domain: spi phys

text:
Romain Fiévet
email: romain.fievet@onera.fr

Hughes Deniau

Jean-Philippe Brazier

Estelle Piot

Numerical study of hypersonic boundary-layer transition delay through second-mode absorption

Numerical time-domain impedance boundary conditions are developed within a highorder spectral difference flow solver. They are designed to accurately replicate the acoustic response of ultrasonically-absorbent coatings over a broad spectral range. The coupling of this class of high-fidelity wall boundary conditions with high-order numerical methods has enabled their use in aero-acoustics applications. This solver is used to perform direct numerical simulations (DNS) of a hypersonic boundary layer and analyze the stabilizing effects of acoustically absorbent materials on the laminar-to-turbulent transition. The damping effectiveness of such complex porous coating on key frequencies of interest is investigated, as well as its effect on the mean flow. It is found that the second-mode instability, which dominates the high-Mach number flow regime, is strongly suppressed at the expense of higher far-field noise radiation. The solutions obtained by means of DNS are also found to compare favorably with linearized stability theory.

I. Introduction

One of the primary concerns when designing hypersonic vehicles is the strong aero-thermal load suffered during atmospheric flights. The surface drag and heating requires the use of thermal protection system which constraints the vehicle weights, reducing its maximal payload and flight performance. Mitigating these fluid-structure exchange processes is, therefore, a key design consideration. Being diffusive by nature, they are known to be mitigated when the boundary layer remains laminar as opposed to being turbulent. [START_REF] Reed | Drag prediction and transition in hypersonic flow[END_REF][START_REF] Zhong | Direct numerical simulation on the receptivity, instability, and transition of hypersonic boundary layers[END_REF][START_REF] Fedorov | Transition and stability of high-speed boundary layers[END_REF] Hence, the laminar-to-turbulent transition of hypersonic boundary layers (HBL) has been the focus of a large body of studies which has shed light on the underlying physics.

In absence of bypass forcing, as is common assumption for high-altitude high-speed flights, transition is caused by the spatial growth of unstable waves. [START_REF] Malik | Boundary layer transition in hypersonic flows[END_REF] At supersonic speeds, linear stability compressible theory was used by Mack 5 to predict the existence of an infinite number of modes, amongst which some are unstable. The first unstabe mode can be considered as the hypersonic equivalent of Tollmien-Schlichting waves and is predominant below Mach 4. The second (and higher) unstable modes are acoustic in nature (i.e. compressible and inviscid) and dominate the transition process beyond Mach 4. The latter modes occur when instabilities' phase speeds become supersonic relative to the flow velocity at the critical layer, defined as the location where the local flow velocity equals the sound speed. Such instabilities have been experimentally [START_REF] Fedorov | Transition and stability of high-speed boundary layers[END_REF][START_REF] Morkovin | Transition at hypersonic speeds[END_REF][START_REF] Mack | On the inviscid acoustic-mode instability of supersonic shear flows. Part 1: two-dimensional waves[END_REF] and numerically [START_REF] Zhong | Direct numerical simulation on the receptivity, instability, and transition of hypersonic boundary layers[END_REF][START_REF] Egoroc | Receptivity of a hypersonic boundary layer over a flat plate with a porous coating[END_REF][START_REF] Li | Direct Numerical Simulation of Hypersonic Boundary Layer Transition over a Blunt Cone[END_REF][START_REF] Knisely | Sound radiation by supersonic unstable modes in hypersonic blunt cone boundary layers. I. Linear stability theory[END_REF][START_REF] Knisely | Sound radiation by supersonic unstable modes in hypersonic blunt cone boundary layers. II. Direct numerical simulation[END_REF] visualized as waves trapped between the wall and sonic line, producing rope-like structures around the critical layer. In fact, the wall-normal gradient of the acoustic impedance forms an impedance well [START_REF] Kuehl | Thermoacoustic Interpretation of Second-Mode Instability[END_REF] inside the boundary layer which physically acts as an acoustic wave-guide. The resonating acoustic modes trapped inside it grow as they propagate and energy is progressively transferred from the mean laminar kinetic energy into acoustic energy until turbulence breaks down. [START_REF] Kuehl | Thermoacoustic Interpretation of Second-Mode Instability[END_REF] Interestingly, wall cooling further destabilizes the higher modes contrary to the first viscous mode [START_REF] Mack | Stability of Time Dependent and Spatially Varying Flows[END_REF][START_REF] Mack | Effect of cooling on boundary-layer stability at Mach number 3[END_REF] as it exacerbates the impedance well effect. As the second mode has a higher amplitude than the higher modes, [START_REF] Mack | Stability of Time Dependent and Spatially Varying Flows[END_REF] it is of critical interest to design second-mode canceling systems to delay the laminar-to-turbulent transition in HBL.

One of the most promising second-mode canceling mechanism is the use of porous coatings. The pores act as acoustic resonators and can be designed to filter out specific frequencies. It was theorized by Fedorov and NSCBC) are presented. Second, the development of the TDIBC is thoroughly described. A third section presents the hypersonic cone configuration and mean flow results. The fourth and last section presents the results of the stability analysis.

II. Numerical methods

The high-order spectral difference code used in this project, JAGUAR (proj ect of an Aerodynamic solver using General U nstructured grids And high ordeR schemes), is actively co-developed by ONERA and CERFACS (European Center of Research and Formation in Advanced Scientific Calculus) with the help of IMFT (Institute of Fluid Mechanics in Toulouse). Ongoing developments include combustion modules, real gas effects, further maturation of characteristic boundary conditions treatment, embedded automatic differentiation and adjoint methods (Cardesa and Airiau [START_REF] Cardesa | Automatic differentiation of a spectral difference code for sensitivity analysis[END_REF] ). Benchmarking and validation test cases are available in the literature [START_REF] Cassagne | High-order Method for a New Generation of Large Eddy Simulation Solver[END_REF][START_REF] Hamri | Evaluation of the JAGUAR solver with benchmark test cases[END_REF] . The SD method is presented first for the sake of completeness. Then, the governing equations for characteristic boundary conditions with the Navier-Stokes equations as implemented in JAGUAR are presented.

A. Spectral Difference method

Discontinuous finite element methods (DFEM) have gained much traction in the past years thanks to their ability to achieve locally arbitrarily-high orders of accuracy on unstructured grid. Such accuracy was previously only attainable with large-stencil finite difference (FD) schemes naturally restricted to structured grids, which prevented the use of complex geometries with high-order schemes.

The SD method resolves the strong form of the Navier-Stokes equations inside each mesh element, as in FD. The global algorithm of the p-order SD method is presented in Fig. 1 for an element in a one-dimensional (1D) mesh and briefly explained below.

During initialization, each element is transformed into a standardized [0,1] domain. Then, p + 1 solution and p + 2 flux points (respectively called SP and FP) are distributed inside the element using the Gauss-Legendre quadrature points. Note that FP are placed at the element boundaries at 0 and +1. The SP store the conservative variables which are marched in time while the FP serve as a base to reconstruct a continuous flux. However, the method does not assume that the conservative variables form a continuous solution across the computational domain.

At the start of a time sub-iteration, the conservative variables solution U stored in the p+1 SP are used to reconstruct a p-order Lagrange polynom which represents the continuous solution across the element (globally discontinuous). It is then used to evaluate a continuous flux F (U ) at the p+2 FP. A Riemann problem is resolved at the element boundaries which provides element-to-element communication and ensures flux continuity across the whole computational domain. A (p+1)-order flux polynom is reconstructed from the flux values at the FP. The flux polynom is then derived at the solution point to march the solution in time.

F(x)

Position of the solutions points and the flux points

In agreement with the literature the solution points are chosen to be the following Gauss points:

Position of the solutions points and the flux points

In agreement with the literature the solution points are chosen to be the following Gauss points: Global algorithm of the p=2 SD method in 1D reproduced from Cassagne et al. [START_REF] Cassagne | High-order Method for a New Generation of Large Eddy Simulation Solver[END_REF] Vanharen et al. [START_REF] Vanharen | Revisiting the spectral analysis for high-order spectral discontinuous methods[END_REF] have assessed the accuracy of the SD method for various polynom orders and compared it with spectral-like resolution 6 th -order compact FD schemes. [START_REF] Lele | Compact finite difference schemes with spectral-like resolution[END_REF][START_REF] Bogey | A family of low dispersive and low dissipative explicit schemes for flow and noise computations[END_REF] They showed that the 6 th -order SD method is at least as accurate, albeit a local method suited for unstructured grids contrary to these FD methods. A spectral-like resolution implies that the numerical scheme causes minimal dissipation and dispersion errors. This is particularly important when resolving problems where a large range of wavenumbers are physically relevant to the problem, as in laminar-to-turbulent transition and aeroacoustics. Finally, a 6-stage explicit Runge-Kutta scheme is used to march the solution in time.

B. Characteristic Boundary Conditions for Navier-Stokes equations

As explained in Sec. I, the use of non-reflecting numerical boundary conditions is of paramount importance in laminar-to-turbulent transition studies. In Fiévet et al., [START_REF] Fiévet | Strong compact formalism for characteristic boundary conditions with discontinuous spectral methods[END_REF] NSCBC were successfully used with the SD method: the reader is referred to their study for a thorough description of their implementation accompanied by validation test cases. A brief overview is offered below to introduce useful notations for Sec. III.

