upon reaction of the peri-bridged phosphonium with gold(I) iodide/bromide may be explained by the following pathways (Scheme 8): (i) direct oxidative addition of the strained P-C bond to gold, (ii) backward-formation of the peri-halo naphthyl phosphine (by C-P oxidative addition to copper followed by C-X reductive elimination), copper to gold exchange and oxidative addition of the C-X bond to gold. Of course, these two pathways may operate competitively or concomitantly.

Scheme 8. Schematic representation of the two possible pathways proposed to account for the formation of the cyclometallated gold(III) complexes (P,C)AuX2 (X = I, Br).

We surmise that the direct oxidative route (i) prevails for the iodide salt 2b which contains only trace amounts of copper (as CuI2 -counter-anion). No intermediate is detected during the course of the reaction in this case. As mentioned above, intermolecular oxidative addition to gold is rare, but not unprecedented. In particular, a few examples have been reported with strained C-C bonds as those of biphenylene and benzocyclobutanone. 32,33 The reaction of Pt(PPh3)4 with a W(CO)5-coordinated peri-bridged naphthyl phosphine, as described by Mizuta and Miyoshi, also deserves to be mentioned here. 21a,b Oxidative addition of the P-C bond to Pt induced ring-expansion and afforded original heterodinuclear complexes.

In the case of the CuBr2 -salt 2e, the reaction seems to operate by the multi-step route (ii) instead. This is apparent from the characterization of the phosphine Au(I) complex 9. Of note, the backformation of the peri-bromo phosphine from 2e demonstrates the reversibility of the C-Br oxidative addition / P-C reductive elimination sequence at copper.

The experimental work was paralleled with a thorough computational study, with the aim (i) to shed light into the mechanism of formation of the peri-bridged phosphonium mediated by copper and identify the factors influencing it (solvent, halogen), (ii) to rationalize the difference observed between copper and gold, and (iii) to gain more insight into the P-C bond cleavage and ringexpansion upon reaction with gold(I) halides.

Mechanism of formation of the peri-bridged naphthyl phosphonium mediated by copper

Previous DFT calculations performed in the gas phase supported the 2e redox sequence made of P coordination, C-I oxidative addition and P-C reductive elimination as a viable process for the reaction of the peri-iodo naphthyl phosphine 1-I with CuI. To describe and analyze more reliably the experimental observations, solvent effects were taken into account, considering both DCM ( = 8.93) and MeCN ( = 35.69) using the universal Solvation Model based on solute electron Density (SMD). The corresponding energy profiles are given in Figure 3 (see also Figures S18 andS19). 25 Once the phosphine is coordinated to copper, oxidative addition of the C-I bond proceeds via a 3centered transition state TS1 (Figure 4a). This step is very mildly affected by the reaction conditions. In all cases, it is about thermoneutral (2.7, 1.9 and -1.4 kcal/mol in the gas phase, DCM and MeCN) and its activation barrier is low (9.3, 9.5 and 7.8 kcal/mol, respectively). More pronounced differences were found on the P-C coupling step following the (P,C)Cu(III)I2 intermediate II, in line with the high polarity of the corresponding transition state TS2 (Figure 4b) and resulting product.

From II, the formation of the phosphonium salt 2 is only slightly downhill in energy in the gas phase (-1.9 kcal/mol), but well favoured thermodynamically in DCM ( G -14.2 kcal/mol) and MeCN ( G -13.7 kcal/mol). The overall transformation is thermoneutral in the gas phase (0.8 kcal/mol) but noticeably exergonic in DCM (-12.3 kcal/mol) and MeCN (-15.1 kcal/mol). Of note, the energy barrier for the reductive elimination at copper is significantly reduced with polar solvents: from 18.5 kcal/mol in the gas phase, it decreases to 7.6 kcal/mol in DCM, and to 6.3 kcal/mol in MeCN. Overall, the energy span for this Cu(I)/Cu(III) sequence is remarkably small. The activation barriers for the C-I oxidative addition and P-C reductive elimination are both under 10 kcal/mol, in line with the very fast reaction observed experimentally between 1-I and CuI at room temperature even under catalytic conditions. It is noteworthy that the cleavage of the C-I bond is slightly more demanding than the formation of the P-C bond (by 1.5 to 1.9 kcal/mol), making the copper(III) complex II a fleeting intermediate. [START_REF] Maeda | Stable organocopper(III) species have been prepared using macrocyclic N-containing ligands: (a) H. Furuta[END_REF] This is consistent with the fact that no other complex than the starting phosphine Cu(I) species was observed experimentally.

