Marine Bossert

I Trimaille

L Cagnon

B Chabaud

C Gueneau

P Spathis

P E Wolf

E Rolley

Surface tension of cavitation bubbles

We have studied homogeneous cavitation in liquid nitrogen and normal liquid helium. We monitor the fluid content in a large number of independent mesopores with an ink-bottle shape, either when the fluid in the pores is quenched to a constant pressure or submitted to a pressure decreasing at a controlled rate. For both fluids, we show that, close enough to their critical point, the cavitation pressure threshold is in good agreement with the Classical Nucleation Theory (CNT). In contrast, at lower temperatures, deviations are observed, consistent with a reduction of the surface tension for bubbles smaller than two nanometers in radius. For nitrogen, we could accurately measure the nucleation rate as a function of the liquid pressure down to the triple point, where the critical bubble radius is about one nanometer. We find that CNT still holds, provided that the curvature dependence of the surface tension is taken into account. Furthermore, we evaluate the first-and second-order corrections in curvature, which are in reasonable agreement with recent calculations for a Lennard-Jones fluid.

Cavitation, the formation of a vapor bubble in a metastable liquid below its saturated vapor pressure P sat , is important in many fields, ranging from engineering (ultrasonic cleaning, propeller damage. . . ) to natural sciences (blocking of sap ascent. . . ). While, for decades, the theoretical framework for cavitation has been the Classical Nucleation Theory (CNT), recent experiments using hexane [START_REF] Doebele | Direct observation of homogeneous cavitation in nanopores[END_REF], heptane [START_REF] Arvengas | Cavitation in heavy water and other liquids[END_REF], ethanol [START_REF] Arvengas | Cavitation in heavy water and other liquids[END_REF], argon [START_REF] Vinogradov | Explosive cavitation in superheated liquid argon[END_REF], nitrogen [START_REF] Baidakov | Homogeneous nucleation in liquid nitrogen at negative pressures[END_REF] have shown that measured cavitation rates are many orders of magnitude larger than predicted by CNT.

This discrepancy is generally attributed to a dependance of the surface tension σ on the interface curvature. Indeed, in the CNT, the liquid-vapor interface is assumed to be infinitely sharp, with a surface tension σ ∞ equal to that for a planar interface, so that the energy barrier for nucleation reads:

E CNT b = 16π 3
σ 3 ∞ (P v -P l ) 2 (1)
where P v and P l are the pressures inside and outside the bubble, respectively. In the above experiments, however, the interface thickness is not negligible compared to the nanometric size of the critical nucleus. As a result, the * etienne.rolley@phys.ens.fr

CNT may overestimate the energy barrier, accounting for the measured cavitation rates. The modification of the energy barrier can be computed from density functional models [START_REF] Debenedetti | Metastable liquids: concepts and principles[END_REF]. A simpler approach consists in keeping the CNT formalism, in particular equation 1, while allowing the surface tension σ to depend on the radius of curvature R. It was recently demonstrated that such an approach is successful for cavitation in water [START_REF] Menzl | Molecular mechanism for cavitation in water under tension[END_REF].

To first order in curvature, the dependence of the surface tension is expected to be of the form:

σ ∞ /σ(R) = 1 + 2δ ∞ /R (2)
(with R < 0 for a bubble), as demonstrated a long time ago by Tolman [START_REF] Tolman | The effect of droplet size on surface tension[END_REF] . Accordingly, δ ∞ in the above equation is generally called the Tolman length. Beyond first order, it is only recently that many theoretical works have attempted to calculate σ(R) for a bubble or a droplet, using direct molecular dynamics simulations [START_REF] Van Giessen | Determination of curvature corrections to the surface tension of a liquid-vapor interface through molecular dynamics simulations[END_REF][START_REF] Sampayo | Communications: Evidence for the role of fluctuations in the thermodynamics of nanoscale drops and the implications in computations of the surface tension[END_REF][START_REF] Baidakov | Molecular dynamics simulation of cavitation in a lennard-jones liquid at negative pressures[END_REF], various density functional calculations [START_REF] Wilhelmsen | Evaluation of finite-size effects in cavitation and droplet formation[END_REF][START_REF] Rehner | Surface tension of droplets and tolman lengths of real substances and mixtures from density functional theory[END_REF] or both [START_REF] Julin | A thermodynamically consistent determination of surface tension of small lennard-jones clusters from simulation and theory[END_REF][START_REF] Block | Curvature dependence of surface free energy of liquid drops and bubbles: A simulation study[END_REF]. For Lennard-Jones (LJ) fluids, a consensus has emerged that, first, δ ∞ is negative and of the order of -0.1d lj (d lj is the LJ length scale). Second, as soon as d lj /|R| > 0.1 which is the case in most nucleation experiments, the second order correction cannot be neglected so that equation 2 does not capture the R-dependence of σ.

Experimentally testing these predictions through cavitation measurements is a prerequisite for using the concept of an effective surface tension σ(R), and the simple and powerful tools of capillarity in situations where the interface is highly curved, ranging from homogeneous nucleation of droplets to sessile nanodroplets [START_REF] Shin | Growth dynamics and gas transport mechanism of nanobubbles in graphene liquid cells[END_REF][START_REF] German | Laplace pressure of individual h2 nanobubbles from pressure-addition electrochemistry[END_REF], meniscii in mesopores [START_REF] Donne | Intrusion and extrusion of liquids in highly confining media: bridging fundamental research to applications[END_REF] or liquid interfaces under an AFM tip [START_REF] Kim | Direct evidence for curvature-dependent surface tension in cap-illary condensation: Kelvin equation at molecular scale[END_REF]. Experimental data are however scarce. Interpretation of most cavitation experiments in simple fluids only yields the effective surface tension for a single value of R, equal to the radius R * of the critical nucleus [START_REF] Vinogradov | Explosive cavitation in superheated liquid argon[END_REF][START_REF] Baidakov | Homogeneous nucleation in liquid nitrogen at negative pressures[END_REF]. This single value σ(R * ) is not enough to test the validity of equation 2 and to determine first and second order corrections in curvature. More detailed experiments have been performed in ethanol and water [START_REF] Azouzi | A coherent picture of water at extreme negative pressure[END_REF][START_REF] Bruot | Curvature dependence of the liquid-vapor surface tension beyond the tolman approximation[END_REF]. The analysis of the pressure dependence of the cavitation rate points to the relevance of the second order correction, but its estimate may be specific to these fluids and has been obtained only in a restricted temperature range.

In a recent work, we demonstrated that, by using porous materials made of assemblies of individual pores with an ink-bottle shape, it is possible to accurately measure the cavitation rate of hexane at room temperature and a given liquid pressure in a single shot experiment [START_REF] Doebele | Direct observation of homogeneous cavitation in nanopores[END_REF]. Here, we extend this new technique to the study of cavitation in two simple cryogenic fluids, helium and nitrogen, over a large range of temperatures. By monitoring the fluid content in the pores when decreasing the liquid pressure, we can determine the cavitation threshold, as well as the cavitation rate around this threshold. At high temperatures, we show that our results are in good agreement with the CNT; in other words, the critical radius is large enough to ensure σ(R) = σ ∞ . At low temperatures, our measurements evidence a decrease of the surface tension, with respect to its bulk value, due to the curvature of the critical bubble. Furthermore, for nitrogen, the optical technique used to monitor the fluid content allows to accurately measure the pressure dependence of the cavitation rate, hence to probe the second order correction to σ. Using the procedure introduced by Bruot and Caupin [START_REF] Bruot | Curvature dependence of the liquid-vapor surface tension beyond the tolman approximation[END_REF], we have extracted from the experimental data the first and second order contributions to σ(R). These contributions are found to be of the same order of magnitude and both are in reasonable agreement with Density Functional Theory (DFT) predictions.

MEASUREMENT OF THE CAVITATION THRESHOLD

Cavitation is studied in membranes of porous alumina [START_REF] Doebele | Direct observation of homogeneous cavitation in nanopores[END_REF] or porous silicon [START_REF] Bossert | Evaporation process in porous silicon: Cavitation vs pore blocking[END_REF]. The native independent pores are roughly cylindrical, with a diameter and a length which can be respectively tuned in the ranges 10-50 nm and 1-10 µm by varying the synthesis conditions. Their typical volume V p is of the order of 10 -21 m 3 (see Materials and Methods). In order to obtain the ink-bottle shape necessary for evaporation to proceed by cavitation, the pore aperture is reduced by Atomic Layer Deposition [START_REF] Bossert | Evaporation process in porous silicon: Cavitation vs pore blocking[END_REF].

Cavitation is monitored by measuring sorption isotherms, that is the amount of fluid in the pores as a function of the vapour pressure P v in the gas reservoir. For nitrogen, we use optical interferometry in white light (WLI) which yields the optical thickness L of the sample. We have checked that the variation of L is proportional to the amount of fluid inside the pores [START_REF] Bossert | Stress or strain does not impact sorption in stiff mesoporous materials[END_REF]. WLI is more precise and less prone to drift than volumetry. While it only probes the illuminated part of the sample, the latter still contains a huge number of pores (of order of 10 9 , about one tenth of their total number).

A typical optical isotherm is shown in figure 1, where the difference ∆L ≡ L(P v ) -L(P v = 0) is plotted as a function of the vapour pressure P v . First, P v is increased up to the saturation pressure P sat in order to fill the pores. Then P v is decreased at a constant rate. In a first step, the optical thickness barely decreases: the liquidvapor meniscii are pinned at the aperture of the pore and the sample remains saturated with liquid. At some point (P v ≲ 0.8P sat in fig. 1), ∆L starts to decrease, corresponding to the recession of meniscii in the largest apertures. This recession regime occurs in a wide pressure range as there is some dispersion in the aperture diameters, due to both the initial dispersion in pore diameter and the ALD process. Finally, the sharp decrease of ∆L over a narrow pressure range around 0.67P sat is the signature of cavitation in the pores whose aperture radius is smaller than a temperature-dependent critical value. The cavitation threshold P * v is defined as the pressure at mid-height of the cavitation step. Here and below, we use the suffix * for quantities evaluated at this threshold. The corresponding liquid pressure at the threshold, P * l , is derived from phase equilibrium between the vapour and the metastable liquid (see Material and Methods). Away from the threshold, P l , the liquid pressure inside the pores, is similarly derived from P v . In the following, we will use P l to measure the departure from bulk phase equilibrium P l = P sat . I). Upon decreasing the pressure, meniscii start to recede in the larger apertures at PV ≃ 0.8Psat, until cavitation occurs around 0.67Psat. Inset: sketch of meniscus receding (right) and cavitation (left) in an ink-bottle pore. large temperature range in several porous samples with various pore diameters (see Table I in Material and Methods). Figure 2 shows a series of isotherms measured on the porous alumina sample B. As the temperature increases, the threshold P * v and the fraction of pores emptying through cavitation increase. This is due to the temperature variation of σ ∞ which favours cavitation with respect to meniscus recession at high temperature [START_REF] Bossert | Evaporation process in porous silicon: Cavitation vs pore blocking[END_REF].