Specifically, the implementation of the NSCBC in JAGUAR follows the methodology of Kim and Lee 38 which apply characteristic boundary conditions through the modification of flux derivatives at the boundaries. While originally intended for FD methods, their method is well suited for SD as well since characteristics can be evaluated with great accuracy at the boundary flux points using the local high-order extrapolation scheme.

The non-reacting Navier-Stokes equations are written in three-dimensional (3D) generalized coordinates (ξ,η,ζ) as:

∂U ∂t + ξ x ∂E c ∂ξ + ξ y ∂F c ∂ξ + ξ z ∂G c ∂ξ + ξ x ∂E d ∂ξ + ξ y ∂F d ∂ξ + ξ z ∂G d ∂ξ + η x ∂E c ∂η + η y ∂F c ∂η + η z ∂G c ∂η + η x ∂E d ∂η + η y ∂F d ∂η + η z ∂G d ∂η + ζ x ∂E c ∂ζ + ζ y ∂F c ∂ζ + ζ z ∂G c ∂ζ + ζ x ∂E d ∂ζ + ζ y ∂F d ∂ζ + ζ z ∂G d ∂ζ = 0, (1)
where (E c ,F c ,G c ) and (E d ,F d ,G d ) respectively correspond to the convective and diffusive fluxes in the physical coordinate system (x,y,z). Without any loss of generality, when applying NSCBC at a ξ-normal face, the characteristic decomposition of the ξ-normal convective flux is expressed as:

ξ x ∂E c ∂ξ + ξ y ∂F c ∂ξ + ξ z ∂E c ∂ξ = P U ∆P U -1 ∂U ∂ξ , (2)
with ∆ being the diagonalized matrix of ∂Ec ∂U 's eigenvalues and P U the associated eigenvectors forming the transformation matrix. The vector of characteristic L is defined as:

L = ∆P U -1 ∂U ∂ξ . (3)
Likewise, the transformation matrix is used to define the following quantities:

W = P U -1 U, T c = P U -1 η x ∂E c ∂η + η y ∂F c ∂η + η z ∂G c ∂η + ζ x ∂E c ∂ζ + ζ y ∂F c ∂ζ + ζ z ∂G c ∂ζ , T d = P U -1 η x ∂E d ∂η + η y ∂F d ∂η + η z ∂G d ∂η + ζ x ∂E d ∂ζ + ζ y ∂F d ∂ζ + ζ z ∂G d ∂ζ , D = P U -1 ξ x ∂E d ∂ξ + ξ y ∂F d ∂ξ + ξ z ∂G d ∂ξ . (4)
Hence, Eq. 1 can be rewritten in characteristic space as:

∂W ∂t + L + T c + T d + D = 0 (5)
The convective and diffusive tangent fluxes T c and T d are evaluated from the inner solution, while the normal diffusive fluxes D usually rely on Dirichlet or Neumann conditions to evaluate the momentum and thermal energy gradients. Applying NSCBC consists in resolving systems of L at the boundary FP verifying the desired physical condition. For instance, pure non-reflecting outflow conditions imply no entering wave, i.e. all entering characteristics set to zero.

These equations are used to derive characteristic boundary conditions using the following steps (a ξnormal boundary is considered to illustrate their method). First, Eq. 3 is used to provide an initial guess for the L using flux derivatives extrapolated from the inner solution. Second, the L are modified using classical NSCBC conditions. 32 The new ξ-normal convective characteristics are labeled L * . Third, Eq. 3 is used again to obtain the corresponding corrected ξ-normal convective flux derivative ∂E * c ∂ξ . Fourth, this quantity is returned into Eq. 1 to march the solution in time.

III. Implementation of time-domain impedance boundary conditions

The implementation of TDIBC with the SD scheme follows the works of Monteghetti et al. [START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF] and later Fiévet et al. [START_REF] Fiévet | Strong compact formalism for characteristic boundary conditions with discontinuous spectral methods[END_REF] A summary of both works is presented below for the sake of completeness.

As explained my Monteghetti et al., [START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF] a TDIBC is comprised of three components. First, a physical impedance model has to be chosen to describe the UAC in Laplace space. Second, the said model is discretized and converted into the time-domain. This usually amounts to evaluate a convolution product in time. Last, a semi-discrete formulation of the TDIBC is needed to couple it with the numerical flow solver.

Some useful notations and basic concepts pertaining to acoustics for linearized Euler equations are given first before describing each aforementioned step.

A. Acoustic impedance conditions for the linearized Euler equations

In acoustics, the impedance Z of a medium characterizes its reaction to a traveling acoustic pressure perturbation. For a mono-dimensional (1D) medium where the impedance depends only on the perturbation's angular frequency ω and space, it can be defined as:

z(x, jω) = p (x, jω) u (x, jω) (6)
where p and u are the Fourier-transformed acoustic pressure and velocity perturbations in a unperturbed flow at Mach M 0 . Further, the acoustic pressure perturbation can be decomposed into a sum of right-running and left-running waves:

p (t) = p e jωt = p + e jωt-k 1+M 0 x + p -e jωt+k 1-M 0 x (7)
with wavenumber k 1+M0 = ω/c 0 /(1 + M 0 ) and k 1-M0 = ω/c 0 /(1 -M 0 ). The reflection coefficient β is defined as the ratio between the amplitudes of the reflected and incident waves, it can also be derived from the impedance as:

β[z] = z -ρ 0 c 0 z + ρ 0 c 0 (8)
where ρ 0 and c 0 are the medium's reference density and speed of sound, respectively. Finally, the impedance and scattering operator Z and B respectively define the following positive-real functions in the time-domain:

p (t) = [z u ](t) = Z[u ](t) (9)
and

v + (t) = [β v -](t) = B[v -](t) (10)
where v + = p -ρ 0 c 0 u and v -= p + ρ 0 c 0 u are the reflected and incident wave amplitudes, respectively. They correspond to the characteristics of the linearized Euler equations (LEE) traveling at the speed of sound in opposite directions. Both Z and B define a TDIBC.

As pointed out by Monteghetti et al., [START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF] it is more appropriate to adopt a B-formalism rather than a Z-formalism when constructing a TDIBC due to β being bounded contrary to z. This proved to relax the restriction on the Courant Friedrichs Lewy (CFL) number [START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF] which is of paramount importance to CFD applications.

B. Impedance modeling of an UAC

The first component of a TDIBC consists of its impedance model, which relates the wall pressure fluctuations with the wall normal velocity. In the present study, the UAC is modeled as a set of narrow cavities which form a honeycomb core. Each cavity behaves as an acoustic wave-guide with a rigid end which effectively damps its resonant frequencies as in Jones et al. [START_REF] Jones | Benchmark Data for Evaluation of Aeroacoustic Propagation Codes with Grazing Flow[END_REF] The honeycomb's cells have a thickness l c , diameter d c and porosity τ c . The porosity measures the ratio between the perforated and whole surface. The impedance of the liner reads:

z SDOF = z c σ c coth(jk c l c ) (11)
with the following fractional wavenumber k c (s) (s = jω referring to the Laplace variable herein):

jk c (s)l c = a 0 + a 1 2 √ s + a 1 s. ( 12
)
where a 0 models frequency-independent losses, a1 /2 corresponds to frequency-dependent losses from thermoviscous dissipation and diffraction, and a 1 characterizes the system's mass reactance and does not induce any energy loss. The three coefficients can be estimated with different theoretical and empirical models.

C. Delayed multi-pole representation of an impedance model

The second step of the TDIBC construction consists in discretizing the reflection coefficient β[z(s)] in the time-domain to derive the scattering operator B. To this end, β is first decomposed into an infinite sum of complex poles using an oscillatory-diffusive representation (ODR) as defined by Monteghetti et al. [START_REF] Monteghetti | Design of broadband time-domain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models[END_REF][START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF] For the sake of conciseness, the reader is referred to these papers to obtain a detailed derivation of the ODR. The ODR of the reflection coefficients reads as:

β(s) = β ∞ + n∈I r 1,n s -s n + ∞ 0 µ 1 (ξ) s + ξ dξ + e -sτ n∈I r 2,n s -s n + ∞ 0 µ 2 (ξ) s + ξ dξ . ( 13
)
where the complex oscillatory poles s n are the zeros of z + ρ 0 c 0 and the diffusive poles ξ are arbitrarily chosen to span the spectral range of interest for a given configuration. The delay τ corresponds to the back-and-forth wave traveling time inside the cavity. Finally, the weights r n and µ n are evaluated to fit the analytical function (Eq. 8).

It is important to note that no approximation has been made yet: the model is exactly represented by an infinite sum of oscillatory and diffusive poles. The oscillatory poles form conjugate pairs which guarantees the bounded-reality of the associated scattering operator. Further, an approximated reflection coefficient β is naturally derived by only using a finite number of N ξ diffusive and N s oscillatory poles ξk and sn :

β(s) = β∞ + Ns n=1 r1,n s -sn + N ξ k=1 μ1,k s + ξk + e -sτ Ns n=1 r2,n s -sn + N ξ k=1 μ2,k s + ξk (14)
Importantly, the ODR discretization offers the possibility to optimize the poles and weights values to match an experimental dataset. An example of the effectiveness of such data-driven discretization is provided in Fig. 2. As previously mentioned, the test case considered in this paper and presented in Sec. IV is based on the experimental work of Wagner et al. [START_REF] Wagner | Experiments on passive hypersonic boundary layer control using ultrasonically absorptive carbon-carbon material with random microstructure[END_REF] The spectral properties of the UAC they use are quantified in, [START_REF] Sousa | Numerical investigation of second mode attenuation over carbon/carbon surfaces on a sharp slender cone[END_REF] and are now used to calibrated the TDIBC's poles and weights. The ODR of β shown in Fig. 2 was performed using 4 diffusive poles ξ and 8 oscillatory pairs of poles. This corresponds to a 20-scalar model, including the poles weights and excluding the complex conjugates, which captures the complex broadband response of the UAC with great accuracy up to 1 MHz. All these scalars are given in the appendix in Tab. 2. Using half the number of poles would result in a discretization similarly accurate up to 500 kHz.