Comparison of the peri-iodo, bromo and chloro naphthyl phosphines 1-X

The influence of the halogen atom in the peri-position of the naphthyl-phosphine was then investigated. The energy profiles for the sequence of P-coordination, C-X oxidative addition to Cu and P-C reductive elimination were computed at the SMD(MeCN)-B3PW91/SDD+f(Cu), SDD(I,Br,Cl),6-31G**(other atoms) level of theory for the reactions 1-I + CuI, 1-Br + CuBr and 1-Cl + CuCl in MeCN. The barrier for the oxidative addition to Cu increases progressively in the series C-I < C-Br < C-Cl (7.8, 13.7 and 18.5 kcal/mol) (Figure 5, see also Figures S20-S22) 25 and this step is less favoured thermodynamically for chlorine and bromine than for iodine (-1.4 for I versus 3.1 for Br and 1.8 kcal/mol for Cl). The halogen is not directly involved in the following P-C bond formation and has logically less impact on this step. In all cases, the reductive elimination is downhill in energy, by 13.7 kcal/mol for I, 16.2 kcal/mol for Br and 10.3 kcal/mol for Cl, over the (P,C)-cyclometallated Cu(III) complex II. The corresponding energy barrier increases from iodine (6.3 kcal/mol) to bromine (9.5 kcal/mol), and chlorine (12.8 kcal/mol). Overall, the formation of the phosphonium salts from 1-X + CuX is thermodynamically favoured and the energy balance does not change much with the halogen ( G = -15.1 kcal/mol for I, -13.1 kcal/mol for Br and -8.5 kcal/mol for Cl). [START_REF]The differences in C-X bond strengths are roughly compensated by those in Cu-X bond strengths[END_REF] While the activation barriers for the C-X oxidative addition and P-C reductive elimination are close for iodine, the first step is significantly more demanding for bromine and chlorine (by 4.2 and 5.7 kcal/mol, respectively), making it the likely rate-determining step.

Comparison of copper and gold: formation and ring-expansion of the peri-bridged naphthyl phosphonium

Finally, we sought to better understand the very different behaviour exhibited by copper and gold, ie the formation of peri-bridged naphthyl phosphoniums with Cu versus (P,C)-cyclometallated gold(III) complexes with Au. Computations were carried out in DCM to enable comparison between the two metals under similar conditions used experimentally. The reaction profiles computed for the reactions of 1-I with AuI and CuI are overlayed in Figure 7, revealing striking differences.

Consistent with experimental observations, 18,[START_REF] Toste | assessed the possibility for (P,C)-cyclometallated Au(III) complexes to undergo P-C reductive elimination, but no peri-bridged naphthyl phosphonium was detected[END_REF] the reaction stops at the M(III)I2 stage with gold: the P-C reductive elimination leading to the phosphonium is uphill in energy (+ 7.8 kcal/mol) and its activation barrier is large (38.1 kcal/mol). Conversely, starting from the AuI2 -phosphonium 2, oxidative addition of the strained P-C bond to give the (P,C)AuI2 complex II is downhill in energy. The corresponding barrier (30.3 kcal/mol) is significantly larger than that of the P-assisted C-I oxidative addition from I (16.6 kcal/mol), but can be crossed over at high temperature, in line with that observed experimentally. The formation of the M(III)I2 species II is less favoured thermodynamically with copper than gold, consistent with their intrinsic properties and relativistic effects. [START_REF] Nakanishi | [END_REF] Due to the smaller size of Cu compared to Au, the (P,C)-cyclometallated structure imposes some geometric constraint, but not too much (Figure 8). Of note, the barrier for C-I oxidative addition is lower for copper than for gold (9.5 versus 16.6 kcal/mol). This has probably to do with the structure of the phosphine M(I) precursor I, which is the standard situation for Au, but a low coordinate state for copper. The same trend is observed from 1-Br (Figure 9). As mentioned above, the activation barriers for C-Br oxidative addition to copper and P-C reductive elimination are both readily accessible (13.9 and 11.0 kcal/mol, respectively), and the formation of the peri-bridged phosphonium 2 is thermodynamically favoured over the (P,C)-cyclometallated Cu(III) complex II ( G : -12.5 kcal/mol). Comparatively, the reaction with AuBr stops at the oxidative addition stage, the (P,C)Au(III)Br2 complex is in a thermodynamic sink. It is 11.5 kcal/mol lower in energy than the phosphine Au(I)Br complex I and 9.4 kcal/mol lower in energy than the peri-bridged phosphonium 2. The activation barrier for P-C reductive elimination is very large (44.9 kcal/mol). Even that for the backward reaction, the oxidative addition of the strained P-Csp2 bond to gold (35.5 kcal/mol) is very high. This is probably why the reaction of 2 with AuBr goes through the phosphine complex (1-Br)Au(I)Br and is mediated by copper. Insertion of CuBr2 -in the strained cyclic phosphonium is much less energy demanding than that of AuBr2 -( G ≠ 23.5 kcal/mol for the P-C oxidative addition to Cu). From II, C-Br reductive elimination is also facile at Cu ( G ≠ 10.5 kcal/mol) and reforms the peri-bromo naphthyl phosphine.

The energy landscapes computed for Cu and Au are very different. The formation of the (P,C)cyclometallated gold(III) complex is a strong driving force while the small energy span of the 2e redox sequence at copper explains its ease and reversibility.