Such isotherms have been measured for nitrogen in a

Similar experiments have been performed for normal liquid helium at temperatures above the superfluid transition (≃ 2.17 K). In this case, the adsorbed amount is measured by a continuous volumetric method [START_REF] Bossert | Evaporation process in porous silicon: Cavitation vs pore blocking[END_REF]. During evaporation, the cell is pumped through a fixed impedance so that the rate of decrease of P v varies through the cavitation stage: P * v is then obtained as the pressure where | Ṗv | is minimal (see Supporting Information).

SURFACE TENSION FROM CAVITATION THRESHOLD

In order to obtain the surface tension at the scale of the critical bubble, we first compute the energy barrier at the cavitation threshold. According to CNT, the nucleation rate J(P l ) at a fixed liquid pressure P l is given by

J(P l ) = J 0 exp -Eb(Pl)/kbT (3)
with J 0 being a prefactor which is controlled by the dynamics of the critical bubble [START_REF] Debenedetti | Metastable liquids: concepts and principles[END_REF][START_REF] Blander | Bubble nucleation in liquids[END_REF]. For nitrogen, the pressure is decreased at a constant rate A = -dP l /dt. Assuming that the energy barrier can be linearized as

E b = E * b + α (P l -P * l )k b
T in the vicinity of the cavitation threshold, the probability Σ(P l ) that cavitation has not occurred at pressure P l in a pore of volume V p is (see Supporting Information):

Σ(P l ) = exp - V p Aα J(P l ) (4)
As shown in Materials and Methods, Σ(P l ) =1/2 corresponds to the middle of the cavitation step, i.e. to P l = P * l . The energy barrier E * b at the threshold is thus:

E * b = k b T ln [J 0 V p /(Aα ln 2)] (5)
For the volumetric measurements with helium, the flow rate is imposed instead of the pressure ramp and the above expression is modified as:

E * b = k b T ln [J 0 V p /τ ] ( 6
)
where the characteristic time scale of cavitation, τ , is the duration of the cavitation step (see Supporting Information). In Eqs.5 and 6, A (or τ ) and V p are known. For Eq.5, we approximate α by its CNT expression

∂E CNT b /∂P l | T /(k B T
). As we will show below, this is correct within 20% (see Fig. 6). Since α enters eq.5 through a logarithmic factor, our approximation has a negligible impact on E * b . The last parameter is the prefactor J 0 ; for our experimental conditions, as usual for a cryogenic liquid not too close to the critical point [START_REF] Vinogradov | Explosive cavitation in superheated liquid argon[END_REF][START_REF] Baidakov | Homogeneous nucleation in liquid nitrogen at negative pressures[END_REF]and for water [START_REF] Menzl | Molecular mechanism for cavitation in water under tension[END_REF] close to room temperature, the dynamics of the critical bubble is controlled by the viscous flow in the liquid. We use the corresponding expression

J 0 = n l σ∞ η σ∞ kbT 1/2
(1-P l /P v ), where n l is the number density of the liquid and η its viscosity (see Ref. [23? ] and Supporting Information).

Plugging the above values in eqs.5 or 6 yields E * b in the range of 40-55 k b T for both nitrogen and helium (see Fig. S2 of the Supporting Information). Because the experimental parameters V p and A or τ vary from run to run, comparing our results with CNT is easier in terms of the effective surface tension σ which, for a given fluid, should only depend on the temperature and on the radius of the critical bubble. Due to the thickness of the bubble interface, different choices can be used for the bubble radius and for the surface tension [START_REF] Tröster | Numerical approaches to determine the interface tension of curved interfaces from free energy calculations[END_REF]. When using the so-called stress radius R * s and the associated surface tension σ(R * s ), for which the Laplace equation

R * s = 2σ(R * s )/(P * l -P * v
) is satisfied, the energy barrier is given by the CNT expression (eq.1), provided σ ∞ is replaced by σ(R * s ) [START_REF] Tröster | Numerical approaches to determine the interface tension of curved interfaces from free energy calculations[END_REF]. Equating this CNT expression to the measured barrier thus yields the effective surface tension:

σ(T, R * s ) = 3 16π E * b (P * v -P * l ) 2 1/3 = σ ∞ (T ) E * b E CNT * b 1/3 (7)
The ratio σ/σ ∞ is plotted in Fig. 3 as a function of the normalised radius |R * s |/d lj as obtained from σ(T, R * s ) and the Laplace equation for both nitrogen and helium, using respectively for d lj the values 0.375 nm and 0.26 nm for nitrogen and helium. For a given fluid, increasing values of R * s correspond to increasing temperatures, reflecting the fact that σ ∞ decreases when T increases while E b only weakly depends on T .The plot of σ/σ ∞ as a function of the normalised temperature T /T c can be found in Fig. S4 of the Supporting Information. [START_REF] Vinogradov | Explosive cavitation in superheated liquid argon[END_REF] and nitrogen [START_REF] Baidakov | Homogeneous nucleation in liquid nitrogen at negative pressures[END_REF] can be analysed in the same way and yield a decrease of σ for small nucleus consistent with our measurements, though with a much larger scatter (see Fig. S7 in Supporting Information).

We stress that Fig. 3 cannot be used to test Eq. 2, or to determine the respective first and second order corrections in curvature. Indeed, in Fig. 3,R * s varies together with T , whereas, because the Tolman length may depend on temperature [START_REF]Such a temperature dependence of the Tolman length would naturally explain the difference, for a given normalised radius[END_REF] , a test of eq.2 requires varying R * s at a fixed temperature. Because R * s = 2σ(R * s )/(P * l -P * v ), this amounts to vary P * v . In the following, we achieve this requirement by varying the cavitation rate.

DETERMINATION OF THE GENERALIZED TOLMAN LENGTH FROM THE PRESSURE DEPENDENCE OF THE CAVITATION RATE

The pressure dependence of the cavitation rate, or that of the energy barrier, α = (∂E b /∂P l | T ) /(k B T ), can be determined by using two different methods for nitrogen.

Relaxation after quenching

The first method is similar to the one we previously reported for hexane [START_REF] Doebele | Direct observation of homogeneous cavitation in nanopores[END_REF]. Starting from a sample fully filled with liquid, it consists in quenching the sample down to a pressure P l close to P * l by decreasing the pressure at a fast rate down to P l and then closing the cell (see Materials and Methods). As shown in Figs. 4 A &B, the optical signal ∆L then decreases with time t to a limiting value ∆L 0 as exp(-t/τ exp ) (figure 4 C) . ∆L 0 reflects the contribution of the fluid adsorbed on the walls of the "emptied" pores at equilibrium, and can be accurately determined along the condensation branch of the isotherm (dashed line in fig. 1), at least when the condensation and evaporation branches are well separated. Close to the cavitation step, the number of filled pores is then proportional to ∆L -∆L 0 .

As a result, the number of filled pores exponentially decays with the characteristic time τ exp (P l ), and the nucleation rate J(P l ) equals 1/(τ exp V P ). Repeating the experiment at different pressures, we determine the pressure dependence of the nucleation rate. We find that J varies exponentially with P l (figure 4 D), allowing to compute the value of α through eq. [START_REF] Vinogradov | Explosive cavitation in superheated liquid argon[END_REF].

We performed such relaxation experiments at two different temperatures. The corresponding data points for α are shown in red in figure 6. This method is simple, but requires a very long term temperature stability. As discussed below, analysing the shape of a single isotherm provides a more efficient way to determine the value of the parameter α.

Analysis of the isotherm shape

The magnification of the cavitation step of figure 1 is shown in the inset of figure 5. As explained above, after subtracting the contribution ∆L 0 of the fluid adsorbed in empty pores (dashed blue line, measured during condensation), the signal is proportional to the number of filled pores. When the pressure is decreased along the isotherm, this number decreases through two mechanisms, namely cavitation and meniscus recession. Assuming that the cavitation probability does not depend on the aperture size, the ratio Ψ between the actual number of filled pores and the number that would be expected without cavitation at a given pressure is:

Ψ = ∆L

p ∆L p + ∆L v (8)
where the optical thicknesses ∆L p and ∆L v are defined in figure 5. Ψ(P l ) obeys (see Materials and Methods):

dΨ dP l = J(P l )V p A Ψ ( 9
)
This is exactly the equation obeyed by Σ(P l ) (see Supporting Information, eq. 6). Hence Ψ(P l ) is expected to show the characteristic double exponential dependence of equation 4. Figure 5 shows that this is indeed the case, and the fit provides the value of α. The uncertainty on α mainly originates from that on the extrapolated contribution of the meniscus recession.

The result of this analysis can be directly compared to that of the relaxation experiment by computing the nucleation rate J(P l ) = (A/V p ) d ln Ψ dPl from the corrected isotherm shape Ψ(P l ). As shown in figure 4D, both methods yield the same curves J(P l ) at T = 73 K, which validates our method for correcting the isotherm shape for the contribution of the meniscus recession. The radius dependence of α, normalized by its CNT value, is shown in Fig. 6. Again, we find that the CNT is correct for large bubbles. Moreover, α decreases with |R S |, the correction reaching 20% for the smallest bubbles. Such a reduction is qualitatively consistent with a reduction of the surface tension at small scales. , normalised by its CNT value calculated from eq. 1. Blue disks: this work, from the isotherm shape. Red: this work, from relaxation after quenching. Green open square are data for argon condensation, derived from the work of Sinha et al. [START_REF] Sinha | Argon nucleation in a cryogenic supersonic nozzle[END_REF]. Note that argon values for α are plotted as a function of the critical EDS radius RE since the stress radius RS cannot be derived from [START_REF] Sinha | Argon nucleation in a cryogenic supersonic nozzle[END_REF].

For helium, a similar analysis can be used to extract the value of α from the volumetric measurements. However, the uncertainty on α is of order ± 10% and we can only conclude qualitatively that the parameter α is smaller than the CNT prediction for small radii (see Supporting Information). In the following, we thus focus on the results for nitrogen.

Nucleation rate has also been measured for argon in condensation experiments [START_REF] Sinha | Argon nucleation in a cryogenic supersonic nozzle[END_REF][START_REF] Iland | Argon nucleation in a cryogenic nucleation pulse chamber[END_REF]. Comparing the nucleation rate J for two different supersaturations S = P V /P sat allowed Sinha et al. [START_REF] Sinha | Argon nucleation in a cryogenic supersonic nozzle[END_REF] to estimate the number n * of atoms in the critical nucleus using the nucleation theorem: n * = ∂(ln J)/∂(ln S)| T [START_REF] Kashchiev | Nucleation work, surface tension, and gibbs-tolman length for nucleus of any size[END_REF]. From this estimate, it is possible to derive α as α = n * /n * CNT . The corresponding values are quite consistent with our measurement for nitrogen bubbles, as shown in fig. 6 .