The underlying interest of the ODR of β lies in the transformation of the convolution product (Eq. 10) into a sum of auxiliary functions solutions of ordinary differential equations (ODE) as demonstrated Frequency (Hz) by Monteghetti et al. [START_REF] Monteghetti | Design of broadband time-domain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models[END_REF] This powerful result permits to approximate the scattering operator B (i.e. the TDIBC) with ODE using the same explicit time-marching scheme as the flow conservative variables U . The evaluation of B based on β reads as:

B(t) = β∞ v -(t) + Ns n=1 r1,n φ(t, -s n ) + N ξ k=1 μ1,k φ(t, ξk ) + n∈I r2,n φ(t -τ , -s n ) + N ξ k=1 μ2,k φ(t -τ , ξk ) (15)
where the φ are evaluated using:

   ∂ t φ = -ζφ(t, s) + v -(t) (t > 0, ζ ∈ C) φ(t, 0) = φ(0, s) = 0 (16)
for any pole ζ. The delay is resolved by transporting at the medium's speed of sound c 0 the φ variables over another spatial coordinate θ of length equal to τ /c 0 . To this end, another set of auxiliary functions ψ is introduced:

ψ(t, ζ, l τ ) = φ(t -τ , ζ). ( 17
)
The additional auxiliary functions ψ are solutions of the following transport equation:

       ∂ t ψ(t, ζ, θ) = c 0 ∂ θ ψ(t, ζ, θ) (t > 0, θ ∈ [0, l τ ], ζ ∈ C) ψ(t, ζ, 0) = φ(t, ζ) ψ(0, ζ, θ) = 0 (18)
This amounts to resolving an advection problem for each pole at each FP located on a TDIBC face. The discretization scheme of the θ-space determines the delay and maximum frequency that can be resolved through Eqs. 18. Its length l τ verifies l τ = τ c 0 which determines the cavity depth for a given delay and speed of sound. The number of elements per wavelength (EPW), N e , determines the cut-off frequency f max = (N e × EPW)/(2τ ) with a minimum of EPW = 2 needed to avoid aliasing. The numerical flux is evaluated using a 6 th -order SD scheme with upwinding. Since there is no guarantee that the ODR yields a passive TDIBC outside the ξ-bounds, it is suggested to choose f max ≤ max(| ξ |) to naturally filter out higher frequencies. A performance study was carried by Fiévet et al. [START_REF] Fiévet | Strong compact formalism for characteristic boundary conditions with discontinuous spectral methods[END_REF] who found that EPW = 8 resulted in dissipative losses on the order 0.01% of the incidence wave's amplitude for a delay equal to 10 periods. They found that the computational cost of the advection problems was greatly mitigated by the use of array-wide operations on contiguous data, clever CPU-partitioning and use of the same time-marching scheme as the main simulation.

Alternatively, Douasbin et al. [START_REF] Douasbin | Delayed-time domain impedance boundary conditions (D-TDIBC)[END_REF] showed that the delay operator can be approximated by a multi-pole function, in similar fashion as β's ODR. Their approach is more efficient memory-wise than the direct resolution presented above which requires to store information at all the θ-elements' solution points. On the other hand, the direct approach uses only one real hyper-parameter τ while the multi-pole approach uses an unknown number of complex poles and weights determined by an iterative process. The pole's numbers, values and corresponding weights are adjusted until the target delay is sufficiently well resolved. The use of complex poles also causes instabilities outside the poles bounds and/or when an insufficient number of poles is used. Importantly, the direct approach permits to resolve all delays up to τ = l τ /c 0 by simply extracting ψ at a shorter θ, and resolving a higher delay simply requires to add θ-elements in order to increase l τ . This allows for a real-time feedback which will prove necessary for future adjoint-based optimization applications, hence this method was chosen.

To conclude, the work presented in this section permits to recast the scattering operator into a numericallyapproximated discrete form embedded within the time-marching scheme of a CFD solver.

D. Coupling of the TDIBC with the Navier-Stokes equations

The third and last TDIBC component is the coupling with the flow solver, i.e. the evaluation of the conservative variables numerical flux F (U ) at the TDIBC faces. To this end a semi-discrete formulation was derived [START_REF] Fiévet | Strong compact formalism for characteristic boundary conditions with discontinuous spectral methods[END_REF] to be used with the Navier-Stokes equations. For the sake of simplicity, the method is presented for a 1D case where W + and W -respectively define the Navier-Stokes (u + c)-traveling incident and the (u -c)-traveling reflected characteristics. c is the flow speed of sound and way vary in space time. They read as:

   ∂W + = ∂p + ρc ∂u ∂W -= ∂p -ρc ∂u (19)
The previous steps detailed in Sec. B and C permit to evaluate in time and space the scattering operator B valid for the linearized Euler equations. As B is a linear function of v -, it follows naturally that:

∂ t W + = B ∂ t W -. ( 20
)
which is reorganized using Eq. 5 into:

[L + D] + = B [L + D + T c + T d ] --[T c + T d ] -. (21)
The scattering operator is then comprised of a weighted sum of auxiliary functions φ and ψ evaluated with Eqs. 18 and 16 where v -has been replaced by W -. This coupling method injects the TDIBC by adjusting the numerical flux at the wall using Eq. 5. In this sense, the TDIBC is embedded within the NSCBC and can be considered as a subclass.

E. Test cases

The robustness of the NSCBC-embedded and ODR-based TDIBC was extensively verified. [START_REF] Fiévet | Strong compact formalism for characteristic boundary conditions with discontinuous spectral methods[END_REF] Yet, in the context of this study, the robustness of the TDIBC specifically derived from Fig. 2 (and Tab. 2) is investigated. To this end, an impedance tube test case is run. It consists of a single-period right-running harmonic wave of frequency f convected onto the TDIBC. The other end of the domain is treated as a non-reflecting pressure outflow. The computational domain is discretized with 50 cells per wavelength and four frequencies f = [25, 50, 100, 200] kHz are investigated.

The reflected wave profiles are sampled and compared with the analytical solution derived from resolving the monodimensional Riemann problem of the linearized Euler equations as in Monteghetti. [START_REF] Monteghetti | Design of broadband time-domain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models[END_REF] The comparisons between these profiles are presented in Fig. 3 for all four frequencies considered and serve as a testament to the TDIBC robustness used herein.

IV. Stability analysis of a hypersonic boundary layer

The high-fidelity numerical methods presented above, integrated within the same flow solver, are now used to simulate a HBL and study the UAC stabilization properties. To this end, an unperturbed hypersonic boundary-layer flow field is first developed. Then, linear stability theory (LST) is used to predict the range of frequency of interests corresponding to the most unstable modes along the streamwise direction. Last, the HBL is excited by the means of a wall blowing mechanism in order to trigger harmonic instabilities and study their interaction with a UAC simulated by a TDIBC. The TDIBC constructed in Sec. III was defined by a series of poles, weights, reflectivity and delay calibrated to match the broadband properties of the carbon fiber porous material developped at the German Aerospace Laboratory. [START_REF] Wagner | Experiments on passive hypersonic boundary layer control using ultrasonically absorptive carbon-carbon material with random microstructure[END_REF][START_REF] Wagner | Ultrasonic absorption characteristics of porous carbon-carbon ceramics with random microstructure for passive hypersonic boundary layer transition control[END_REF] This UAC was successfully used to delay the laminar-to-turbulent transition on a conical surface in an experiment at the High-Enthalpy Goettingen facility mimicking a high altitude Mach 7.4 flight. Specifically, the similarly designed numerical experiment of Sousa et al. [START_REF] Sousa | Numerical Investigation of Second-Mode Attenuation over Carbon/Carbon Porous Surfaces[END_REF] replicated 28 km-altitude Mach 7.4 flight conditions. The numerical experiment considered herein is designed to match as closely as possible the conditions of these aforementioned studies.

A. Description of the numerical experiment protocol

The far-field conditions of all numerical simulations presented from this point on are those given in Tab. 1. The model's geometry is chosen to be a 7 • half-angle 120 cm-long dihedral. While the second-mode properties are known to differ from a cone to dihedral model, this allows to decrease the simulation's computational cost by adopting a 2D Cartesian mesh instead of a 3D cylindrical mesh. As the objectives of this seminal study are not to perform a direct comparison with experimental data, this is deemed acceptable. A schematic of the numerical configurations used throughout this study is presented in Fig. 4. Its protocol comprises the following orderly steps:

Re x [m -1 ] M ∞ p ∞ [Pa] T ∞ [K] u ∞ [m/
1. A precursor solution of the dihedral flow is run with the high-enthalpy solver CEDRE [START_REF] Scherrer | Research on Supersonic Combustion and Scramjet Combustors at ONERA[END_REF][START_REF] Refloch | CEDRE Software[END_REF] until time and grid convergence are reached. It uses a second-order spatial method and a second-order implicit time scheme. Its purpose is to serve as the unperturbed reference flows and initial state to subsequent runs. The wall isothermal temperature T w = 300 K for the remainder of the study.