Conclusion

In summary, the reaction of peri-halo naphthyl phosphines with Cu(I) was found not to stop after Pcoordination and oxidative addition of the C-X bond as for gold. P-C reductive elimination readily proceeds to give strained peri-bridged naphthyl phosphonium salts. The reaction is general, as shown by the formation of the diamino-substituted, spiro-bicyclic phosphonium 4 and diphenylsubstituted phosphonium 7 (for which a bis-phosphine copper(I) complex featuring weak contacts between copper and the peri-iodine atoms was authenticated crystallographically). Not only naphthyl phosphines featuring C-I bonds in peri position react with copper(I) salts to give phosphoniums, but also the bromo and chloro derivatives 1-Br and 1-Cl although more forcing conditions are required. Remarkably, the reaction is amenable to catalysis and actually stands as the first Cu-catalyzed phosphine-aryl coupling. With the peri-iodo naphthyl phosphine 1-I, it proceeds very fast under mild conditions (5 minutes at room temperature) with only 1 mol% of copper source.

The reaction most likely involves a Cu(I)/Cu(III) cycle made of P-coordination, C-X oxidative addition and P-C reductive elimination, as supported by DFT calculations. In line with experimental observations, polar solvents such as DCM and MeCN are preferred. They make the formation of the phosphonium salt more thermodynamically favourable and lower the activation barrier for the P-C reductive elimination step. The energy span for the Cu(I)/Cu(III) sequence is remarkably small, with activation barriers for C-I oxidative addition and P-C reductive elimination both under 10 kcal/mol. The influence of the halogen atom (I, Br, Cl) was also examined computationally. Accordingly, C-X oxidative addition is more demanding with bromine and chlorine, and represents the ratedetermining step. The reaction profiles computed for gold are very different, explaining the striking difference observed between the two metals experimentally. The (P,C)-cyclometallated Au(III)X2 complexes are in thermodynamic sinks. With gold, P-C reduction elimination is uphill in energy and its activation barrier is very high.

Because of the strain induced by the four-membered ring structure and the rigid naphthyl backbone, peri-bridged naphthyl phosphonium salts can be expected to be quite reactive. First studies in this direction have shown the possibility of ring-expansion by reaction of gold(I) precursors, leading to (P,C)-cyclometallated gold(III) complexes. Mechanistic studies have evidenced two contributing scenarios in which the insertion of the gold atom into the strained P-C bond proceeds either directly by oxidative addition to gold or involves the reversibility of the Cu(I)/Cu(III) sequence and ultimately oxidative addition of the C-X bond to gold assisted by P-chelation.

Peri-bridged naphthyl phosphonium salts represent a new type of -conjugated phosphoniums, besides phospholium, benzophospholium and phosphaphenalium salts. 39 Future work will aim at exploring further their chemistry and studying related peri-P-bridged naphthyl compounds. Besides their reactivity, their photophysical properties certainly deserve to be investigated. Indeed, the incorporation of main-group elements in -extended systems has been established as a very powerful strategy to alter their electronic and optical properties. 40 are acknowledged for computational facilities. E.D.S.C. thanks CDAPP for funding part of his postdoctoral contract.

Figure 1 .

 1 Figure 1. Top: Schematic representation of the reductive coupling between L and X-type ligands at transition metals. Bottom: Phosphine-aryl coupling promoted by transition metals (a) and aryl/aryl exchange between phosphines and transition metals (b).

Figure 2 .

 2 Figure 2. Copper-mediated couplings between L-type ligands and aryl groups.

Scheme 6 .

 6 Scheme 6. Reactions of the peri-bridged naphthyl phosphonium 2b with AuI (isolated yield in parentheses) and alternative synthesis of the ensuing (P,C)AuI2 complex 8 from 1-I. Below: graph showing the rates of conversion of 2b/1-I as determined by 31 P NMR spectroscopy.

1 Scheme 7 .

 17 Scheme 7. Reaction of the peri-bridged phosphonium 2e with AuBr(SMe2) leading to the (P,C)cyclometallated dibromo gold(III) complex 10. Below: 31 P NMR spectroscopic monitoring.

Figure 3 .

 3 Figure 3. Energy profiles computed in the gas phase and in DCM/MeCN solutions for the C-I oxidative addition / P-C reductive elimination sequence. Free energies in kcal/mol computed at

Figure 4 .

 4 Figure 4. Transition states TS1 and TS2 computed respectively for the C-I oxidative addition (a) and P-C reductive elimination (b) at the SMD(DCM)-B3PW91/SDD+f(Cu),SDD(I),6-31G**(other atoms) level of theory. Key distances in Å.

Figure 5 .

 5 Figure 5. Energy profiles computed for the C-X oxidative addition (X : I, Br, Cl) / P-C reductive elimination sequence at the SMD(CH3CN)-B3PW91/SDD+f(M),SDD(I,Br,Cl), 6-31G**(other atoms) level of theory. Free energies in kcal/mol.

Figure 6 .

 6 Figure 6. Transition states (TS1) computed for the C-I, C-Br and C-Cl oxidative additions to copper at the SMD(MeCN)-B3PW91/SDD+f(Cu),SDD(I,Br,Cl),6-31G**(other atoms) level of theory. Key distances in Å.