Generalized Tolman length

In order to determine the curvature dependence of σ for nitrogen, we use the fact that α can be related to R e , the radius of the equimolar dividing surface (EDS) defined such that there is no excess mass associated with the interface. The EDS is different from the surface of tension (of radius R s ), though one expects that both surfaces lay in the region of the atomistically diffuse interface. Similarly to other work on nucleation [START_REF] Azouzi | A coherent picture of water at extreme negative pressure[END_REF][START_REF] Sinha | Argon nucleation in a cryogenic supersonic nozzle[END_REF][START_REF] Kim | Isothermal nucleation rates in supersonic nozzles and the properties of small water clusters[END_REF], we obtain the equimolar volume V * e of the critical nucleus through the nucleation theorem which can be written as: [START_REF] Kashchiev | Nucleation work, surface tension, and gibbs-tolman length for nucleus of any size[END_REF] V

* e = ∂E b ∂∆P T (10)
where ∆P = P l -P v . Applied to a spherical critical bubble of radius R e , this leads to:

4π 3 |R * e | 3 = α 1 1 -ρ v /ρ l ( 11
)
where ρ v and ρ l are the density of the bulk vapour and liquid phases at the cavitation threshold.

It is customary to call δ the (algebraic) distance between the two surfaces:

δ = R * e -R * s ( 12
)
δ, rather than δ ∞ , is called the Tolman length by some authors. In order to avoid any confusion, we will call below δ the generalized Tolman length. The Tolman length δ ∞ is properly defined as the asymptotic value of δ in the limit of zero curvature. As conjectured by Tolman [START_REF] Tolman | The effect of droplet size on surface tension[END_REF], the natural assumption, which should be valid at least at large radius, is to consider δ as a constant: δ ≃ δ ∞ . Equation 2 is valid only at this level of approximation.

One can thus probe the validity of this first order approximation by computing independently δ from R * e and R * s on the one hand, and, using equation 2, δ ∞ from σ and R * s on the other. Although both parameters are found to have the same order of magnitude (about -0.3 d LJ ), δ is significantly smaller than δ ∞ (see Fig. S5 in Supporting Information). This implies that δ cannot be considered as a constant and σ cannot be described by a first order expansion in curvature. This conclusion makes sense in view of the small radii of cavitation bubbles (2.5 -10 d LJ ) in the temperature range where α is measured.

CURVATURE DEPENDENCE OF THE SURFACE TENSION

Our results show the necessity to go beyond the above first order approximation, and to allow for a dependence of δ on curvature. Simulations suggest the following dependence [START_REF] Tröster | Numerical approaches to determine the interface tension of curved interfaces from free energy calculations[END_REF]:

δ = δ ∞ + c R s (13)
The corresponding expansion of the surface tension reads [START_REF] Baidakov | Spontaneous cavitation in a lennard-jones liquid at negative pressures[END_REF]:

σ ∞ σ(R) = 1 + 2δ ∞ R s + δ 2 ∞ + c R 2 s ( 14
)
Following Bruot and Caupin [START_REF] Bruot | Curvature dependence of the liquid-vapor surface tension beyond the tolman approximation[END_REF], it is straightforward to compute the two parameters δ ∞ and c from the knowledge of the energy barrier E * b and the radius R e (we refer the reader to Ref. [START_REF] Bruot | Curvature dependence of the liquid-vapor surface tension beyond the tolman approximation[END_REF] for the details of the algebra).

The second order expansion is more customarily written in a Helfrich form [START_REF] Van Giessen | Determination of curvature corrections to the surface tension of a liquid-vapor interface through molecular dynamics simulations[END_REF][START_REF] Wilhelmsen | Evaluation of finite-size effects in cavitation and droplet formation[END_REF][START_REF] Rehner | Surface tension of droplets and tolman lengths of real substances and mixtures from density functional theory[END_REF][START_REF] Block | Curvature dependence of surface free energy of liquid drops and bubbles: A simulation study[END_REF]:

σ(R) = σ ∞ 1 - 2δ ∞ R s + κ R 2 s ( 15
)
where κ is the average curvature-elastic modulus, involving both the bending rigidity and the rigidity constant associated with Gaussian curvature. Identification of eqs. 14 and 15 yields κ = σ ∞ (3δ 2 ∞ -c). The results for nitrogen are summarised in Fig. 7 where we compare the asymptotic Tolman length δ ∞ and the rigidity constant κ as a function of temperature to the predictions of two different DFT calculations for a Lennard-Jones fluid. In the middle of the temperature range, we find δ ∞ /d LJ ≃ -0.1 and κ/ϵ LJ ≃ -0.8, in reasonable agreement with the DFT calculations (ϵ LJ = 98k B is the LJ characteristic energy scale ). Within the experimental uncertainty, κ is constant, consistent with the DFT calculations. On the other hand, |δ ∞ | seems to increase with temperature in the whole experimental range, a behavior not expected from the DFT.

We also applied the same analysis to the data that we obtained earlier with hexane at room temperature [START_REF] Doebele | Direct observation of homogeneous cavitation in nanopores[END_REF]. We find δ ∞ /d LJ ≃ -0.15 and κ/ϵ LJ ≃ -0.9, very close to the parameters for nitrogen at the same normalised temperature T /T C = 0.57 (with d LJ = 0.54 nm and ϵ LJ = 400k B for hexane). Bottom: normalised rigidity constant κ/ϵLJ. Lines are the results of DFT calculations: full line from ref. [START_REF] Wilhelmsen | Evaluation of finite-size effects in cavitation and droplet formation[END_REF], dashed line from ref. [START_REF] Rehner | Surface tension of droplets and tolman lengths of real substances and mixtures from density functional theory[END_REF].

Returning to the surface tension, it turns out that the first and second order contributions in eq. 14 are of similar magnitude over the whole explored temperature range (see Fig. S6 in Supporting Information). Thus, the simple functional form proposed by Tolman (σ ∝ 1 1+C t /R ) fails to capture the curvature dependence of the surface tension. This seems to be a general feature for nuclei of nanometer size, as the same conclusion has been drawn for ordinary liquids (ethanol, heptane and water) at room temperature [START_REF] Bruot | Curvature dependence of the liquid-vapor surface tension beyond the tolman approximation[END_REF].

CONCLUSION

As a first practical remark, this work demonstrates that continuous measurements of evaporation in a porous material are a powerful tool to investigate cavitation statistics. Here, we have focused on simple fluids, but our technique can be applied to any volatile fluid.

Using nitrogen and helium, we have shown that the Classical Nucleation Theory correctly predicts the cavitation threshold as well as the parameter α = (1/k b T ) (∂E b /∂P l | T ) when the critical bubbles are large enough, that is at high temperature. At low temperature, for a bubble radius smaller than about 10 times the molecular size, the simple CNT approach is still valid if, instead of the bulk surface tension σ ∞ , one uses an effective surface tension σ(R). In the range R/d LJ < 10, we find that an expansion up to second order in curvature is necessary to account for the experimental results. Presently, the measured values of the Tolman length and rigidity are consistent with DFT, but the accuracy of both experiment and calculation is still not high enough to elucidate what happens moderately close to the critical temperature (T /T C ∼ 0.9).

To summarize, a macroscopic description of critical cavitation bubbles is quantitatively valid down to the nanometer scale if the surface tension is corrected for the interface curvature. Beyond cavitation, our results give a firmer basis to the concept of an effective surface tension σ(R), and suggest that the simple and powerful tools of capillarity can be used in any situations where the liquid-vapour meniscus is highly curved.

MATERIALS AND METHODS

Samples

Details of the synthesis of the samples can be found in Ref. [START_REF] Doebele | Direct observation of homogeneous cavitation in nanopores[END_REF] for porous alumina samples and Ref. [START_REF] Bossert | Evaporation process in porous silicon: Cavitation vs pore blocking[END_REF] for porous silicon samples. In porous alumina samples, the cylindrical pores are independent with a narrow distribution in diameter. In porous silicon, recent studies [START_REF] Bossert | Evaporation process in porous silicon: Cavitation vs pore blocking[END_REF][START_REF] Kondrashova | Scale-dependent diffusion anisotropy in nanoporous silicon[END_REF][START_REF] Thelen | Laser-excited elastic guided waves reveal the complex mechanics of nanoporous silicon[END_REF] have shown that the pore network is more disordered: the pores are still aligned along the etching direction, but constrictions are present along the pores, as well as connections between neighbouring pores. The pores size distribution is also quite wide: for the sample used in this work, the mean transverse dimension < d >= 50 nm, with 23 < d < 80 nm. For nitrogen experiments, we report only measurements where P * V is independent on the pore diameter, so that the fluid-wall interaction is not relevant and the cavitation homogeneous. In practice, we find that this limit is reached when the radius is of the critical nucleus R * s is smaller than about one fifth of the pore diameter (the maximum value R max s of R * s is given in Table 1). This limit is also consistent with an earlier study of cavitation in porous silica [START_REF] Rasmussen | Cavitation in Metastable Liquid Nitrogen Confined to Nanoscale Pores[END_REF].

For helium, the data discussed in this paper are restricted to temperatures smaller than 4.5 K, corresponding to a R * s smaller than 3.5 nm. Based on the theoretical effect of confinement [START_REF] Bonnet | Thermally activated condensation and evaporation in cylindrical pores[END_REF], this ensures a negligible influence of confinement on cavitation.

Cell for optical measurements with nitrogen

In contrast to standard volumetric setups, the dead volume around the sample in the optical cell is very large, about 50 cc. This is not an issue because interferometry only probes the fluid content in the pores. Moreover, it has two important advantages: first, it allows to reach the required low pressure rates with standard flowmeters (Brooks 5850 series). Second, it makes the relaxation experiments at a fixed pressure straightforward: we only need to close the cell once the desired pressure is reached since the subsequent liquid evaporation from the porous sample causes a negligible pressure increase (Fig. 4 B).

Volumetric measurements with helium

Helium cavitation is studied by connecting the experimental cell filled with liquid helium to a vacuum pump through a controlable microvalve, and measuring the cell pressure using a room-temperature pressure gauge connected to the cell through a separate 0.2 mm diameter capillary. Thermomolecular effects were computed to be negligible in the range of temperatures and pressures studied, so that the pressure read by the gauge equals the cell pressure. The cavitation isotherm is obtained from the time dependence of this pressure, and an analysis similar to that used above for nitrogen yields the pressure dependence of the cavitation rate (see Supporting Information).