2. The wall-normal profiles are extracted along the wall. A ONERA in-house linear stability theory (LST) solver, MAMOUT, [START_REF] Brazier | Transition prediction on Reentry-F trajectory with PSE at chemical equilibrium[END_REF] is then used to determine the frequency range of interest to be investigated with the DNS, alongside the corresponding second-mode growth region.

3. The precursor's solution within the aforementioned region is extracted and used to define the DNS' boundary conditions and initial state. A first unperturbed DNS is then run to develop a time-converged, unperturbed HBL with JAGUAR. Importantly, it uses characteristic boundary conditions all along the isothermal no-slip wall wall: that is the TDIBC uses a fully-reflective (B=1) hard wall condition.

4. The UAC is then activated, and its impact on the mean field is quantified. In absence of any timeinstability, this case is expected to yield a solution identical to that obtained with the hard wall boundary conditions. This is a direct consequence of the TDIBC's well-posedness which verifies lim f →0 [ (β(f )), (β(f ))]=(1,0).

5.

Then, based on the LST results, the laminar HBL is excited at particular frequencies of interest f by the means of a wall blowing mechanism. A 0.5 mm wide harmonic source of amplitude v 0 = 4.3e -5 × U ∞ = 0.1 m.s -1 is then centered at a pertinent location identified by the LST. It replaces the fully-reflective hard wall conditions at the corresponding wall flux points and simply reads as:

W + = W -+ 2πv 0 f cos(2πf t). (22)
6. Once the solution has converged, the oscillation amplitudes are measured and the HBL response is compared with the LST, for all the frequencies f investigated, as to evaluate JAGUAR's capability in resolving the spatial growth rate.

7. The TDIBC is then activated in the region of interest identified by the LST. Once all transient waves have been washed out of the computational domain, the sound pressure level and other metrics of interest are sampled.

8. Finally, the previous two steps are repeated for the most unstable frequency with a reduced UAC surface in order to assess its impact on the pressure perturbation spatial growth and time-averaged solution as well.

B. Unperturbed direct numerical simulation of HBL

The CEDRE precursor simulation is run over 10000 cycles of 1 µs each using a first-order implicit timeintegration scheme. The fluxes are evaluated using a second-order flux reconstruction scheme with a HLLC Riemann solver [START_REF] Toro | Restoration of the contact surface in the HLL-Riemann solver[END_REF] at the cell faces. The configuration is shown in Fig. 4. The mesh was generated with a streamwise spatial increment δx = 1 µm at the leading edge and a 1.01 growth factor. Likewise, the cells at the walls have a size of δy = 1 µm and a 1.04 growth factor. As a result, this 2D structured mesh is composed of approximately 2.2 million elements. The air is treated as a thermally perfect gas which uses a 7-th order polynomial expression for the specific heat at constant pressure.

Once the solution has converged in time, the wall-normal velocity, temperature and pressure profiles at wall coordinate x ∈ [0.2, 1.2] are extracted into the LST solver MAMOUT. Further information about this solver are given in Esquieu et al. [START_REF] Brazier | Transition prediction on Reentry-F trajectory with PSE at chemical equilibrium[END_REF][START_REF] Esquieu | Flow and Stability Analysis of a Hypersonic Boundary Layer over an Axisymmetric Cone Cylinder Flare Configuration[END_REF] Excitation frequencies f ∈ [180, 200, 220, 240, 260, 280, 300, 360, 420] kHz. The resulting stability analysis is presented in Fig. 5 where the N-factors profiles are plotted for all frequencies along x. The N-factor is defined as the natural logarithm of the pressure oscillation ratio, or as the integral of the exponential spatial growth rate -α i of the most unstable mode. The following observations can be made. Frequencies lower than 180 kHz are not expected to substantially destabilize the boundary layer within one meter of the leading edge. Frequencies higher than 420 kHz will not cause a substantial growth anywhere in the HBL, with a maximum N-factor below 4 located around 30 cm from the leading edge. As a result, the region of interest to be investigated with the DNS was chosen to encompass the region x ∈ [45, 90] cm where the limiting frequencies f = 180 and 420 kHz are inactive while all intermediate frequencies should experience partial growth. Following step 3's instructions, as listed in Sec. A, a first unperturbed DNS is run using hard wall characteristic boundary conditions. It is initialized from the precursor simulation and uses its solution as well to impose the non-reflecting inflow/outflow boundary conditions. It uses the numerical methods presented in Sec. II with a 4 th -order SD method and 6 th -stage Runge-Kutta explicit time-marching scheme. It ran for over half a millisecond until all transients were washed out using about 500 cpuh. The structured mesh is clustered as shown in Fig. 4 and uses N x × N y = 900 × 180 = 162000 elements, or 2592000 SP. It possesses a uniform streamwise spatial increment of 0.5 mm and a first cell size at the wall of 10 µm. At the inlet plane, i.e. x = 45 cm, the boundary layer height δ equals 2.25 mm which results in a computational domain spawning 200 δ and having at least 84 cells across the boundary layer. It is reminded that each cell is then further discretized into 9 solution points (3 along each direction). The air is again treated as a thermally perfect gas. Naturally, it uses the same 7-th order polynomial expression as the precursor simulation to evaluate the specific heat at constant pressure as a function of temperature.

The fourth step listed in Sec. A consists in activating the TDIBC and observe its passive effect on the unperturbed DNS. Based on the N-factor profiles, it was decided to start the TDIBC at x=50cm (where the 260 kHz unstable mode starts its growth) and end it at x = 85 cm (5 cm before the computational domain end). The brutal activation of the TDIBC causes some weak oscillations of normal velocity at the wall which quickly damp as the boundary condition equilibrates. Once the solution has stabilized, the solutions from all three simulations ran up to this point are compared. Figures 6 and7 present various wall-normal profiles of, respectively, the streamwise u and wall-normal v instantaneous (as opposed to time-averaged) velocities for all three cases. Interestingly, while the streamwise profiles appear to collapse on top of each other, this is not the case for the weak v profiles. In the latter case, the precursor simulation presented slightly higher amplitude profiles. This is caused by weak a discontinuity which invariably appears at the sonic point located on the inlet plane in all DNS. The unperfect inflow conditions are responsible for the inevitable appearance of this artificial shocklet (pressure jump inferior to 5 Pascal). In response, the flow is slightly deflected away from the wall, which triggers a weak expansion affecting the velocity profiles. This is further seen in Fig. 8 which compares the wall pressure profiles. Indeed, the differences between the DNS and the precursor solution appear to diminish further away from the inlet. On the other hand, both DNS solutions appear to match throughout the computational domain. Importantly, this validates the low-frequency passivity of the TDIBC. Overall, the discrepancies between all cases are deemed to be sufficiently small to render relevant DNS/LST comparisons.

C. Excitation and stability analysis

Having verified the coherence of the DNS unperturbed flow with the precursor solution, a source is then placed along the wall as discussed in Sec. A (step 5). It is located at x = 47.5 cm, right upstream of the 260 kHz unstable mode's growth region. The HBL response to the following frequencies is now investigated: f ∈ [180, 200, 220, 240, 260, 280, 300, 420] kHz.

As the source activates, a strong transient response is observed in all cases. It is wonderfully revealed by plotting the wall pressure envelope signals in the space-time presented in Figs. 9 and 10 for the 180 and 300 kHz excitation frequencies, respectively. These space-time diagram are built from 451 wall static pressure probes placed every millimeter from x -x s = 0 to 45 cm, where x s = 45 cm is the distance of the DNS inlet plane from the leading edge. First, the 180 kHz case's envelope presents a wave-packet format which indicates a nonlinear dynamic response to the initial perturbation. Eventually, this wave-packet convects through space at the constant speed of sound and exits the computational domain. Interestingly, the pressure oscillation quickly dampens after its passage at every x-station: this confirms the LST results which predicted the x ∈ [45, 90] region to be stable under 180 kHz perturbations. Strikingly, this is not the case for the 300 kHz case: not only is the wall static pressure envelop growing in space, but strong oscillations remain once a permanent response has been established across the computational domain. The LST predicted that the 260 kHz case would, amongst the interrogated frequencies, be the most unstable throughout the region spawned by the DNS. The static pressure contours should, therefore, be mostly responsive to its perturbations. Figure 11 presents snapshots of static pressure perturbations p illustrating the strong growth of waves by many orders of magnitudes over a few tens of centimeters only. Pressure oscillations also appear around the sonic line which is often observed and indicative of secondmode transition [START_REF] Tian | Reverse Design of Ultrasonic Absorptive Coating for the Stabilization of Mack Modes[END_REF][START_REF] Knisely | Significant Supersonic Modes and the Wall Temperature Effect in Hypersonic Boundary Layers[END_REF] Finally, Fig. 12 quantifies the growth of wall pressure perturbations and provides a 13 of 21 American Institute of Aeronautics and Astronautics Paper 2020-2061 comparison with the previously discussed LST results. A superposition of the N-factor plots shown in Fig. 5 with the profiles obtained from the DNS offers little insight (left figure) as the DNS perturbations are only initiated at x -x s = 2.5cm. Hence, a quick rescaling along the y-axis is necessary, and reveals how the profiles collapse remarkably well despite the small discrepancies shown in Fig. 8. It can be noted that the 420 kHz is not plotted in this figure: it was observed to naturally decay and never cross the N-factor positive plane. This is an important feature predicted by the LST that the DNS was able to capture. A comparison of wall-normal pressure perturbation amplitude profiles shown in Fig. 13 for the 260 kHz excitation case further highlights the good agreement between the solvers. Importantly, all these results are a testimony to the DNS capability in mirroring the LST growth rates for the hard wall case. This opens up the way towards the study's next step: perturbations damping through the use of the TDIBC.