Figure 7 .

 7 Figure 7. Overlay of the energy profiles computed for the reactions of the peri-iodo naphthyl phosphine 1-I with CuI and AuI at the SMD(DCM) B3PW91/SDD+f(M),SDD(I),6-31G**(other atoms) level of theory. Free energies in kcal/mol.

38

 38

Figure 8 .

 8 Figure 8. Structures computed for the (P,C)-cyclometallated M(III) complexes II (M = Cu, Au) at the SMD(DCM)-B3PW91/SDD+f(M),SDD(I),6-31G**(other atoms) level of theory. Key distances in Å.

Figure 9 .

 9 Figure 9. Overlay of the energy profiles computed for the reaction of the peri-bromo naphthyl phosphine 1-Br with CuBr and AuBr at the SMD(DCM)-B3PW91/SDD+f(M),SDD(Br),6-31G**(other atoms) level of theory. Free energies in kcal/mol.

  . at rt with 1 mol% Cu)  2e Cu(I)/Cu(III) redox sequence (P-coordination, C-X oxidative addition, C-P reductive elimination)  ring-expansion by insertion of Au(I)

Table of Contents

 of

Acknowledgments

This work was supported financially by the Centre National de la Recherche Scientifique, the Université Toulouse III -Paul Sabatier and IFP Energies nouvelles. The "Direction du Numérique" of the Université de Pau et des Pays de l'Adour, MCIA (Mésocentre de Calcul Intensif Aquitain) and CINES under allocation A007080045 made by Grand Equipement National de Calcul Intensif (GENCI)


Sébastien Marbach

Christophe Cordier

Rémy Claveau

Thierry Engel

Paul Montgomery

Manuel Flury

White Light Interference Microscopy System Design

Keywords: White Light Interferometry, Köhler Illumination, Linnik, Mirau, Phase Shifting Interferometry, Coherence Scanning Interferometry, Resolution Limit, Optical Metrology

Interference microscopy is a non-destructive full-field imaging method, mainly used to measure the surface topography of different samples. In this work, two designs for improving the signal quality are described. The first consists of an original vertically orientated breadboard interferometer, in a Linnik configuration. The mechanical design of the arms allows the independent control and alignment of the coherence and the focal plane positions for optimizing fringe contrast. A low noise 16-bit camera is used to improve the sensitivity. The second interferometer is based on a Thorlabs tube system, with a Nikon Mirau Objective and a white LED, all controlled with IGOR Pro software or Labview, with the aim of being more compact, flexible and mobile. For both systems, an evaluation of the interferometric signal quality is performed, whereas the difference in lateral resolution by considering the 3D nature of the interferometric system, or only its 2D imaging abilities, is explored.

INTRODUCTION

The characterization of surface topography is often performed with techniques such as stylus profilometry, confocal microscopy, and AFM for small nm-structures [START_REF] Conroy | A comparison of surface metrology techniques[END_REF][START_REF] Upadhyay | Bearing failure issues and corrective measures through surface engineering[END_REF][START_REF] Tomovich | Quantitative surface characterisation using laser scanning confocal microscopy[END_REF] . However, if the sample topography needs to be measured at the micrometer scale laterally, white light interference microscopy is often preferred [START_REF] De Groot | Principles of interference microscopy for the measurement of surface topography[END_REF][START_REF] Niehues | Low coherent Linnik interferometer optimized for use in nanomeasuring machines[END_REF][START_REF] Montgomery | Towards real time 3D quantitative characterisation of in situ layer growth using white light interference microscopy[END_REF] since it presents the advantages of being nondestructive, cost-effective, easy to use, fast and allowing full-field imaging with a nanometric sensitivity along the optical axis. The lateral resolution can be further improved by combining this method with microspheres [START_REF] Wang | Three-dimensional super-resolution morphology by near-field assisted white-light interferometry[END_REF][START_REF] Kassamakov | 3D superresolution optical profiling using microsphere enhanced Mirau interferometry[END_REF][START_REF] Leong-Hoi | High Resolution Microsphere-Assisted Interference Microscopy for 3D Characterization of Nanomaterials[END_REF][START_REF] Marbach | Microsphere-assisted imaging of subdiffraction-limited features[END_REF] . If the sample does not present step heights larger than half of the central illumination wavelength (with a microscope working in reflection), phase shifting microscopy (PSM) can be performed to obtain the topography, based on quantitative phase determination [START_REF] Coquoz | High-speed phase-shifting common-path quantitative phase imaging with a piezoelectric actuator[END_REF] . With steps greater than half of the wavelength, it becomes necessary to take advantage of the short coherence length of the source. Since the fringe intensity is maximum only when the optical path difference between the arms is null, it is possible to use these white-light fringes as an optical probe. By making a scan over the sample roughness height and topography can be thus obtained. This method is commonly known as White Light Scanning Interferometry (WLSI), or Coherence Scanning Interferometry (CSI) [START_REF] Montgomery | Real time surface morphology analysis of semiconductor materials and devices using 4D interference microscopy[END_REF][START_REF] Coupland | Measurement of Step Surfaces Using White Light Interferometry[END_REF][START_REF] Montgomery | The metrology of a miniature FT spectrometer MOEMS device using white light scanning interference microscopy[END_REF] . White light interferometry can also be used to perform characterization over the depth when the sample is transparent [START_REF] Benatmane | 3D analysis of buried interfaces using interference microscopy[END_REF][START_REF] Montgomery | Tomographic analysis of medium thickness transparent layers using white light scanning interferometry and XZ fringe image processing[END_REF] , a method often used in biology, under the name of Full-Field Optical Coherence Tomography (FF-OCT) or Full-Field Optical Coherence Microscopy (FF-OCM) [START_REF] Dubois | High-Resolution Full-Field Optical Coherence Tomography with a Linnik Microscope[END_REF][START_REF] Dubois | Ultrahigh-resolution full-field optical coherence tomography[END_REF] , for instance to provide photothermal [START_REF] Nahas | Detection of plasmonic nanoparticles with full field-OCT: optical and photothermal detection[END_REF] or elastographic [START_REF] Nahas | 3D static elastography at the micrometer scale using Full Field OCT[END_REF] information, or to detect skin cancer [START_REF] Krauter | Full-field swept-source optical coherence tomography with phase-shifting techniques for skin cancer detection[END_REF][START_REF] Dubois | Line-field confocal optical coherence tomography for high-resolution noninvasive imaging of skin tumors[END_REF] .