Calculation of the liquid pressure P l For a vapor pressure P v in the vapor reservoir outside the pores, the pressure P l of the metastable liquid in the pores is calculated assuming that the chemical potentials of the two phases are equal. The chemical potential of the vapour is calculated using the NIST values for fugacity (for nitrogen) or by numerical integration over pressure of the NIST volume data (for helium). The chemical potential of the metastable liquid phase is calculated assuming that the liquid compressibility is equal to the compressibility at saturated vapour pressure. This is consistent with the fact that nucleation occurs far from the spinodal. In principle, the measurement of the optical thick-ness L on the saturation plateau of the isotherm could be used to determine the metastable liquid compressibility for nitrogen, and to check its sensitivity to the confinement [START_REF] Dobrzanski | Elastic properties of confined fluids from molecular modeling to ultrasonic experiments on porous solids[END_REF]. However, because we cannot preclude that the L variation is partially due to the emptying of the larger pores, we can only obtain an upper bound for the compressibility (see Supporting Information). This upper bound is enough to estimate the uncertainty on P l .

For both nitrogen and helium, the typical effect on P l of the liquid compressibility is less than 3 %, with an uncertainty due to the assumption of a constant compressibility of about 1%.

Equation for Ψ(P l )

Above the cavitation step, the variation dN r of the number of filled pores due to evaporation by meniscus recession in the pressure interval dP v is dN r = a v dP v , the coefficient a v being proportional to the slope of the isotherm (red dotted line in Fig. 5). As the cavitation step is very sharp, one can assume that, if cavitation would not occur, the number of pores N ′ (P v ) remaining filled would be the extrapolation of the behaviour above the cavitation step, hence follows the red dotted line, so that dN ′ = a v dP v , or in P l variable, dN ′ = a l dP l .

If we now include cavitation, the actual number of filled pores is N = ΨN ′ < N ′ . If recession and cavitation are independent processes, the actual number of pores emptying trough recession is simply decreased by a factor Ψ: dN r = Ψ(a l dP l ). The total number dN of pores emptying through cavitation and recession is thus:

dN = J(P l )V P N A + a l Ψ dP l (16)
Using this equation to compute the derivative of Ψ = N/N ′ yields equation 9. Note that the identity with the equation for Σ implies that Σ=1/2 for Ψ=1/2, i.e. the point at mid-height of the cavitation step.

FIG. 1 .

 1 FIG.1. Nitrogen optical isotherm at 90 K for the porous alumina sample A (mean pore diameter 33 nm, see tableI). Upon decreasing the pressure, meniscii start to recede in the larger apertures at PV ≃ 0.8Psat, until cavitation occurs around 0.67Psat. Inset: sketch of meniscus receding (right) and cavitation (left) in an ink-bottle pore.

FIG. 2 .

 2 FIG. 2. Nitrogen optical isotherms for the porous alumina sample B (mean pore diameter 19 nm, see tab. I). The cavitation pressure and the height of the cavitation step increase with temperature.

Figure 3

 3 Figure 3 exhibits two important features. First, the surface tension equals its bulk value for large critical bubbles (|R * s |/d LJ > 10); this implies that the CNT is valid in this regime and that the evaluation of J 0 is correct. Second, for small bubbles, σ becomes significantly smaller than σ ∞ , in agreeement with expectation. For nitrogen, the relative deviation reaches -0.2 for |R * s |/d LJ ≃ 3, and -0.15 for helium for |R * s |/d LJ ≃ 5. Data from earlier cavitation experiments by Baidakov and co-workers in superheated argon[START_REF] Vinogradov | Explosive cavitation in superheated liquid argon[END_REF] and nitrogen[START_REF] Baidakov | Homogeneous nucleation in liquid nitrogen at negative pressures[END_REF] can be analysed in the same way and yield a decrease of σ for small nucleus consistent with our measurements, though with a much larger scatter (see Fig.S7in Supporting Information).

FIG. 3 .

 3 FIG. 3. Normalised surface tension as a function of the normalised bubble curvature |R * s |/dLJ. Data for helium are measured above the superfluid transition. dLJ for helium and nitrogen are given in the text. The temperature range is 65-120 K for nitrogen and 2.3-4.5 K for helium. The uncertainty on σ∞ is about 1% and is not taken into account in the evaluation of the error bars (see Supporting Information).

FIG. 4 .

 4 FIG. 4. (A) Optical isotherms measured in nitrogen at 73 K on the porous alumina sample C (see tab. I) during quenching at various pressures. The total measurement time is about 48 h. (B) Close-up of the isotherms showing the relaxations at constant pressures 239.7, 240.36, 241.07, 241.86, 242.41, 243.24, 244.04, 245.51 mbar. (C) Relaxation of ∆L -∆L0 at constant pressures (same values as in B). ∆L0 stands for the optical thickness of empty pores (∆L0 ≃ 0.12 µm, a small variation is allowed from run to run to account for the small long term drift). (D) Nucleation rate J as a function of the liquid pressure: from the relaxation time measured in fig. C (squares -the size of the symbols sets the uncertainty) and from the shape of the isotherm measured at constant pressure rate (line).

FIG. 5 .

 5 FIG. 5. Fraction Ψ of pores that have not cavitated at pressure Pl: experimental points (crosses) and fit by the double exponential of the equation 4 (red solid curve). Inset: close-up of of the cavitation step of fig. 1. The dotted curves represent the signal ∆L0 for "empty" pores (blue) and the extrapolated signal (red) that would be observed without cavitation.

FIG. 6 .

 6 FIG. 6. Plot of α = 1 kbT ∂Eb ∂Pl T

FIG. 7 .

 7 FIG.7. Top : normalised asymptotic Tolman length δ∞/dLJ. Bottom: normalised rigidity constant κ/ϵLJ. Lines are the results of DFT calculations: full line from ref.[START_REF] Wilhelmsen | Evaluation of finite-size effects in cavitation and droplet formation[END_REF], dashed line from ref.[START_REF] Rehner | Surface tension of droplets and tolman lengths of real substances and mixtures from density functional theory[END_REF].

TABLE I .

 I Characteristics of porous samples.

	sample	mean pore diameter pore lenth R max s
		(nm)	(µm)	(nm)
	porous silicon	50	2	5.5
	porous alumina -A	33	5.4	2.8
	porous alumina -B	19	7	2.2
	porous alumina -C	11	8	1.1
	porous alumina -D	36	85	3.5

Reviews 8, 021317 (2021).

ACKOWLEDGMENTS

The authors thank J. Puibasset for stimulating discussions and careful reading of the manuscript, and F. Souris for having developed the experimental cell for helium. This work, including the funding of MB, has been financially supported by Agence Nationale de la Recherche through the project CavConf, ANR-17-CE30-0002.


domain: spi phys

text:
Lauren Schatz

Jared R Males

Carlos Correia

Benoit Neichel

Vincent Chambouleyron

Johanan Codona

Olivier Fauvarque

Jean-François Sauvage

Thierry Fusco

Michael Hart

Schatz Lauren

Correia Carlos

Neichel Benoit

Chambouleyron Vincent

Codona Johanan

Fauvarque Olivier

Sauvage Jean-François

Fusco Thierry

Hart Michael

Janin-Potiron Pierre

Johnson Robert

Joseph D Long

Mateen Mala

Three-sided pyramid wavefront sensor, part 1: simulations and analysis for astronomical adaptive optics

Keywords: Adaptive optics, Wavefronts, Wavefront sensors, Modulation, Telescopes, Stars, sensors

come

Introduction

Within the next decade, the world will see a new generation of Giant Segmented Mirror Telescopes (GSMTs). The Giant Magellan Telescope 1 (GMT) under construction at Las Campanas Observatory, Chile, will have seven 8.4-meter mirrors, forming a 24.5-meter primary mirror. The Thirty Meter Telescope 2 (TMT), and the European Extremely Large Telescope 3 (E-ELT) at Cerro Paranal, Chile, have highly segmented primary mirrors. The 39-meter E-ELT primary mirror will be comprised of hundreds of 1.4-meter hexagonal segments. The TMT will have 492 [START_REF] Sanders | The thirty meter telescope (tmt): An international observatory[END_REF] 1.44-meter hexagonal segments to form the 30-meter primary mirror. The GSMTs in combination with extreme adaptive optics (ExAO) systems will have the angular resolution and light-collecting power to image Earth-like planets around M-dwarf stars. [START_REF] Males | GMagAO-X: extreme adaptive optics & coronagraphy for gmt at first light[END_REF] There are several ExAO instruments in operation on current generation telescopes. Current instruments include the Subaru Coronagraphic Extreme Adaptive Optics instrument (SCExAO), [START_REF] Jovanovic | The subaru coronagraphic extreme adaptive optics system: enabling high-contrast imaging on solar-system scales[END_REF] the Spectro-Polarimetric High-contrast Exoplanet Research instrument (SPHERE), [START_REF] Beuzit | Sphere: a planet finder instrument for the vlt[END_REF] the Gemini Planet Imager (GPI), [START_REF] Macintosh | First light of the gemini planet imager[END_REF] the Keck Planet Imager and Characterizer (KPIC), [START_REF] Jovanovic | The keck planet imager and characterizer: demonstrating advanced exoplanet characterization techniques for future extremely large telescopes[END_REF] and most recently the Magellan Extreme Adaptive Optics System (MagAO-X). [START_REF] Males | MagAO-X first light[END_REF] All of these instruments have common features: a high actuator count deformable mirror running at extreme speeds (at least 1kHz), a high-performance wavefront sensor (WFS), and a high-contrast coronagraph. These instruments are pathfinders for ExAO systems on the GSMTs.

Detector size, speed, and noise all impose limits on WFS performance. The sampling of the wavefront limits the number of modes for which the adaptive optics (AO) system can correct.

The read-out speed of the detector sets an upper limit on the speed of the AO loop. Detector noise degrades the measurement of the wavefront error, forcing slower loop speeds to compensate, resulting in poor correction when using faint guide stars. Higher wavefront sampling requires a larger format detector which takes longer to readout. This trade-off motivates the need for a new wavefront sensor architecture to build first-light ExAO instruments for the GSMTs.