V. Analysis of the stabilizing effect of the UAC

The TDIBC presented in Sec. III is now activated and is initially located as shown in Fig. 4. It is calibrated to replicate the broadband response of the DLR's UAC, as was previously explained.

A. Suppression of the wall pressure oscillations by the UAC Just as the activation of the source term in the HBL marked the start of a transient regime (see Figs. 9 and 10), the same occurred when activating the TDIBC of the UAC. This time however, its effect was a strong and quick suppression of the instabilities as seen in Fig. 14 for the 260 kHz case with the wall pressure oscillation amplitude at the outlet plane being resolutely damped.

The stabilizing effect is further studied for the 260 kHz case of interest by looking at the power spectral density of the wall pressure perturbations at x -x s = 4,45 cm, which correspond to positions upstream/downstream of the TDIBC. They are respectively shown in Figs. 15 and16. Importantly, the TDIBC has no recursive effect on the upstream source, which would have raised questions on the pertinence of a direct comparison between the TDIBC and hard wall cases. The power spectral density peak value has been decreased by 6 orders of magnitude: this corresponds to a decrease in wall pressure oscillation amplitude by a factor 1000. Similar damping is observed for the other frequencies as presented in Fig. 17, albeit the amplitude of the pressure perturbation has not grown as much as the with the 260 kHz excitation. Notably, the limiting 180 and 420 kHz excitation cases result in a similarly weak wall pressure power spectral density at the domain's outlet, being outside the range of unstable frequency in this particular region as predicted by the LST.

The impact of the UAC on the instability growth is, once again, quantified by evaluating the N-factor along the streamwise direction. These are plotted in Fig. 18 for the same frequencies shown in Fig. 12. Once again, the 420 kHz does not cross into the positive N-factor plane and was not plotted for clarity purpose. The left figure directly compares the profiles with and without UAC, while the right figure shows the difference between these profiles. This difference corresponds to the integration in space of a ∆α i,UAC directly corresponding to the UAC impact on the pressure growth rate. Interestingly, these profiles appear strikingly linear and almost parallel across the TDIBC region which is reminded to be x ∈ [50, 85]. This indicates that the stabilization factor the UAC provides is roughly equivalent to a constant exponential pressure growth rate α i,UAC = -20. This is very encouraging as it indicates that the TDIBC, homogeneously defined in space and time by its poles and weights (see Tab. 2), also presents a homogeneous effect on the growth rate of unstable modes. However, more surprisingly, it is observed to be practically independent of the frequency for the [180,300] kHz range. A glance at the UAC broadband properties (see Fig. 2) confirms a relatively constant reflectivity bounded between 0.55 and 0.65 around the first anti-resonance for this frequency range. This could explain the aforementioned low spread in damping coefficients.

B. Activation of supersonic mode and radiation of noise

The effect of the UAC on the static pressure field further away from the wall is illustrated in Fig. 19 which corresponds to the contours shown in Fig. 11 once the TDIBC has been activated and all transient washed out. It can be noticed that, besides the contour's bounds decreasing by two orders of magnitude, there appears to be some pressure wave radiation towards the far-field. It originates from within the boundary layer and these waves were not present in the hard wall case. According to Knisely et al., [START_REF] Knisely | Sound radiation by supersonic unstable modes in hypersonic blunt cone boundary layers. I. Linear stability theory[END_REF][START_REF] Knisely | Sound radiation by supersonic unstable modes in hypersonic blunt cone boundary layers. II. Direct numerical simulation[END_REF] the subsidence of pressure oscillations further away from the wall suggests that a fast supersonic mode might become dominant over the slow acoustic mode. This would suggest that the UAC is particularly effective at suppressing the slow mode and less so with the higher ones.

Having observed the effect the UAC has on the acoustic far-field, one wonders if it also has any impact on the time-averaged static pressure contours. These are now scrutinized for the hard wall and UAC-260 kHz excitation cases: they are presented, respectively, in Figs. 20 and21. First, it is reminded that the 260 kHz excitation hard wall case resulted in pressure perturbations on the order of hundreds of Pascals (see Fig. 11). Yet, there appears to be no mark of these strong oscillations on the time-averaged contours, as it should be for a linear super-imposed process. On the other hand, the case with the TDIBC exhibits strong oscillations attached to the wall whose wavelength matches the 260 kHz excitation frequency. Surprisingly, the amplitude of these time-averaged oscillations are also much higher than the instantaneous perturbations shown in Fig. 19. Again, it is reminded that the unperturbed solution using the UAC and hard wall numerical boundary conditions were identical. Therefore, the excited UAC has triggered a strong nonlinear response from the instantaneous and mean flow altogether.

To conclude, a particular trade-off between enhanced stability and noise has been found. It is due to the UAC radiating noise under excitation, amounting to a sort of wall singing phenomenon, and to the formation of high-amplitude standing waves attached to the UAC.

C. Effect of the UAC positioning

The 260 kHz case was further studied by investigating the effect of displacing the TDIBC further downstream on the suppression of the second-mode instabilities. Three new test cases are run where the TDIBC starts at x = 60,70,80 cm instead of 50 cm. The end location is kept at x = 85 cm. This sensitivity study is motivated by the need to understand how much surface has to be treated to achieve a target reduction of HBL instabilities. Coincidently, new LST calculations are also carried at 260 kHz which use the constant impedance resistivity and reactivity calculated from Sousa et al. [START_REF] Sousa | Numerical Investigation of Second-Mode Attenuation over Carbon/Carbon Porous Surfaces[END_REF] at this particular frequency. This permits to evaluate the LST ability in locally resolving these complex flows on its own. The N-factor profiles from the DNS and LST for all configurations are presented in Fig. 22. First, all DNS and LST seem to react similarly near the TDIBC edge, independently of its placement. Second, the longer the UAC, the stronger the suppression of instability waves. Third, the LST and DNS solutions appear to diverge and the discrepancies appear to build in space. This is probably due to the modification of the time-averaged supporting pressure field by the TDIBC observed in the DNS. The LST is not able to resolve this nonlinear response and over-predicts the suppression capabilities of the UAC. Finally, this study concludes by observing the impact the UAC placement has on the perturbations further away from the wall. To this end, the x = 90 cm wall-normal profiles of pressure oscillation amplitudes, p [dB] are plotted in Fig. 23. This corresponds to the outlet plane, 5 cm after the end of any TDIBC. As expected, the longest the UAC, the more suppressed the oscillations are close to the wall. However, further away, there does not appear to be a linear relation between UAC length and radiation noise levels. In fact, placing a small portion of UAC later downstream once the instabilities have already grown considerably tends to backfire, as is done with the UAC-80 case. This suggests that further studies are needed to optimize the use of UAC for passive control is hypersonic boundary layers.

VI. Conclusions

The second-mode stabilizing capacity of ultra-sonically absorbent coatings for hypersonic boundary layers has been studied by the means of direct numerical simulations.

These high-fidelity simulations used a high-order spectral difference method, which is unprecedented for such applications to the author's best knowledge. This numerical scheme is able to retain a high-order of accuracy on unstructured meshes, hence its importance in simulating complex geometries. Until recently, however, this family of schemes was missing proper boundary conditions required to resolve aeroacoustics problem. Notably, non-reflecting boundary conditions required to prevent spurious reflections from polluting the inner domain acoustic were only recently implemented with SD. [START_REF] Fiévet | Strong compact formalism for characteristic boundary conditions with discontinuous spectral methods[END_REF] Coincidently, time-domain impedance boundary conditions recently derived by Monteghetti et al., [START_REF] Monteghetti | Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations[END_REF] proven to be broadband-accurate, were also enabled with SD. [START_REF] Fiévet | Strong compact formalism for characteristic boundary conditions with discontinuous spectral methods[END_REF] The conjunction of both types of boundary conditions has resulted in the development of a high-order numerical flow solver aimed at investigating the underlying physics of hypersonic boundary-layer transition and eventually designing UAC-based passive control devices.

Specifically, the solver ability in resolving the growth rate of unstable mode was demonstrated through comparison with linear stability theory. The use of UAC, modeled through TDIBC, was also proven to be a formidable passive control device and promising technology. Further, the suppression of second-mode instability was observed to coincidentally trigger the radiation of sound outside the boundary layer.

These results highlight the need to further develop numerical tools to guide the design of optimal coatings mitigating the trade-off. To this end, the implementation of adjoint-based optimization tools closely coupled with these high-order methods are the subject of ongoing work. Overall, this seminal study is a testament to the maturity this integrated numerical platform, JAGUAR, has reached. It serves as a milestone towards the achievement of the aforementioned long-term objectives.