In this paper, two original designs of white light interferometer (WLI) working in the reflective mode are presented. The first system is an innovative vertically orientated breadboard interferometer in a Linnik configuration. This has been designed to meet the need for a high-quality interferometric signal, not through signal processing but with hardware improvements such as the use of high quality objectives, an accurate piezo-system and a low noise camera. The design process, from computer design until the achievement of the setup is explained. The choices were made with the aim of reducing noise sources as much as possible to enhance the fringe signal. Just enough degrees of freedom were used to allow the adjustment of certain key parameters without making overly complicated alignment of the interferometer. *marbachs@unistra.fr; phone +33 3 68 85 46 14; mflury@unistra.fr, phone + 33 3 88 14 47 47; http://icubeipp.unistra.fr/en/index.php/Home

The second WLI is based on a Thorlabs tube system in the Mirau, configuration. The purpose of this setup is to be low cost, more compact and to make faster measurements. The performance of these devices is illustrated through topographic measurement and with the lateral resolution. Indeed, since interferometric system supplies results in 3D, it is not sufficient to describe the lateral resolution of the system based only on its 2D imaging abilities, with the classical Abbe or Rayleigh criteria. Usually, the lateral resolution that these systems can achieve, while providing relevant information on the measured height, is worse than the 2D lateral resolution [START_REF] Leach | ISO Definition of Resolution for Surface Topography Measuring Instruments[END_REF][START_REF] De Groot | Interpreting interferometric height measurement using the instrument transfer function[END_REF][START_REF] Lehmann | Fundamental aspects of resolution and precision in vertical scanning white-light interferometry[END_REF] . This is why Leach et al. [START_REF] Leach | ISO Definition of Resolution for Surface Topography Measuring Instruments[END_REF] , De Groot et al. [START_REF] De Groot | Interpreting interferometric height measurement using the instrument transfer function[END_REF] , and Lehmann et al. [START_REF] Lehmann | Fundamental aspects of resolution and precision in vertical scanning white-light interferometry[END_REF] , propose to take into consideration the global 3D optical transfer function of the interferometer, to define the lateral resolution. A criterion of lateral resolution that depends on the height measured is exploited to highlight this aspect.

WHITE LIGHT INTERFERENCE MICROSCOPY PROPERTIES

White Light Interference Microscopy consists in acquiring successive interferometric images, with the introduction of a known optical path difference (OPD) between each image. Experimentally, the OPD is modified by adjusting the distance travelled by the light between the objective and the sample, thanks to a piezoelectric system mounted on the objective or under the sample. The piezo translator is moved step by step along the optical axis z to image the entire sample surface with the fringe pattern. At each step, an XY interferometric image is acquired and stored in an XYZ stack of interferometric images, as illustrated in Fig 1(a). Then, pixel by pixel the interferometric signal along the depth z is extracted. At a given pixel (x,y) from the XYZ stack, the intensity I measured as a function of the depth z can be represented as in Fig 1(b). Since the maximum of the fringe signal corresponds to a null OPD, this fringe can be used as a reference. With the position of the piezo system being determined with nanometer sensitivity, determining the position of the fringe maximum in the image, allows the determination of the topography of the sample. To be more robust, the position of the piezo can be determined at the maximum of the fringe envelope. In this case, the piezo step should be small enough to correctly sample the interferometric signal and so be able to detect with accuracy the maximum of the fringe envelope. The microscope objective has a given Numerical Aperture (NA) so that the reflected light can be collected with an angle θ up to the maximum angle θmax accepted by the objective, and defined in air as the arcsinus from the NA. Beams coming back from the reference arm and the object arm with an angle θ between them can interfere together. So all the waves included in the cone of light with an apex angle of 2 . θmax, must be considered in the interferometric signal measured. In such a configuration, the interfringe distance, or the length between two consecutives fringe maxima or minima, known as the effective wavelength λeff, depends not only on the illumination wavelength λ, but also on the NA of the microscope objectives. Since it is difficult to theoretically calculate λeff of the system with precision, this is normally measured experimentally from an interferogram after system calibration with a reference step. λeff takes into account the whole response of the interferometric system and is important for accurate PSM measurements by introducing the correct phase shifts.