The GSMTs all plan to use pyramid wavefront sensors (PWFS) in ExAO instruments. The PWFS acts like a Foucault test in 2 dimensions. Light from the telescope is focused onto a glass pyramid tip where it is split and then the pupil plane is reimaged onto a detector. The result is copies of the telescope pupil that contain intensity fluctuations that are related to the wavefront phase. All current PWFS on telescopes use a four-sided pyramid (4PWFS), resulting in four pupil copies. The Planetary Systems Imager [START_REF] Fitzgerald | The planetary systems imager for TMT[END_REF] for the TMT will use a non-modulated PWFS [START_REF] Guyon | Wavefront control architecture and expected performance for the TMT planetary systems imager[END_REF] in combination with lower-order wavefront control to reach and maintain high contrast. The Multi-AO Imaging Camera for Deep Observations (MICADO) [START_REF] Davies | The MICADO first light imager for the ELT: overview, operation, simulation[END_REF] for the E-ELT is a pathfinder instrument for performing high contrast imaging on GSMTs that uses a PWFS. The High Angular Resolution Monolithic Optical and Near-infrared Integral field spectrograph (HARMONI), [START_REF] Neichel | The adaptive optics modes for harmoni: from classical to laser assisted tomographic ao[END_REF] also for the E-ELT, has a PWFS in the single-conjugate adaptive optics (SCAO) mode to deliver diffraction-limited performance for the E-ELT's core spectroscopic capability. The Giant Magellan Extreme Adaptive Optics System (GMagAO-X) [START_REF] Males | GMagAO-X: extreme adaptive optics & coronagraphy for gmt at first light[END_REF] is being developed as a first light ExAO instrument for the GMT and will use a PWFS.

As an alternative to the conventional 4PWFS, we are exploring a three-sided PWFS (3PWFS) as a GSMT-ExAO wavefront sensor to reduce the required number of detector pixels. The 3PWFS uses fewer detector pixels than the 4PWFS, and therefore should be less sensitive to read noise. However, the 3PWFS has not been well studied, and no 3PWFS has been tested in a closed-loop adaptive optics system. The goal of this paper is to analyze the refractive three-sided PWFS and demonstrate its operation in simulations. We start by deriving a Slopes calculation for 3PWFS signals and test the validity in a real adaptive optics closed-loop system on the LOOPS [START_REF] Janin-Potiron | Adaptive optics with programmable fourier-based wavefront sensors: a spatial light modulator approach to the loops testbed[END_REF] test bench at the Laboratoire d'Astrophysique de Marseille. Then we use the diffraction theory of the Foucault knife edge test to predict the expected signal and sensitivity of the PWFS to Fourier modes. Finally, we use the Object Oriented MATLAB Adaptive Optics toolbox (OOMAO) [START_REF] Conan | Object-oriented matlab adaptive optics toolbox[END_REF] to simulate an end-to-end model of an adaptive optics system using a PWFS with modulation and compare the performance of the 3PWFS to the 4PWFS.

The Pyramid Wavefront Sensor

The optical design of the PWFS consists of four components: a focusing optic, the glass pyramid, and a relay lens to image the pupils on the detector. [START_REF] Ragazzoni | A pyramid wavefront sensor with no dynamic modulation[END_REF] Figure 1 shows a schematic of the operation of a PWFS. The starlight is first brought to a focus on the tip of a glass pyramid that splits the focal plane into parts. The apex angle of the pyramid imparts a tilt to separate each of the sections.

The result is copies of the telescope pupil that are separated spatially on a detector. The number of pixels across each pupil determines the number of modes the instrument is sensitive to. Current on-sky adaptive optics systems that have a PWFS use a four-sided PWFS. The Magellan Adaptive Optics System (MagAO), [START_REF] Close | Status of MagAO and review of astro-nomical science with visible light adaptive optics[END_REF] the Large Binocular Telescope Interferometer (LBTI), [START_REF] Esposito | Adaptive optics at the lbt telescope: from ngs to lgs and interferometry[END_REF] and MagAO-X use an achromatic double four-sided pyramid. The SCExAO instrument uses two crossed roof prisms as its pyramid optic. The PWFS is highly sensitive but suffers from a low dynamic range. A modulator is used to increase the linear range of the PWFS. Modulation is achieved by oscillating a piezo-actuator-driven mirror to drive the PSF on the pyramid tip into a circular pattern with a radius quoted in units of λ/D, where λ is the wavelength, and D is the diameter of the entrance pupil. This increases the effective spot size on the pyramid tip which increases the sensor's linearity at the cost of sensitivity. [START_REF] Guyon | Limits of adaptive optics for high-contrast imaging[END_REF]

Fig 1

The optical design of a PWFS. Light from the telescope is focused onto a glass pyramid tip where it is split and then the pupil plane is reimaged onto a detector. The result is copies of the telescope pupil that contain intensity fluctuations that are related to the wavefront phase. [START_REF] Tokovinin | Ao tutorial at ctio[END_REF] The intensity measured on the PWFS detector must be processed to recover only the intensity signal due to the wavefront phase error. First, the image is dark subtracted and gain corrected to remove detector artifacts. A threshold mask is then applied to mask out any pixels that do not contain a signal. The wavefront sensor response to a flat wavefront is subtracted off to leave only the intensity pattern in each pupil caused by the wavefront phase error. There are two ways we consider handling this intensity to reconstruct the wavefront. The first is the Raw Intensity method (RI), in which the intensity due to only the phase error is used as-is. The second method is the Slopes Maps (SM) method where the intensity in each pupil is combined to estimate the sine of the wavefront derivative, or slope. [START_REF] Vérinaud | On the nature of the measurements provided by a pyramid wave-front sensor[END_REF] The equation for the 4PWFS is,

S x = I 1 + I 2 -I 3 -I 4 I 1 + I 2 + I 3 + I 4 (1)
S y = I 1 -I 2 -I 3 + I 4 I 1 + I 2 + I 3 + I 4
where S x , S y are the local wavefront slopes in the x and y direction, and I 1 ...I 4 are the intensity values of the pixel corresponding to the same location in each pupil. A better way to normalize the S x and S y slopes it to normalize by the mean value across all valid pixels on the wavefront sensor detector, instead of normalizing pixel by pixel. This method gives a better slope estimate in low-light conditions and when the beacon brightness is fluctuating. The intensity patterns in the pyramid pupils contain both the X and Y spatial information of wavefront gradient from the diffraction off of the pyramid edges. The RI signal processing method uses the pyramid signal as-is. The benefit of the Raw Intensity method is that it is relatively unaffected by any errors in aligning the pyramid pupils on the detector of a real system. In the Raw Intensity method, each pupil pixel is considered independently in the reconstruction of the wavefront, whereas in the Slopes Maps method a single pixel is referenced to pixels in the same location in the other pupils. Misregistration of the pixels in the Slopes Method results in uncorrected phase errors and so a tighter tolerance on the optical alignment is needed. The major challenge in using the Raw Intensity technique is obtaining a good flat wavefront reference image. The signals in the pyramid pupils are then extracted into a single column vector of intensity values.

Performance Comparison

All current PWFSs on telescopes use a 4PWFS. ExAO systems need a high sampling of the PWFS pupils to optimize performance, and as a result, require larger detectors. We are interested in exploring the 3PWFS as an alternative PWFS for an ELT-ExAO wavefront sensor to overcome these limitations. The 3PWFS only has three copies of the pupil and therefore uses fewer detector pixels than the 4PWFS and should be less sensitive to read noise. This will result in increased SNR for a detection that is read noise dominated. The SNR equation for a CCD camera used for astronomical measurements is, 23

SN R = R × t (R × t) + (RN 2 × n pix ) + n pix (N s + N D ) ( 2
)
where R is the photon arrival rate, t is the integration time, RN is the read noise, and n pix is the number of pixels used for wavefront sensing. The terms N s and N D refer to the sources of noise from sky background and dark current.

Using equation 2 we calculate the SNR for the 3PWFS and 4PWFS based on guide star magnitude. In Figure 2 the SNR for the 3PWFS and 4PWFS was calculated for different guide star magnitudes, for a 24.5 meter primary mirror, and 0.001 second integration time. The number of pixels in each pupil was assumed to be 20,000. The 3PWFS used a total of 60,000 pixels, and for the 4PWFS 80,000 pixels were used. Taking the ratio of the SNR curves shows the higher SNR from the 3PWFS in low light conditions. In bright light conditions, the dominant error is photon noise, and the SNR for both the 3PWFS and 4PWFS is the same. As the light level dims and read noise starts to dominate, the SNR of the 3PWFS measurement is higher than the 4PWFS and the ratio converges to a value of 4 3 .

The 3PWFS pupils do not have a natural alignment geometry on a pixel grid detector. The misregistration between pixels in the same location of each of the pupils degrades the performance of the 3PWFS correction when using the Slopes Maps method. The 3PWFS additionally has the potential for a larger null space than the 4PWFS as the wavefront is reconstructed using three points of measurement instead of four. To assess the performance of the 3PWFS compared to the 4PWFS, a simulation was developed using the Object Oriented Matlab Adaptive Optics toolkit (OOMAO). [START_REF] Conan | Object-oriented matlab adaptive optics toolbox[END_REF] The OOMAO toolkit is an end-to-end adaptive optics model that can simulate dif- ferent combinations of guide stars, turbulent atmospheres, wavefront sensors, deformable mirrors, and science cameras. Light is propagated using Fraunhofer diffraction. The PWFS is simulated using a single tip/tilt phase mask that is segmented into N parts. Figure 3.A and 3.C show the masks for the 3PWFS and 4PWFS. The masks are scaled and rotated according to a user input for rotation and pyramid apex angle that controls the separation of the pyramid pupils. After scaling, the mask is converted into a phase mask that is applied at the focal plane to simulate a pyramid tip. using Slopes Maps. We extended the PWFS class in OOMAO to include a 3PWFS, taking care that the amplitude of the tip/tilt phase in the 3PWFS phase screen generation matched that of 4PWFS.

We included our derivation of the Slopes Maps equation for the 3PWFS (derived in Section 3.1 below) and added the Raw Intensity method for both the 4PWFS and the 3PWFS. For both the Slopes Maps and Raw Intensity method the signal was normalized by the mean value across all valid pixels on the wavefront sensor detector, instead of normalizing pixel by pixel.

The threshold mask that determines which pixels are used for wavefront sensing is determined by a calibration step that generates a PWFS with high modulation and a propagated wavefront that contains no phase error. The result is highly uniform pupils on the detector. The detector image is thresholded to mask out any pixels with a low signal. The thresholded image is then converted into a binary mask, where the pixels that contain the pupil signals have a value of 1, and all other pixels have a value of 0. This mask is common to both the SM and RI methods, assuring that in both cases the same pixels are used for wavefront sensing.

3PWFS Slopes Maps

Derivation

The Slopes Maps technique was derived for the 4PWFS using the same intensity centroid calculation as a Shack-Hartmann wavefront sensor. The Shack-Hartmann uses a quad-cell intensity calculation to track the movement of spots to calculate the X and Y gradient of the wavefront phase in post-processing. In the geometric optics approximation, the PWFS acts as a slope sensor, and the Slopes Maps calculates the wavefront gradients in a similar way to the Shack-Hartmann. In the diffractive optics derivation, the wavefront slopes are calculated by the diffraction of the pyramid edges and encoded directly into the intensity pattern. We do not need to perform a Slopes Maps calculation for the pyramid signal because we are directly measuring a function that is related to the gradient phase error, but it is still beneficial to do so. The Slopes Maps is the natural recombination of the pyramid signals that subtracts off the constant Intensity pattern, so a reference image is not required. In a closed-loop system, the wavefront signal is driven towards zero slopes which arises when the PSF is centered on the pyramid tip with no phase errors. The Slopes Maps technique suffers from misalignments of the pyramid pupils; any misregistration in the sampling of the pupils with respect to each other results in a loss of performance.