Figure 2 . 1 :Figure 2 . 2 :Figure 2 . 1 :Figure 2 . 2 :Figure 2 . 1 :Figure 2 . 3 :Figure 2 . 4 :FFigure 2 . 2 :

 2122212221232422 Figure 2.1: Step 0: Solution points (N), solution (•) and 2 nd order polynomial on the segment Figure 2.2: Step 1: Extrapolation of solution (•) at the flux points (H)

Figure 2 . 5 :

 25 Figure 2.5: Step 4: Building a new polynomial at flux points (at degree p + 1) Figure 2.6: Step 5: Di erentiation of the flux polynomial at solution points The last step consists in di erentiating the polynomial to compute the flux increment at each solution point, Fig. 2.6.

Figure 2 . 5 :

 25 Figure 2.5: Step 4: Building a new polynomial at flux points (at degree p + 1) Figure 2.6: Step 5: Di erentiation of the flux polynomial at solution points The last step consists in di erentiating the polynomial to compute the flux increment at each solution point, Fig. 2.6.

1

  Figure 1. Global algorithm of the p=2 SD method in 1D reproduced from Cassagne et al.34

Figure 2 .

 2 Figure 2. Discrete ODR of the UAC from Sousa et al. 25 using 4 diffusive and 8 oscillatory poles.

Figure 3 .

 3 Figure 3. Comparison between analytical and DNS pressure perturbations signals from an impedance tube numerical experiment ran at different frequencies.

Figure 4 .

 4 Figure 4. (Top) Configuration of the Mach 7.4 dihedral precursor simulation. (Bottom) Configuration of the HBL direct numerical simulation with excitation and UAC stabilization which uses characteristic boundary conditions calibrated from the precursor simulation. Only 1/24 mesh elements are displayed.

Figure 5 .

 5 Figure 5. N-factor streamwise profiles obtained from the precursor simulation using linear stability theory.

Figure 6 .Figure 7 .

 67 Figure 6. Comparison of instantaneous wall-normal profiles of unperturbed streamwise velocity at various streamwise locations across the hypersonic boundary layer.

Figure 8 .

 8 Figure 8. Comparison of unperturbed instantaneous wall static pressure profiles along the wall starting at xs = 45 cm.

Figure 9 .

 9 Figure 9. Space-time envelop of a wall static pressure 180 kHz-perturbation introduced at x -xs = 2.5cm as it excites the boundary layer (colored by static pressure p [Pa]).

Figure 10 .

 10 Figure 10. Space-time envelop of a wall static pressure 300 kHz-perturbation introduced at x -xs = 2.5cm as it excites the boundary layer (colored by static pressure p [Pa]).

Figure 11 .Figure 12 .

 1112 Figure 11. Static pressure perturbations snapshots for (top) x ∈ xs + [20, 30] and (bottom) x ∈ xs + [30, 40] for the hard wall with a 260 kHz-excitation case.

Figure 13 .Figure 14 .

 1314 Figure 13. Wall-normal pressure perturbations amplitude at different streamwise locations for the LST and DNS solutions when using a hard wall boundary condition.

1 ]Figure 15 . 1 ]Figure 16 .

 115116 Figure 15. (Top) Wall pressure time-signals at x -xs = 4 cm and (bottom) their corresponding power spectral densities for the 260 kHz-excitation case over the hard and ultrasonically absorbent walls.

Figure 17 .Figure 18 .

 1718 Figure 17. Power spectral densities of the x = 85 cm wall pressure signals for various excitation frequencies over the hard and ultrasonically absorbent walls.

Figure 19 .

 19 Figure 19. Static pressure perturbations snapshots for (top) x ∈ xs + [20, 30] and (bottom) x ∈ xs + [30, 40] for the UAC wall with a 260 kHz-excitation case.

Figure 20 .

 20 Figure 20. Time-averaged contours p -6712 [Pa] for (top) x ∈ xs + [20, 30] and (bottom) x ∈ xs + [30, 40] for the hard wall with a 260 kHz-excitation case.

Figure 21 .

 21 Figure 21. Time-averaged contours p -6712 [Pa] for (top) x ∈ xs + [20, 30] and (bottom) x ∈ xs + [30, 40] for the UAC with a 260 kHz-excitation case.

Figure 22 .

 22 Figure 22. N-factor profiles for the 260 kHz-perturbation case for various wall boundary conditions using the (+) LST and the (dashed) DNS.

Figure 23 .

 23 Figure 23. Wall-normal profile of pressure perturbations measured in dB at x=90 cm (outlet plane) for various UAC positions.

Table 1 .

 1 Far-field conditions for the hypersonic dihedral simulation.

	s]

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Acknowledgement

The first author is funded by the STAE foundation (Science and Technology for Aeronautics and Space), grant number CDT-R050-L00-T00. This work is part of the STAE-funded 3C2T project (control of compressible, transitional and turbulent boundary layer) involving ONERA, CERFACS and the IMFT under the supervision of Christophe Airiau. The authors also thank Loïc Jecker for his kind help in configuring the precursor simulation using the code CEDRE.

Poles Functions weights (s n , -ξk ) n,k (rad.s -1 ) (r 1,n , -μ 1,k ) n,k (rad.s -1 ) (r 2,n , -μ 2,k ) n,k (rad.s -1 ) -1.256637e+3 5.660893e+0 4.277775e+0 -2.148819e+4 6.807834e+2 9.184530e+2 -3.674429e+5

3.589210e+4 1.261821e+4 -6.283185e+6 -1.331916e+6 -4.117427e+4 (-1.974050e+5,3.1685357e+5) (-5.737098e+4,-8.382613e+3) (1.115722e+4,-2.097452e+4) (-3.544161e+2,1.234667e+6) (-5.028205e+4,-1.018499e+4) (-4.218533e+3,-9.128232e+3) (-4.614602e+5,2.198411e+6) (-5.601881e+4,-1.478211e+4) (-4.509185e+3,-2.663290e+3) (-5.482225e+5,3.181595e+6) (-6.548491e+4,-1.802480e+4) (-2.827210e+3,-2.504131e+2) (-6.231411e+5,4.176283e+0) (-8.044434e+4,-1.886116e+4) (-1.356928e+3,2.770425e+2) (-6.900477e+5,5.178806e+6) (-1.101268e+5,-1.467016e+4) (-3.609999e+2,-1.731994e+2) (-7.510679e+5,6.187110e+6) (-2.339126e+5,-2.490394e+4) (1.160141e+3,-2.148801e+3) (-8.075265e+5,7.199905e+6) (-5.691639e+4,-5.317827e+5) (6.532230e+3,3.590720e+3) Table 2. Scalars defining the ODR representation of the UAC presented in Fig. 2.

Appendix

The table below presents all the scalars used to discretize the TDIBC matching the broadband profile shown in Fig. 2.

Scalars

β ∞ = 8.187879e -1 τ = 5.877685e -6 s


text:
Pierre Bourdon

Hermance Jacqmin

Laurent Lombard

Béatrice Augere

Anne Durécu

Didier Goular

Roland Domel

Didier Fleury

Christophe Planchat

Bastien Rouzé

Target-in-the-loop coherent combining of 7 fiber lasers: first results

teaching and research institutions in France or abroad, or from public or private research centers.

ONERA -The French Aerospace Lab, DOTA, BP 80100, F-91123 Palaiseau, France, Email: pierre.bourdon@onera.fr (2) DSO National Laboratories, 14 Science Park Drive, Singapore 118226, Singapore KEYWORDS: Fiber laser, coherent combining, phase control, target-in-the-loop, turbulence mitigation

ABSTRACT:

We developed a laser testbed coherently combining seven 1.5-µm fiber lasers through active phase control, using frequency-tagging to assess the phase fluctuations to be compensated for. In this paper, we present the testbed and its components as well as the results obtained in direct coherent combining, operated at the output of the lasers, without target-in-the-loop feedback. We then introduce the basic principle and show the first indoor results of target-in-the-loop coherent beam combining experiments at a shorter distance of 15 meters. We then describe the outdoor facility that was chosen for longer range experiments and show qualitative results and qualitative comparisons of the target-in-the-loop coherent beam combining challenge when the target is moved farther away, increasing the range from 15 meters to up to 1 km.

INTRODUCTION

Coherent combining techniques involving active phase control of the laser emitters have demonstrated their potential to power scale continuous-wave fiber lasers through coherent addition of the power emitted by multiple separate amplifiers up to the multi-kilowatt level [START_REF] Flores | Coherent beam combining of fiber amplifiers in a kW regime[END_REF]. Live phase-control using fast electro-optic phase modulators is very effective to achieve low residual phase error and high efficiency coherent beam combining (CBC). CBC has been demonstrated for a large number of lasers [START_REF] Bourderionnet | Collective coherent phase combining of 64 fibers[END_REF] but many realisations use the fact that the laser beams are spatially separated at the output of the lasers, before overlapping after propagation and becoming undistinguishable. While the laser beams are still spatially distinct, it's feasible to measure the phase of each laser and to control this phase in real time to achieve phaselocking.

However, for some applications, such process cannot be applied as phase measurement has to be done once the laser beams have already begun to overlap and to interfere. That's the case of target-in-the-loop CBC where one wants to phaselock the lasers on a remote target, maximizing the power density deposited at long range. In this case, phase measurement at the output of the lasers is useless, as beam propagation induces additional phase shifts that are not accounted for in this measurement. It's necessary to find some way of driving the laser phases, benefiting from the information available in the optical signal backscattered by the remote target.