VERTICALLY ORIENTATED LINNIK INTERFEROMETER

The first design, presented in Fig. 2, is an original vertically orientated breadboard interferometer in a Linnik configuration, and working in the reflective mode. A Linnik interferometer presents the distinctive feature of possessing identical microscope objectives in the reference and object arms, in order to be able to measure the sample topography with lateral-µm-sized elements. The choice of components was made with the aim of reducing and minimizing the different sources of noise as much as possible.

First, Köhler illumination was chosen to optimally illuminate the sample. The main aims are to obtain a uniform and homogeneous sample illumination and identical spatial coherence at each of the sample points [START_REF] Köhler | Ein neues Beleuchtungsverfahren für mikrophotographische Zwecke[END_REF] . This enables the independent control of the illumination area on the sample and the angle of the illumination cone using respectively a field diaphragm (FD) and an aperture diaphragm (AD). A computer-assisted design with CODEV Optical Design Software was first made to dimension the illumination part of the system. After traveling through the lenses acting as Köhler illumination, the light from the white LED is divided into two beams with the same intensity for the reference arm and the object arm. In both arms, the light passes through identical high-quality long working distance apochromatic microscope objectives (Mitutoyo MY20X-804), corrected for infinity, with a x20 magnification, a NA equal to 0.42 and a working distance of 20 mm. In the reference arm, the beam is reflected back by a mirror and in the object arm by the sample. These beams are then merged by travelling again through the beam splitter. Since the objectives are infinity corrected, a relay lens is necessary to produce the image on the camera sensor. For the image acquisition, a PCO Edge 4.2 camera with a scientific-CMOS (sCMOS) monochrome sensor interfaced with the computer in USB 3.0 is used. It also presents a higher dynamic range with the value encoded to 16 bit, and a better signal to noise ratio than with a classic CMOS sensor, while maintaining a high acquisition speed (170 fps for a 640 by 480 pixels image). Moreover, the camera is cooled with a Peltier system, which allows making measurements over a long period without drift. The sample is placed on a piezo electric translator (PI Hera P-621.ZCD), controlled in closed-loop allowing a position sensitivity along the optical axis of +/-1 nm. The piezo is placed on a 2-axis goniometer in order to adjust the inclination of the sample. Since a low coherence source is used, the fringe pattern appears only when the optical path difference between the object and reference arm is shorter than the coherence length of the source (several µm in this case). The whole of the reference arm is placed on a translation table with a µm-displacement sensitivity to allow the independent control and alignment of the coherence plane and the focal plane positions for optimizing fringe contrast. The OPD in the Linnik interferometer also depends on the reference mirror tilt and from the distance separating the optical center of each microscope objective orthogonally to the system optical axis. To be able to optimize the fringe contrast and adjust the fringe spacing on an XY interferometric image by superimposing the coherence and focal planes, the mirror is mounted on a cage. This cage allows its inclination to be adjusted while the object arm objective is mounted on a translation cage.

The goal of this setup is to be able to control the optical centers of the two objectives. A vertically orientated design was chosen to simplify the measurement of samples placed on the piezo system. The mechanical stability was reinforced by using low thermal coefficient materials and an optical table reinforced with struts, mounted on rubber vibration dampers.

The interferometric system is controlled by a home developed LabVIEW 2019 program using the IMAQVision imaging module (National Instruments). To increase the imaging contrast, the aperture diaphragm of the setup is closed. The effective wavelength measured experimentally with this system is equal to λeff =660 nm, and the lateral resolution obtained is about Δx=1000 nm.

COMPACT MIRAU INTERFEROMETER

The second system developed is a compact white light interferometer based on a Mirau microscope objective (Fig. 3). The white light beam coming from the LED source is directly inserted into the beam splitter, which transfers half the light into the Nikon Mirau Objective (x10, NA=0.3). The objective is mounted on a piezo PI PIFOC P-721.C20, controlled in closed loop with a positioning sensitivity of 10 nm, in order to modify the optical path difference. This objective is corrected for infinity, so the light reflected from the sample travels through a tube lens identical to that of the previous system. The image from the sample is recorded by an 8-bit high-speed color Basler camera (ACA640-750uc, about 800 frames per second for 640*860 pixels, USB 3.0 connection). The sample is placed on a goniometer (identical to the previous setup) to modify its orientation. The interferometer is directly included in the Mirau objective which is mounted at the precise position to give a magnification of 10x while respecting the right distance between the tube lens and the objective. The Mirau objective ensures the optimum superposition of the focal and coherence planes and guarantees a better mechanical stability than the Linnik configuration although it has less degrees of freedom. The optical setup is placed on an anti-vibration table with sorbothane isolators, acting as vibration absorbing feet. The acquisition (control of the camera and the piezo device) and the processing of the measurements are performed with IGORPro software or LabVIEW 2017. The effective wavelength measured through this system is equal to λeff =580 nm. Although difficult, since the fringe pattern always appears on the image of the sample, the lateral resolution Δx, measured experimentally is about 1.2µm.