The Slopes Maps for the 4PWFS are already known. In previous work by Costa, [START_REF] Büchler De Matos | Development of a new Infrared Pyramid Wavefront Sensor[END_REF] a Slopes Map calculation for the 3PWFS was derived assuming a geometric optics approximation of the PWFS as a derivative wavefront sensor in the modulation regime. In this paper, we introduce a new Slope Maps method to handling the 3PWFS signals, which is derived using the intensity centroid of an equilateral triangle.

To calculate the Slopes Maps equations for the 3PWFS, we start from the same assumptions used for the 4PWFS. We assume that the maximum sensitivity of the sensor occurs when the PSF is centered at the tip of the four pyramid edges. We seek a Slopes Maps equation that drives the wavefront to zero slope, resulting in uniform intensity in the re-imaged pupils. To derive the centroid equation for the 3PWFS we start with an equilateral triangle, shown in Figure 4. The center of the triangle is at the origin of a Cartesian coordinate system, and the vertices of the triangle are all an equal distance from the origin. The blue circles represent the layout of the pupils on the PWFS detector, and I 1 corresponds to (x 1 , y 1 ), etc. Solving for the coordinates (x 1 , y 1 ), (x 2 , y 2 ), (x 3 , y 3 ) gives the weights in the S x and S y Slopes Maps.

For the 3PWFS we find that the S x and S y Slopes Maps are,

S x = √ 3 2 I 2 - √ 3 2 I 3 I 1 + I 2 + I 3 (3)
S y = I 1 -1 2 I 2 -1 2 I 3 I 1 + I 2 + I 3
The S x slopes only depends on the intensities from the I 2 and I 3 pupil pixels which act as a roof sensor. The result is that care must be taken to properly index the pupils when applying the Slopes Map equation.

Closed-Loop Testbed Verification of the 3PWFS

To test the validity of the Slopes Maps equation for the 3PWFS, and the Raw Intensity method we implemented these signal processing methods into a real closed adaptive optics system using the LOOPS testbed at the Laboratoire d'Astrophysique de Marseille (LAM). The source for the LOOPs is a HeNe laser attenuated by a continuously variable gradient ND filter wheel. There are multiple paths through the loops testbed created by beamsplitters. Different branches can be blocked or passed to create a path where only flat mirrors are illuminated to record a reference PSF or to illuminate only the deformable mirror or phase screen. The phase screen is a milled reflective mirror and simulates a D/r 0 of about 28. The wind speed of the turbulence can be changed by speeding up or slowing down a stepper motor that rotates the phase screen. An ALPAO 69 actuator deformable mirror is used for the closed-loop correction. A phase screen applied on a spatial light modulator (SLM) is used to create the pyramid tip. The wavefront sensing camera is an OCAM 2 K used in single-pixel mode. Each pyramid pupil is 80 pixels in diameter. A full description of the LOOPS testbed can be found in a previous paper by Janin et al. [START_REF] Janin-Potiron | Adaptive optics with programmable fourier-based wavefront sensors: a spatial light modulator approach to the loops testbed[END_REF] On the testbed, a closed-loop correction was achieved for both a 3PWFS and 4PWFS with reconstructors calculated from both the Raw Intensity and Slopes Maps signal handling techniques.

The loop was run at 250Hz, with a modulation radius of 5 λ/D. In all cases, a stable closed-loop was obtained. Figure 5 The PWFS is an extension of the knife edge test: each of the pyramid pupils is the signal from a knife edge test, but includes a mirroring effect of the signal across the facets. The underlying physical processes of the pyramid and the knife edge are the same, so we can use the equations that describe the knife edge test to explore the operation of the PWFS. The diffraction theory of the PWFS has been studied in detail. Vérinaud [START_REF] Vérinaud | On the nature of the measurements provided by a pyramid wave-front sensor[END_REF] extends the knife edge diffraction theory to model the signal of a roof sensor with dynamic circular modulation. Hutterer et. al. [START_REF] Hutterer | Real-time adaptive optics with pyramid wavefront sensors: part i. a theoretical analysis of the pyramid sensor model[END_REF] extends the knife edge analysis into general operators such that modulations of any pattern can be modeled. Shatokhina et. al. [START_REF] Shatokhina | Fast algorithm for wavefront reconstruction in xao/scao with pyramid wavefront sensor[END_REF] presents simplified equations to approximate pyramid signals with and without modulation.

Correia et. al., [START_REF] Correia | Performance limits of adaptive-optics/highcontrast imagers with pyramid wavefront sensors[END_REF] and Fauvarque et. al. [START_REF] Fauvarque | Kernel formalism applied to fourier-based wave-front sensing in presence of residual phases[END_REF] model the pyramid operation as a Fourier filter. In this paper, we use a formalism similar to Vérinaud and limit our analysis to that of a one-dimensional knife edge test for simplicity. We use the knife edge analysis to derive the relationship of the PWFS sensitivity to Fourier modes. Fourier modes have a direct relationship to locations on the focal plane; which is important for coronagraphy because we are trying to maximize the contrast in the dark hole region created by the coronagraph.

In this section, we consolidate the diffraction theory of the knife edge test by Linfoot, [START_REF] Linfoot | On the theory of the zonal foucault test[END_REF] Kat-zoff, [START_REF] Katzoff | Quantitative determination of optical imperfections by mathematical analysis of the Foucault knife-edge test pattern[END_REF] and Wilson [START_REF] Wilson | Wavefront-error evaluation by mathematical analysis of experimental foucault-test data[END_REF] into a single derivation with uniform notation. These three papers build upon the same derivation, and hereafter we refer to the collection of papers as LKW. We then deviate from their derivation to link phase aberrations in the shape of Fourier modes to intensity patterns produced by the knife edge test. We assume a focal plane bisected by a binary amplitude mask representing the knife edge.

Figure 6 describes the steps taken in this derivation. The diagram is in two dimensions for visualization, but in this derivation, we assume a one dimensional case for simplicity. We start with an electric field in the entrance pupil, u 0 (x 0 ) that has a phase error given by cos(nx). A Fourier transform is taken to propagate the electric field to the focal plane where the knife edge, H f (ξ f ) is applied. We don't solve for U (ξ f ), the electric field in this focal plane, and instead, take an inverse Fourier transform to the conjugate pupil plane where the knife edge test signal is found.

The electric field at this pupil plane is given by u(x i ), is the convolution of the electric field in the entrance pupil propagated to this pupil plane, u 0 (x i ), and the inverse Fourier transform of the knife edge, h i (x i ). The phase error in the electric field is expanded in a power series, then linearized, and the modulus squared is taken to find the field intensity. By subtracting the constant term, we are left with the intensity pattern due to the phase errors only.

To start we define the electric field in the entrance pupil as u 0 (x 0 ). We take the Fourier transform of the electric field to get to the focal plane, where we will apply the knife edge. The Fourier transform is here defined as:

F{ξ x } = ∞ -∞ e -i2πxξx f (x)dx (4)
where ξ x is the spatial frequency corresponding to the spatial coordinate x. [START_REF] Gaskill | Linear systems, fourier transforms, and optics[END_REF] F{•} denotes the Fourier transform operator and F -1 {•} will denote its inverse. The equation of the electric field in the focal plane, U (ξ f ) is:

U (ξ f ) = U f (ξ f )H f (ξ f ) (5)
where,

H f (ξ f ) = 1/2 + 1/2sgn          1, for ξ f > 0 0, for ξ f < 0 (6)
is the binary mask function of the knife edge, and U f (ξ f ) is the Fourier transform of u 0 (x 0 ).

Taking the inverse Fourier transform brings us to the conjugate pupil plane. The inverse Fourier

transform of H f (ξ f ) is h i (x i ) = F -1 [H f (ξ f )] = 1 2 δ(-x i ) + i 2π 1 x i (7)
We take the results of Equation 7and convolve it with u 0 (x i ), which is the Fourier transform of U f (ξ f ) in Equation 5, to get the electric field at the pupil plane, u(x i ). The resulting equation is Equation 5a from Wilson. 31

u(x i ) = u 0 (x i ) i 2πx i + 1 2 δ(-x i ) = ∞ -∞ u 0 (x ) 1 2 δ(x -x i ) + i 2π 1 x i -x dx (8)
In high contrast imaging, we are interested in the effect of phase errors at different spatial frequencies as these correspond to the contrast at specific locations in the post-coronagraphic focal plane. To find the response of a knife edge to an error of the form cos(nx), where n is the spatial frequency, we need to modify the derivation by LKW to reach a form that allows us to easily compute the expected signal of the Foucault test from a phase error in the form of a Fourier mode.

We make two modifications to LKW: we assume the infinity pupil approximation, and consider only the linear component of the electric field. The assumptions made by LKW lead to an integral that describes the intensity signal of a knife edge test that is difficult to solve. Katzoff attempts a modified approach for solving the integral for phase errors in different forms. This approach did not produce a good match for the intensity signal from a phase error in the shape of a Fourier mode.

This leads us to revisit the derivation and modifying the assumptions to better reflect the knife edge approximation to the pyramid wavefront sensor. In the derivation by LKW the infinite limits of the integral are set arbitrarily to -1 to 1 since the factor dx /(x i -x ) is nondimensional. [START_REF] Katzoff | Quantitative determination of optical imperfections by mathematical analysis of the Foucault knife-edge test pattern[END_REF] They found that the intensity pattern given by the infinite approximation is valid for within the pupil boundaries but gave an infinite irradiance at locations on the pupil edge. We are only interested in the intensity pattern inside the pupil boundary. We assume an infinite aperture approximation and take the integral from -∞ to ∞. LKW expand the electric field and consider the quadratic component as well as the linear component of the electric field to the knife edge signal. We are interested in only the linear component of the knife edge signal. In our expansion of the electric field, we only consider the linear component and substitute the result into Equation 8.