One approach is to try and maximize the intensity of this backscattered signal, as maximum power density deposited on the target corresponds to a maximum of this backscattered signal too. It's been done, for instance, very early on, during the first experiments of TIL-CBC on a glint target [START_REF] Pearson | Coherent optical adaptive techniques: design and performance of an 18-element visible multidither COAT system[END_REF], and, later on, up to 7 km using a stochastic parallel gradient descent algorithm [START_REF] Weyrauch | Experimental demonstration of coherent beam combining over a 7 km propagation path[END_REF], a classical approach to maximize optical intensity by step-bystep optical wavefront correction.

In 2009, our team demonstrated a more practical and promising approach that can be performed using frequency-tagging for CBC phase control [START_REF] Jolivet | Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence[END_REF]. Frequency-tagging each optical channel at a specific frequency to assess the phase fluctuations to be compensated for is an efficient technique for CBC, also known as LOCSET [START_REF] Shay | First experimental demonstration of self-synchronous phase locking of an optical array[END_REF]. Frequencytagging has the advantage of "engraving" specific information on the signal emitted by each optical channel through low-depth modulation. This tagging process enables to easily retrieve the phase information from each channel, within the complex interference signal generated once the laser beams have overlapped. Moreover, this phase information retrieval can also be achieved using the backscattered signal from a remote target. We also demonstrated that a simple evolution of the configuration of detection used for direct CBC could transform a standard CBC system into a fully operational TIL-CBC device [START_REF] Jolivet | Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence[END_REF].

The tests we performed in 2009 were in the laboratory with artificially generated atmospheric turbulence. In this paper, we present the work done recently to build a testbed coherently combining seven 1.5-µm fiber lasers through active phase control, using frequency-tagging. We also present the first results obtained on TIL-CBC operating this testbed. These results are obtained propagating the laser beams through the atmosphere, hence through real atmospheric turbulence. In a first part of this paper, we present the laser testbed and its operation in direct coherent combining, operated at the output of the lasers, without the target in the loop feedback. In a second part, we present the first results obtained on TIL-CBC when propagating through real turbulence and increasing the propagation range from 15 meters to 1 kilometer.

THE LASER TESBED FOR CBC AND TIL-CBC

The laser configuration is a standard 7-channel master oscillator power amplifier (MOPA) configuration at 1.5 µm wavelength.

A single-frequency low power (40 mW) master oscillator (MO) is split and amplified into 7 channels. Each of the 7 Er-Yb doped fiber amplifiers delivers up to 3 W at 1543 nm. One of the fiber amplifiers is not phase controlled and is free to fluctuate in phase while in-between the MO and 6 of the amplifiers, we use a fast electro-optic phase modulator with a 150-MHz bandwidth to control the phase of each channel and simultaneously tag each of these 6 channels at a specific frequency around 20 MHz with a lowdepth phase modulation.

In this self-referenced LOCSET configuration, the 6 phase-controlled channels follow the phase fluctuations of the seventh channel that's not modulated (see figure 1).

Master Oscillator The 7-channel CBC setup delivers a nice interference pattern in the far-field. We confirmed that in direct CBC, i.e. using a fast photodiode to capture part of the central lobe of interference in the far-field, it was easy to lock the phases of the 7 laser channels and stabilize the position of the interference pattern.

In LOCSET, frequency tagging of the i th channel at a specific frequency ν i results in an intereference signal that generates an error signal after demodulation at the same frequency given by equation ( 1): [START_REF] Flores | Coherent beam combining of fiber amplifiers in a kW regime[END_REF] where φ i is the i th amplifier chain output phase in the photodetector plane, and φ u the unmodulated reference beam phase in the photodetector plane. E i and E u are the respective electric field amplitudes at the output of the i th amplifier and of the unmodulated amplifier. β i and β u are the respective modulation depths of the i th amplifier channel and of the unmodulated channel. Driving the 6 error signals simultaneously to zero in real time results in maintaining equal phases for the 7 channels. Any sudden phase shift is immediately compensated for, thanks to the very fast electro-optic phase modulators and fast detection.

CBC efficiency measured as the fraction of the total power located in the central lobe of the interference pattern was higher than 53 % (see figure 2). It's very close to the experimental maximum that can be achieved with Gaussian laser beams (theoretical maximum is 63 %) taking into account the opto-mechanical constraints limiting the effective fill-factor in the near-field. Residual phase error has been measured and is lower than λ/40, confirming again the high efficiency of the CBC process and the high speed of the feedback loop.

% P

The opto-mechanical head where the outputs of the 7 fiber amplifiers are collimated and sent through the atmosphere is completely adjustable (see figure 3).

Piezo-electric motors are used to set the transverse position of the output fiber tip with respect to the collimating lens, in order to align the 7 collimated beams in the same direction. The focus of the collimating lenses is manually tuned beforehand. Figure 3 presents pictures of the first part of the testbed dedicated to laser emission. For TIL-CBC, a second stage of the testbed is dedicated to long range focusing of the laser beams and to collection and detection of the backscattered signal from the remote target. We chose to use a bi-static configuration (see figure 4) where the long range focusing and the reception apertures are separated, to limit the risk of narcissus effect, and decrease the optical complexity of the detection line with respects to a monostatic configuration. 1, the detection line now receives the signal backscattered by the remote target.

( ) ( ) ( ) ( ) ( )       - + - ∝ ∑ - = 1 1 0 1 _ sin sin N j i j j j i u u i i i error J E E J E t i φ φ β φ φ β … … n-

Figure 4. Schematics of the TIL-CBC bi-static configuration. Instead of collecting a fraction of the interference signal between the laser beams at the output of the laser system as in figure

EXPERIMENTAL DEMONSTRATIONS OF TIL-CBC AT SHORT RANGE

The first experimental tests of TIL-CBC were conducted in the laboratory. As the experiments took place inside closed doors, atmospheric turbulence was very weak and it was easy to achieve similar results in TIL-CBC as in direct CBC.

The target we used was a small diameter high reflectance area obtained by sticking high reflectance tape on a plate and setting up a blackened (i.e. low albedo) aperture in front of this high reflectance target to limit its size.

Indoor one-way TIL-CBC experiments could be conducted up to 15 meters. We used a large aluminium mirror to fold the optical path back and operate round-trip TIL-CBC experiments in the laboratory up to 30 meters (see figure 5).

Figure 5. Schematics of the experimental setup for 30-m range TIL-CBC with optical path folding mirror to double the range of the experiment. The interference pattern is monitored with a camera located at the same distance from the laser as the target.

In both cases, TIL-CBC operated extremely well, which is not surprising as turbulence was extremely weak.

Next step was to move the testbed to another laboratory to be able to shoot the laser beam outside, propagating through real outdoor turbulence.

Once again, the results were excellent and TIL-CBC operated well. Due to beam wander induced by the outdoor turbulence, TIL-CBC was a little less stable, but we never observed TIL-CBC collapse due to turbulence. These short range one-way outdoor experiments were conducted up to 45 meters. Using the same metallic mirror as indoors to fold back the optical path, we doubled this range up to 90 meters and still observed very efficient TIL-CBC at this distance (see figure 6). We were also able to perform beam-steering, as TIL-CBC was locked on the small diameter target: when moving the target to the right or to the left, the interference pattern will follow the movement of the target. That's just a standard capability of direct frequency-tagging CBC that can be extended to TIL-CBC this way: in direct CBC, when the detector is moved, the interference pattern moves to follow in order to maximize the power density deposited on the detector.

EXPERIMENTAL DEMONSTRATIONS OF TIL-CBC AT LONG RANGE

The testbed has been designed to be mobile and we transferred it to an Onera laser site where we are able to shoot lasers safely up to 1-km range (see figure 7).

The site is equipped with equipment to monitor the weather and atmospheric conditions, and also to measure turbulence strength through the average value of the C n ² over 1 km. The experiments took place during summer in the south west of France. As the laser beams propagated horizontally at 1.5 m from the ground, the level of turbulence was quite high and the average value of C n ² over 1 km was often close or even much higher than 10 -13 m -2/3 . Despite these detrimental turbulence conditions, when the C n ² value was lower than 10 -13 m -2/3

X1

, we were able to perform efficient TIL-CBC up to 1 km. A transparent wedge plate was placed before the target, to reflect a small fraction of the combined beams power to a black screen located at the same distance as the target. Proceeding like this, we could record video sequences of the interference pattern displayed on black screen and observed with an InGaAs Raptor camera, without interfering with the TIL-CBC process in the target plane. Even if turbulence induced perturbations clearly appear to decrease the efficiency of the CBC locking process, TIL-CBC still operates and results in an increased power density deposited on the remote target at 1 km.

CONCLUSION AND FUTURE WORK

In this paper, we present a mobile testbed for coherent beam combining of 7 fiber lasers emitting at 1.5 µm. This testbed operates through frequency-tagging phase difference monitoring. It can concentrate more than 53 % of the overall power in the central of the interference pattern generated in the far-field, after overlap of the laser beams.

The testbed has been designed to use the backscattered signal coming from a remote target as a reference for phase difference monitoring and phase locking. It was operated in a target-in-theloop configuration, first at short distance in the laboratory, and then at longer distance outdoors, propagating the beams through real, weak to strong atmospheric turbulence.