SYSTEM PERFORMANCE

Interferometric measurement

An example of a CSI measurement made with the Linnik system was carried out on a reference standard consisting of a square groove etched in quartz with a depth of 380 nm. To obtain a good compromise between the measurement speed and the measurement accuracy, a piezo step of 25 nm was chosen over a dynamic range of 8 µm. At each step, 10 interferograms were averaged, to reduce the temporal noise. The resulting images were stored and then processed, with the previous algorithm described. An additional median (3 by 3 pixels) and averaging filter (2 by 2 pixels) were applied to further eliminate noise in the image. The 3D structure measured is illustrated in Fig 4(a), and a profile extracted in Fig 4(b). A typical height of 375 nm was found, but with heights varying between 350 nm and 400 nm, due to the piezo step chosen which can only provide a sensitivity of 25 nm, together with surface defects in the grating.

For the sake of comparison, a PSM measurement of the same standard was made with the Mirau setup described in section 4. The sample was axially moved 4 times, with successive piezo steps of 1/8 of the effective wavelength, to introduce phase shifts of π /2. As before, for each step, the interferogram stored was the average of 10 interferograms.

The phase was obtained with the 5-step phase shifting algorithm, unwrapped, and converted into topographic information. The height obtained was then processed with identical median and averaging filters as before. The result obtained was smoother, as shown in Fig. 4(c), due to the method (PSM) used. An extracted profile is also presented in Fig. 4(d). The typical height measured is 379.1 nm, with a standard deviation of 4.35 nm. This results in an average error of +/-1.15 %, with the standard deviation as a reference. It should be noted that similar results are obtained with both systems in PSM and CSI.

Signal to Noise Ratio (SNR)

An evaluation of the interferometric signal was performed, through the determination of different types of signal to noise ratio (SNR). The interferometric signal along each pixel was first processed, with the suppression of the offset, and then by taking its absolute value. The first signal to noise ratio definition (SNR1) is proposed in equation ( 1). SNR1 consists of the ratio of the mean value of the processed interferometric signal maximum (µmax), divided by the mean value of the noise amplitude at each pixel (µnoise).

(1) max 10

1 10.log noise SNR µ µ   =    
A second SNR typically used in signal processing [START_REF] Lehmann | Fundamental aspects of resolution and precision in vertical scanning white-light interferometry[END_REF] , and named SNR2, is written in equation ( 2). SNR2 is the ratio of the mean square of the processed signal 2 signal RMS on the variance of the noise 2 noise σ .

(

The value is determined for these 2 SNR with the Mirau system, the Linnik configuration and the Linnik Configuration with 10 images averaged for each piezo step. The results in dB is presented in Table 1. For SNR1 and SNR2, the values for the Linnik configuration (particularly for averaging 10 images) are better than those for the Mirau configuration, which can be explained by the low-noise 16-bit camera used with the Linnik system. If it is admitted that a value greater than 20 dB for SNR2 corresponds to high signal quality, the results demonstrates that a good interferometric signal is obtained for both setups.

Lateral resolution of topographic system

As mentioned in the introduction, the 3D response of the topographic measurement system should be considered to describe the performance of the optical setup. For this reason, the ability of both systems to measure the heights of gratings with different spatial frequencies was checked. To do so, a topographic reference standard (RS-N gratings from SiMETRICS Gmbh), made of a series of square profile gratings with different periods etched in a silicon wafer, allowed a rigorous study to be performed. Since the height of the gratings given by the manufacturer is 192 nm, they were measured and processed with a 5-step phase shifting algorithm with 10 images averaged at each step. Due to the very straight edges of the square shaped grooves, at these positions a diffractive phenomenon, known as the batwing effect [START_REF] Leong-Hoi | High Resolution Microsphere-Assisted Interference Microscopy for 3D Characterization of Nanomaterials[END_REF][START_REF] Lehmann | Fundamental aspects of resolution and precision in vertical scanning white-light interferometry[END_REF] , can appear in the phase. This can lead to misinterpretation by the 2D unwrapping algorithm, and so to wrong measurements of the topography, thus requiring special attention to processing. The measured height as a function of the period gratings Λ is represented in Fig. 5(a), in red, blue and black respectively for the Mirau interferometer, Linnik interferometer and the characterization made by the manufacturer. In Fig. 5(b), the error E defined in equation (3), between the heights measured, hmeas, and the height characterized by the manufacturer, hchar, is plotted with the same color code defined previously.