The electric field at the entrance aperture is now:

u 0 (x i ) = exp(-2πim(x i )) = 1 -2πim(x i ). (9)
We substitute Equation 9into Equation 8 to find the electric field in the pupil plane u(x i ), which yields:

u(x i ) = 1 2 (1 -2πim(x i )) + i 2π ∞ -∞ (1 -2πim(x ))dx x i -x (10)
Expanding the integral, we have:

u(x i ) = 1 2 (1 -2πim(x i )) + i 2π ∞ -∞ dx x i -x + ∞ -∞ m(x )dx x i -x (11)
where the integrals are now in the form of a Hilbert transform. [START_REF] Villa | Foucault test: shadowgram modeling from the physical theory for quantitative evaluations[END_REF] The Hilbert transform is defined as:

H(y(t)) = -1 π P V ∞ ∞ y(t )dt t -t (12)
which has the property that the Hilbert transform of a constant is 0. [START_REF] Poularikas | Handbook of formulas and tables for signal processing[END_REF] In this equation P V stands for the principal value. The P V is used to evaluate integrals that have a discontinuity, in our case this is when x i = x . [START_REF] Johansson | The hilbert transform[END_REF] When the wavefront error is given by m(x) = cos(nx), Equation 10becomes,

u(x i ) = 1 2 (1 -2πi cos(nx i )) + ∞ -∞ cos(nx )dx x i -x (13)
To get the integral in the form of a Hilbert transform we multiply by π π . The Hilbert transform of H(cos(nx)) = cos(nx

+ π 2 ) = -sin(nx). ∞ -∞ cos(nx )dx x i -x = π * [ 1 π ∞ -∞ cos(nx )dx x i -x ] = -π sin(nx i ) (14)
The equation for the electric field in the pupil plane of the knife edge signal is,

u(x i ) = 1 2 -iπ cos(nx i ) -π sin(nx i ) (15)
and we can now take the modulus squared to get the intensity.

I(x i ) = 1 4 + π 2 -π sin(nx i ) (16)
Subtracting the constant terms which represent the intensity response I r (x i ) to a perfect wavefront, gives the intensity pattern due to the phase errors only. In this derivation, we find an intensity pattern that is proportional to -sin(nx) resulting from a phase error of the form cos(nx). In the next section, we demonstrate that this is the expected response.

The diffraction theory of the Foucault test predicts an intensity pattern that is the Hilbert transform of the wavefront phase. The result is an intensity pattern that is related to the wavefront derivative but is not quite the derivative. In the case of a derivative wavefront sensor such as the Shack-Hartmann, the operation of the wavefront sensor on the phase to signal measurement is d dx cos(nx) = -n sin(nx). In the case of the pyramid, the reference subtracted result is d dx cos(nx) = -sin(nx), which is similar to the derivative wavefront sensor but without the scaling by spatial frequency. The result is that the knife edge and therefore the unmodulated pyramid sensitivity does not depend on spatial frequency, because all spatial frequencies will produce an equally strong signal. In the Shack-Hartmann case, the strength of the signal scales linearly with spatial frequency.

Verification

The approximations we used to derive Equation 15 are supported by our simulation of the response of a PWFS to a cosine phase error. Using OOMAO we apply a Fourier mode as a phase error in the pupil plane, propagate through the pyramid onto the detector, and examine the resulting intensity pattern on the detector as well as the calculated slopes. The derivation in Section 4.1 is an approximation valid when the wavefront sensor is operating in the linear regime. We first consider the modulated case to ensure the linearity of the pyramid signal. We then consider the unmodulated pyramid to confirm that the derivation is still a good approximation of the unmodulated PWFS signal. The PWFS are under 5 λ D modulation to ensure that the pyramid response is linear for the amplitude of Fourier mode applied, and we are not including any noise. We apply one radian of phase error that is in the form of cos(3x) to both the modulated and unmodulated PWFS. The phase error in radians is shown in Figure 7, where Figure 7.A is the cos(3x) phase error, and Figure 7.B is a cross-section of the center function in the x-direction. We compare the resulting intensity patterns to our prediction in Equation 16. For an intensity pattern of the form cos(3x) shown in Figure 8.A, we would expect an intensity pattern in a pyramid pupil to be 1/4 + π 2 -π sin(3x) The resulting intensity patterns from the 3PWFS and 4PWFS are shown in Figure 9 A and B.

In Figure 9 C and D we take a cross-section from one of the pupils from each wavefront sensor and scale the signal. We then plot this against the predicted scaled intensity pattern. We find that the mathematical prediction is a good match for the measured intensity pattern inside the pupil of a modulated pyramid when the signal is linear.

To gain further clarity we use the Slopes Maps equation that subtracts off the constant intensity and turns the signal into a pure X and Y measurement of phase. The Slopes Maps naturally subtracts the flat wavefront response and in that way is self-referencing. More details on the Slopes calculation for the 3PWFS are in Section 3. The Slopes Maps for the 3PWFS and the 4PWFS are shown in Figure 10, as well as the cross-sections that are indeed a -sin(nx) function.

The previous simulations were performed with 5λ/D modulation. We can consider the 0 modulation case, which is a more direct relation to the results from the derivation. The unmodulated PWFS signal is more nonlinear than the modulated pyramid and therefore the fit between the prediction and the measured signal will be poor. In the case of the Slopes Maps signal, we found in simulation that the self-referencing is degraded from a nonlinear signal, and to return to a closer estimate of the signal the flat wavefront reference signal should be subtracted off. The fit worsens for the Slopes Maps without reference subtraction when there is a misregistration of the pupils, as will always be the case for the 3PWFS. For the 3PWFS subtracting a reference improved the mean 5 Performance Comparison of the 3PWFS and 4PWFS

Simulation

In Section 4 we derived the response of the Foucault knife edge to a Fourier mode. We then confirmed in simulation that both the 3PWFS and 4PWFS produce a signal that is well described by the predictions of our derivation. These signals are used to reconstruct the wavefront in an AO closed-loop correction. Using OOMAO an end-to-end simulation of an adaptive optics system was written to compare the performance of the 3PWFS and 4PWFS using both Raw Intensity and Slopes. The goal was to measure the quality of correction from the AO closed-loop by calculating the Strehl Ratio produced by each wavefront sensor as a function of guide star magnitude. The AO correction was optimized by measuring Strehl for each guide star magnitude, and scanning over loop gains from 0.1 to 1.8. In our simulations, we do not include system latency which resulted in an optimum loop gain of over 1 for bright guide stars. A maximum gain of 1.8 was chosen to ensure that the loop gain corresponding to the best correction was chosen. We plotted Strehl versus loop gain for each guide star magnitude as a diagnostic tool to verify that Strehl was decreasing at high loop gains and that we did not need to scan to higher values. In this simulation the Strehl ratio was calculated using OOMAO's built-in Strehl calculator, which uses the OTF calculated from a PSF with no phase aberration, and a PSF with AO compensated phase aberration. The details of the simulation are summarized in the flow chart in Figure 12. To characterize the sensitivity of each sensor to read noise the following experiment was performed twice, once at 0.5 e -read noise to match that of an OCAM2K camera, and once at 12 e -read noise to match the noisiest camera on the Comprehensive Adaptive Optics and Coronagraph Test Instrument (CACTI) at the University of Arizona Extreme Wavefront Control Lab (XWCL). Listed below is the experiment performed in OOMAO:

• Guide star magnitude was varied incrementally from 0 to 12 th magnitude in steps of 2.

• At each magnitude closed-loop data was taken with different loop gains from 0.1 to 1.8 in steps of 0.1. Loop gains of above 1 were used because the simulated system has no latency allowing for higher gains to be used to increase performance.

• At each loop gain the performance was tested by closing the loop on 11 different atmospheric realizations and recording 500 closed-loop PSF frames.

• Strehl values were calculated from the 5,500 PSFs and an average Strehl ratio is reported for a given loop gain at a given magnitude.

• The loop gain that gave the highest Strehl value for that guide star magnitude was used in the final calculation of Strehl vs guide star magnitude.

• The result is a plot of Strehl versus guide star magnitude, where the value of Strehl has been optimized over the loop gain shown in Figure 13 5.2 Results

In Figure 13, no significant difference in performance was found in the comparison case of the wavefront sensors with 0.5 e -read noise. Most on-sky adaptive optics systems use a 4PWFS

with Slopes (4PWFS SM) so we use this wavefront sensor as our reference for comparison. The performance of each wavefront sensor was found to be within a percent of Strehl from the 4PWFS SM, across all stellar magnitudes. For the simulations at 12 e -read noise in Figure 13 an increase of 0.0359 Strehl was found for the 3PWFS using Raw Intensity (3PWFS RI) over the 4PWFS SM at a stellar magnitude of 10. At the same magnitude, the 4PWFS RI also outperformed the 4PWFS SM, but the increase was only 0.0122 Strehl. This simulation successfully showed the increase in performance from the 3PWFS at low light levels where the effects of read noise are stronger.

The overall performance of each wavefront sensor at each guide star magnitude is given in Figure 13. In Figures 14 and15 From these results, we can conclude that the 3PWFS is a viable wavefront sensor with performance comparable to a 4PWFS.

Discussion

The diffraction theory of the Foucault test predicts an intensity pattern that is the Hilbert transform.

The resulting signal is related to the gradient of the wavefront phase, but without the scaling by spatial frequency. The PWFS is an extension of the Foucault test, and the diffraction theory accurately predicts the intensity patterns seen by a PWFS. Similar to the Foucault test, the sensitivity of a PWFS with no modulation is independent of spatial frequency as found in previous work by

Guyon [START_REF] Guyon | Limits of adaptive optics for high-contrast imaging[END_REF] because all spatial frequencies produce an equally strong signal.

Codona et al. [START_REF] Codona | Comparative performance of a 3-sided and 4-sided pyramid wavefront sensor[END_REF] used the AOsim3 wave optics package to compare the end-to-end performance of a 3PWFS to a 4PWFS with no modulation at 633 nm wavelength. In this simulation, the Raw Intensity signal handling method was used. In open-loop with no noise, the study showed that the would explore the optimum PWFS design based on the system parameters such as telescope size and seeing conditions. The scope of our simulations is limited but still agree that the 3PWFS is less sensitive to read noise, however, the amount of improvement gained is still under question. Both the 3PWFS and 4PWFS discussed in this paper are refractive, and image all pupils onto the same detector. Moving towards a reflective PWFS that would image each pupil onto its own detector could have a more substantial benefit. Each detector would be smaller, resulting in faster readout speeds with less added read noise. Low-read noise detectors are expensive, so using a reflective PWFS would increase system cost and complexity.

Comparing the results from our simulations we see that for 0.5 e -read noise and 12 e -read noise results, the RI method has slightly better performance than the SM method for both the 3PWFS and 4PWFS in low light conditions. A possible explanation comes from Section 4.2. A better fit for the Slopes Maps method was found for the unmodulated 3PWFS when a flat wavefront reference was subtracted. A main benefit of the SM method is that it is self-referencing and does not inherently need the flat wavefront reference as these references can be difficult to take. If the PWFS signal does not match the linear response signal expected for a given phase error, due to noise, nonlinearity, or pupil misregistration the self-referencing of the SM method might not be as accurate. This is supported by our results in Figure 14 and Figure 15 which suggest that the SM method suffers more from the addition to noise as the more significant deviations in performance between the RI and SM method occur at faint guide stars.

We have found that the 3PWFS is a viable wavefront sensor that is able to fully reconstruct a wavefront and produce a stable closed-loop. Our simulations assumed a perfect wavefront sensor with the absence of manufacturing errors. A high-quality four-sided pyramid optic is difficult and expensive to manufacture. Common errors on four-sided pyramid manufacturing include roofing of the pyramid tip or chipping of the pyramid tip which results in a loss of Strehl. The 3PWFS is an exciting opportunity for systems that face a trade-off between cost and quality. A real 3PWFS could be cheaper and offer better performance than a 4PWFS. As a next step to developing the potential of the 3PWFS, the University of Arizona Extreme Wavefront Control Lab (XWCL) in partnership with Hart Scientific has manufactured a three-sided pyramid optic with a tip less than 5µm in size. The optic has been integrated into a 3PWFS that is a visiting instrument on the Comprehensive Adaptive Optics and Coronagraph Test Instrument (CACTI). A future experiment will test the performance of the 3PWFS against the currently integrated 4PWFS under different strengths of turbulence.

Conclusion

The PWFS is an extension of the Foucault knife edge test to two dimensions. By examining the diffraction theory of the Foucault test, which is much simpler than a full pyramid, we gain insight to the physical processes behind the PWFS. In Section 4 we consolidated the diffraction theory of the knife edge test by Linfoot, [START_REF] Linfoot | On the theory of the zonal foucault test[END_REF] Katzoff, [START_REF] Katzoff | Quantitative determination of optical imperfections by mathematical analysis of the Foucault knife-edge test pattern[END_REF] and Wilson 31 into a single derivation with uniform notation. Expanding upon their results, we linked phase aberrations in the shape of Fourier modes to intensity patterns produced by the knife edge test. The result found for a phase error in the shape of a Fourier mode is an intensity pattern described by the Hilbert transform of the phase function. We considered an example cos(nx) phase pattern, and mathematically derived that the resulting intensity pattern is proportional to -sin(nx), which is functionally the derivative of the phase without the dependence on spatial frequency. We confirmed this result by considering the intensity and slope signals from a simulated 3PWFS and 4PWFS. We used this result to motivate the signal processing of the pyramid measurements and introduced a new Slope Map method to handling the 3PWFS signals, which is derived using the centroid of an equilateral triangle. This method provided a stable closed-loop correction on the LOOPS testbed.

As a further test of the 3PWFS and slopes maps equations, we performed an end-to-end simulation of an adaptive optics system to compare the performance of the 3PWFS and 4PWFS as well as the Raw Intensity and Slopes Maps signal processing techniques. We found that in the absence of read noise the performance of the wavefront sensors is within 0.01 Strehl. When we included read noise into the simulation, we saw a break in this trend at lower guide star magnitudes and found a gain of 0.036 Strehl the 3PWFS using Raw Intensity over the 4PWFS using Slopes Maps at a stellar magnitude of 10. At the same magnitude, the 4PWFS using Raw Intensity also outperformed the 4PWFS Slopes Maps, but the gain was only 0.0122 Strehl. We conclude that the 3PWFS is a viable wavefront sensor with similar performance in the scope of our study. Our simulations only considered a single modulation radius and r 0 of atmospheric turbulence. Further work should explore the respective performances at different turbulence strengths and different modulation radii.

Fig 2

 2 Fig 2 Ratio of the SNR values of the 3PWFS and 4PWFS. In low light conditions when the SNR is dominated by readnoise, the ratio converges to[START_REF] Sanders | The thirty meter telescope (tmt): An international observatory[END_REF] 3 .

Figure 3 .

 3 Figure 3.C and 3.D shows the resulting pyramid pupils on the simulated wavefront sensor camera,

Fig 3

 3 Fig 3 Details of the simulated PWFS in OOMAO. A) and B) The simulated 3PWFS and 4PWFS pyramid masks in OOMAO. These masks are used as a phase screen in the pupil plane to emulate the focal plane splitting and separation done by a real glass pyramid. C) and D) The pupils from the 3PWFS and 4PWFS respectively on the simulated detector. In OOMAO the user can change the size, separation, and intensity values of the pupils through user-defined inputs.

Fig 4

 4 Fig 4 Equilateral triangle with the center at the origin. Solving for the coordinates of the vertex points gives the weights for the pixel Intensities used in the Slopes Maps calculation. The blue circles represent the layout of the pupils on the detector.

  shows example images from the LOOPS test-bed of (A) the pyramid signal of the 3PWFS under turbulence in a closed-loop, (B) the pyramid signal of the 4PWFS under turbulence in a closed-loop, (C) the LOOPS PSF with no turbulence, and (D) the LOOPS PSF in a closed-loop with the 3PWFS using the Slopes Map technique. Further testing and analysis of the 3PWFS was performed on the Comprehensive Adaptive Optics and Coronagraph Test Instrument (CACTI) at the University of Arizona. Details of the experimental setup and results from a performance comparison test between the 3PWFS and 4PWFS are discussed in an upcoming paper expected late 2021, (Schatz et al., 2021b).

Fig 5

 5 Fig 5 Example images from the LOOPS test-bed of A) the pyramid signal of the 3PWFS under turbulence in closedloop, B) the pyramid signal of the 4PWFS under turbulence in closed-loop, C) the LOOPS PSF with no turbulence, and D) the LOOPS PSF in closed-loop with the 3PWFS using the Slopes Map technique.

Fig 6

 6 Fig 6 Diagram of the steps taken in the derivation. The entrance pupil with electric field u 0 (x 0 ) is Fourier transformed to the focal plane. At the focal plane the knife edge by multiplying the binary mask function of the knife edge, H f (ξ f ), by the Fourier transform of the electric field in the entrance pupil. An inverse Fourier transform taken to return to the pupil plane where the pyramid signal is detected.

Fig 7 A

 7 Fig 7 A. Applied cosine phase error in radians in pupil plane to be propagated through the PWFS. B. Scaled cross section of the phase error.

Fig 8

 8 Fig 8 Predictions from the Foucault derivation. A. The applied cos(3x) phase error. B. The intensity pattern predicted for a single pupil. C. The expected pyramid signal after subtracting constant terms and scaling.

Fig 9

 9 Fig 9 Resulting intensity patterns on the PWFS detector from A. the 3PWFS, and B. the 4PWFS. Figures C. and D. are cross sections of the intensity pattern plotted against the mathematical prediction. For C. The cross-section was taken from center of the middle-left pupil. For D. The cross section was taken from the center of the top two pupils.

Fig 10 A

 10 Fig 10 A. and B. Calculated slopes from the pyramid pupils. C. and D. Cross section of the Slopes plotted against the predicted signal.

Fig 11 A

 11 Fig 11 A. and B. The wavefront sensor detector signals. C. and D. cross-section of the wavefront sensor Slopes Maps plotted against the prediction. E. and F. the cross-section of the Slopes Maps with the reference signal subtracted.

Fig 12

 12 Fig 12 Experimental details of the simulation done in OOMAO. Light starts at the natural guide star and is propagated through the atmosphere, telescope and wavefront sensor. The wavefront sensor measures a correction that is then applied by the deformable mirror. The resulting AO corrected PSF is recorded on a noiseless science camera.

  comparison plots were made by subtracting the Strehl values for 4PWFS SM from the other wavefront sensors, to help visualize the performance of each wavefront sensor.

Fig 13

 13 Fig 13 Strehl vs Guide Star magnitude in R-band for each PWFS. The steep drop-off in performance of the lower curves is due to the effects of 12 e -read noise. At a guide star magnitude of 10, the read noise starts to matter, and the 3PWFS out performs the 4PWFS.

Fig 14

 14 Fig 14 Comparison plot of wavefront sensor performance vs the 4PWFS using Slopes at 0.5 e -read noise in R-band.

Fig 15

 15 Fig 15 Comparison plot of wavefront sensor performance vs the 4PWFS using Slopes at 12 e -read noise in R-band.

  They found that for a detector with 3 e -read noise, that there was a gap performance improvement by 1 guide star magnitude in favor of the 3PWFS. Codona et. al. considered other factors including system latency. Our simulations are in R band and were optimized only for loop gain. Using a detector with 12 e -read noise we found that the performance of the wavefront sensors are once again comparable, and we found only 0.02 increase in Strehl from the 3PWFS at 10 th magnitude. The differences between the two simulations could explain the discrepancies in the results. Codona et al. simulate a smaller telescope aperture that is 1.5m in diameter with a central obscuration. In addition, they have worse seeing conditions with r 0 =5 cm, but have a larger 12x12 actuator deformable mirror and pyramid pupils that are 34 pixels in diameter. Further work

5 read noise that the performance of the wavefront sensors is within 0.01 Strehl. Our results differ from Codona et al. when including read noise.

Acknowledgements

We thank everyone involved who made the collaboration between the University of Arizona and the Laboratoire d'Astrophysique de Marseille possible. Part of this work was completed during a research visit by Lauren Schatz to the Laboratoire d'Astrophysique de Marseille that was supported by the ARCS Foundation Scholarship. Thank you to Alexander Rodack for his computer help, and Logan Pearce for sharing her Python plotting style sheet. This work has been supported in part by the Air Force Research Laboratory, Directed Energy Directorate, under contract FA9451-19-C-0581. The opinions, findings, and conclusions expressed in this paper are those of the authors and do not necessarily reflect those of the United States Air Force. This work benefited from the support of the WOLF project ANR-18-CE31-0018 of the French National Research Agency (ANR). It has also been prepared as part of the activities of OPTICON H2020 (2017-2020) Work Package 1 (Calibration and test tools for AO assisted E-ELT instruments). OPTICON is supported by the Horizon 2020 Framework Programme of the European Commission's (Grant number 730890). Authors are acknowledging the support by the Action Spécifique Haute Résolution Angulaire (ASHRA) of CNRS/INSU co-funded by CNES. Finally, part of this work is supported by the LabEx FOCUS ANR-11-LABX-0013, and received the support of Action Spécifique Haute Résolution Angulaire ASHRA.

ARCS Foundation scholar, and a receiver of the Society of Women Engineer's Ada I. Pressman Memorial Scholarship from 2019 to 2021.

List of Figures

The optical design of a PWFS. Light from the telescope is focused onto a glass pyramid tip where it is split and then the pupil plane is reimaged onto a detector.

The result is copies of the telescope pupil that contain intensity fluctuations that are related to the wavefront phase. [START_REF] Tokovinin | Ao tutorial at ctio[END_REF] 2 Ratio of the SNR values of the 3PWFS and 4PWFS. In low light conditions when the SNR is dominated by read noise, the ratio converges to