(

To describe the performance of the systems, the 3D measurement sensitivities were determined. This was performed by measuring a flat silicon sample using PSM in order to determine the standard deviation of the local height distribution. The sensitivities were estimated as being 1.48 nm, and 1.73 nm respectively for the Mirau and the Linnik interferometers. The errors between the measured groove heights and the data from the company, is not entirely explainable by these sensitivities. These errors could be for instance related to differences between the areas measured here and those measured by the manufacturer. For spatial frequencies smaller than 0.5 cycles/µm, the error is negligible with the Linnik system. However, the errors in the heights measured observed for spatial frequency greater than 0.5 cycles/µm, cannot be explained only by the noise from the system. This error is rather due to the 3D optical transfer function of the system, i.e. taking into account the frequency response of the optical system but in 3 dimensions.

Indeed, the ability of the system to resolve lateral elements in 2D does not guarantee that reliable height information of these elements will be obtained. The 3D response of the interferometer does not depend only on its 2D imaging properties. This is why the lateral limit of measurable features with a 3D system is different, and usually lower, than the lateral resolution that can be achieved considering only its direct imaging capacities, as mentioned by Leach et al. [START_REF] Leach | ISO Definition of Resolution for Surface Topography Measuring Instruments[END_REF] , De Groot et al. [START_REF] De Groot | Interpreting interferometric height measurement using the instrument transfer function[END_REF] , and Lehmann et al. [START_REF] Lehmann | Fundamental aspects of resolution and precision in vertical scanning white-light interferometry[END_REF] . A 3D lateral limit needs to be established, by defining a criterion taking into

  =       ( ) ( ) ( ) ( ) char meas char h h E h Λ - Λ Λ = Λ
consideration the aptitude of the system to perform relevant topographic information as a function of spatial frequency. For example, in Ref. [START_REF] De Groot | Interpreting interferometric height measurement using the instrument transfer function[END_REF] , a proposition is to define the 3D lateral limit as the spatial frequency from which the height measured is smaller than the half of the true height. Considering this proposition as a criterion, the green line is added in Fig. 5(b) corresponding to half of the height given by the manufacturer (and so to an error of 50 %). The intersection of this line (black dot), with the curve obtained with both system allows the definition of a 3D lateral limit of 0.547 cycles/µm (i.e. 1.83 µm) and 0.658 cycles/µm (i.e. 1.52 µm), compared to a 2D lateral resolution experimentally measured of 1.2 µm and 1 µm, respectively for the Mirau and the Linnik interferometer. As expected, these results show a decrease in lateral resolution when considering the 3D response of the system, compared to its 2D response. Error on the height measurement for both system with identical color code, in green the criterion for lateral limit of topographic system (error inferior to 50%) and in black dots the intersection of the criterion with the system.

CONCLUSION

In this work, two original white light interferometers, working in the reflective mode, have been presented and experimentally studied. The first configuration is a vertical orientated breadboard Linnik interferometer. The components choice was based on the results of optical simulation, so as first to obtain experimentally a Köhler illumination and then to minimize the noise by using a 16-bit sCMOS cooled camera and a Piezo working in closed loop with nanometer sensitivity. A sufficient number of degrees of freedom was built in to allow the adjustment of necessary parameters without a too complicated alignment, and so as to give ease of modification. The second system developed was a compact Mirau Interferometer, which is easier to use than the Linnik, giving a high mechanical stability, but less degrees of freedom. The performance of each system was then determined, through classic interferometry measurement (PSM and CSI), and determination of signal to noise ratio values. Finally, the 3D lateral limit was compared to the 2D lateral resolution for both systems, in order to highlight the specificities of the interferometric systems, resulting in a difference in resolution as illustrated with the study conducted on spatial frequency response.

Figure 1 .

 1 Figure 1. (a) XYZ stack of interferometric images, (b) Interferometric intensity extracted in one pixel (x,y) along the optical axis z .

Figure 2 .

 2 Figure 2. Vertically-orientated white light Linnik interferometer working in the reflective mode.

Figure 3 .

 3 Figure 3. Home-made Mirau white light Interferometer.

Figure 4 .

 4 Figure 4. 3D results of a 380-nm step etched in quartz measured using PFSM with the Linnik configuration in (a), using PSM with the Mirau configuration in (c), together with extracted profiles respectively in (b) and in (d).

Figure 5 .

 5 Figure 5. (a) Height measurement of the standards as a function of the period for the Mirau and Linnik interferometer respectively in red and blue compared with the value given by the manufacturer in black. (b)Error on the height measurement for both system with identical color code, in green the criterion for lateral limit of topographic system (error inferior to 50%) and in black dots the intersection of the criterion with the system.

Table 1 .

 1 SNR value in dB determined for the Mirau, Linnik and Linnik (10 Avg) system. Linnik (10 Avg) consists in averaging 10 interferograms for each piezo step

	Mirau Linnik Linnik (10 Avg)
	SNR1 14.96	19.56	23.61
	SNR2 24.79	28.02	36.47