The longer range outdoor experimental campaign demonstrated that TIL-CBC efficiency is preserved up to more than 310 m. Perturbations induced by strong atmospheric turbulence are significantly affecting TIL-CBC, but TIL-CBC operates at 1 km anyway, however with lower efficiency. Future work is dedicated to processing the data recorded during this experimental campaign to derive more quantitative results.

1 Figure 1 .

 11 Figure 1. Schematics of the experimental setup for frequency-tagging CBC (also known as LOCSET).

Figure 2 .

 2 Figure 2. Interference pattern in the far-field, captured on a 1.5-µm camera located at the same distance as the direct CBC detector.

Figure 3 .

 3 Figure 3. Pictures of the experimental setup parts.

Figure 6 .

 6 Figure 6. TIL-CBC Interference pattern in the far-field at 90 m, captured on a 1.5-µm camera located at the same distance as the TIL-CBC target.

Figure 7 .

 7 Figure 7. Pictures of the laser site where the long-range TIL-CBC experiments took place.

Figure 8 Figure 8 .

 88 Figure 8. Instantaneous images of the interference pattern generated at 310 m when TIL-CBC feedback loop is active and inactive.

Figure 9 Figure 9 .

 99 Figure 9 presents averaged images from the same video sequence. The size of the interference lobes is not different in the instantaneous images and in the averaged image, demonstrating that the interference pattern was locked on a fixed position in the target plane. CBC ON CBC OFF

Figure 10

 10 Figure10presents the averaged image obtained when target distance was 1 km.

ACKNOWLEDGEMENTS

This work was funded by DSO National Laboratories, Singapore.


text:
Keywords: Optical parametric oscillator, frequency converter, nonlinear optics, coherent combining, phase control

Coherent combining (CBC) techniques are very efficient to power scale lasers to the multikilowatt level. However, high power fiber lasers are only available within a finite set of wavelengths, mainly amongst 1 µm, 1.5 µm and 2 µm. Hopefully, nonlinear crystals offer the capability to convert these laser lines to access higher wavelengths in the midinfrared bands, for instance between 3 and 5 µm. In this paper, we present the challenges of applying coherent combining by active phase control to much more powerful frequency converters: optical parametric oscillators (OPO). An experimental demonstration of OPO CBC is also detailed.

COHERENT COMBINING OF OPTICAL PARAMETRIC OSCILLATORS: CHALLENGES AND EXPERIMENTAL DEMONSTRATIONS

P. Bourdon (1) , R. Chtouki (1) , L. Lombard (1) , C. Planchat (1) , M. Raybaut (2) , A. Godard (2) , A. Durécu

(1) (1) ONERA -The French Aerospace Lab, DOTA, BP 80100, F-91123 Palaiseau, France, Email: pierre.bourdon@onera.fr (2) ONERA -The French Aerospace Lab, DPHY, BP 80100, F-91123 Palaiseau, France

INTRODUCTION

Coherent beam combining (CBC) techniques involving active phase control of the laser emitters have demonstrated their potential to power scale continuous-wave [START_REF] Flores | Coherent beam combining of fiber amplifiers in a kW regime[END_REF] and pulsed [START_REF] Lombard | Coherent beam combination of narrow-linewidth 1.5 µm fiber amplifiers in a long-pulse regime[END_REF] fiber lasers through coherent addition of the power emitted by multiple separate amplifiers. Using all-fiber components, very compact configurations of coherently combined fiber amplifiers can be designed, with demonstrated overall emitted power exceeding the kilowatt level [START_REF] Flores | Coherent beam combining of fiber amplifiers in a kW regime[END_REF]. However, high power fiber lasers are only available within a finite set of wavelengths, mainly amongst 1 µm, 1.5 µm and 2 µm. Hopefully, nonlinear crystals offer the capability to convert these laser lines to access higher wavelengths in the midinfrared bands, for instance around 4 µm. But as with lasers, power scaling of frequency converters is limited, essentially by the damage threshold of the nonlinear medium. Coherent combining techniques would be extremely useful to power scale frequency converters beyond this damage threshold limit. Unfortunately, combination techniques are difficult to implement in this case, as active phase control requires fast phase modulators at the midinfrared wavelength emitted by the converter. Such components are not exactly "off-the-shelf" and are not as practical nor as fast as the all-fiber modulators available at more standard wavelengths. Fortunately, taking advantage of the phasematching relations involved in any frequency converting process, one can indirectly control the converted wave phase through control of the pump wave phase. When this pump wave is delivered by fiber amplifiers, such a frequency-converter coherent combining configuration can be achieved using standard off-the-shelf all-fiber phase modulators. Our team successfully applied this approach to perform efficient coherent combination of second harmonic generators and difference frequency generators [START_REF] Odier | Coherent combining of second-harmonic generators by active phase control of the fundamental waves[END_REF], [START_REF] Odier | Coherent combining of fiber-laser-pumped 3.4-µm frequency converters[END_REF], [START_REF] Odier | Coherent combining of mid-infrared difference frequency generators[END_REF]. In this paper, we present the challenges of extending this indirect phase control technique to midinfrared OPO coherent combining. An experimental demonstration of OPO CBC is also detailed. Results will be presented during the conference.

PRINCIPLE AND EXPERIMENTAL SETUP FOR CBC OF PARAMETRIC AMPLIFIERS

Coherent beam combining of non-degenerate three wave mixing nonlinear processes was first demonstrated on devices that didn't require a cavity to operate: difference frequency generators. But the theoretical bases are identical to those of OPO CBC, as both experiments combine parametric amplifiers.

To achieve CBC of DFGs or OPOs, phase control of one of the converted waves, the signal or the idler waves, has to be performed. Indirect phase control benefits from the phasematching condition.

The electric field corresponding to each nonlinearly coupled wave can be written as:

, = - + . (1)
with or p and the complex amplitude of the pulsation wave. If we assume that the pump wave is not depleted when passing through the crystal, which is true on the first millimeters of the nonlinear crystal where parametric amplification is still weak, the coupledamplitude equations describing difference frequency generation are:

= * ∆ = * ∆ (2)
where ∆# = # p -# s -# i . Propagation equation of the idler wave is:

= ' * ' ( ) ) (3)
with ' =

. A similar equation can be written for the signal wave. Maximum power conversion is achieved for perfect phase-matching condition, i.e. ∆k = 0.

This system of equations can be solved, in the case of an unseeded idler wave A i (0) = 0, into:

= * +,-.' ' ) ) = ' * .' ' ) ) * * ,(/-. ' ' ) ) (4)
These solutions correspond to a signal field that keeps its initial phase-offset, and is simply amplified by the interaction, while the idler wave phase-offset depends on both the pump wave's and the signal wave's.

0 = 0 -0 + (5-a)
Similarly, in the QPM case, high conversion efficiency can be achieved in a periodically poled crystal. The relation between the phases of the pump, signal and idler waves can be obtained propagating the waves in the crystal, domain by domain and step by step:

0 = 0 -0 (5-b)
Due to this phase relation, controlling the pump wave phase gives complete control of the idler wave phase. For instance, in the QPM case, a ϕ p0 phase shift induced on the pump wave will compensate for an idler wave phase variation of ϕ p0 -ϕ s0 , where ϕ s0 is the initial phase of the signal wave at the entrance of the nonlinear crystal. Consequently, phase control of DFG and OPO converters is achievable, using all-fiber electrooptic phase modulators to control the pump wave.

EXPERIMENTAL DEMONSTRATIONS OF DFG CBC

We demonstrated experimentally the feasibility of such indirect phase control combining two DFG modules pumped by 1.064-µm Yb:fiber lasers delivering up to 15 W power each.

To fix and maintain the signal wavelength and phase, it is necessary to seed the DFGs with a signal seed at 1553 nm from a laser diode amplified by a commercial erbium-doped fiber amplifier split in two. Using this 1553-nm seeder, we are able to keep both DFG channels at a common and fixed signal wavelength.

The DFG nonlinear crystals are PPLN crystal from Covesion and their length is L = 20 mm. With a 1064-nm pump wavelength and a seeded signal at 1553 nm, quasi-phase matching in the PPLN crystal is obtained with a poling period of 30.49 µm and at an operating temperature of 60 °C for both crystals, generating an idler wave at 3.4 µm. The pump beams are focused down to 100 µm waist in the PPLN crystals. Coherent combining of these DFG modules is achieved using the idler waves interference signal to close the feedback loop (see figure 2).

Figure 2. Interference signal of the DFG idler waves, fluctuating between destructive and constructive interference when CBC is inactive and stabilizing on a constructive interference state when phase-locking is active.

Time-averaged combining efficiency is excellent: the residual phase error is λ/28 rms for the idler wave. This experiment is the proof that CBC of continuous wave DFG using indirect phase control is feasible, hence proving that OPO CBC is also feasible as the nonlinear process involved in both cases is the same: parametric amplification.

The main difference between SHG CBC and DFG CBC is that simultaneous phase locking that has been proved feasible for SHG combination [START_REF] Odier | Coherent combining of second-harmonic generators by active phase control of the fundamental waves[END_REF], cannot be achieved so easily when combining nondegenerate three-wave mixing frequency converters [START_REF] Odier | Coherent combining of mid-infrared difference frequency generators[END_REF].

Even if seeded, the optical path difference between the two signal channels induces a phase difference that is not compensated for, and the signal waves are not phase-locked simultaneously when the idler waves are.

CHALLENGES OF OPO CBC

The main drawback of DFG processes is their extremely low efficiency. For instance, pumping the DFG modules of the CBC experiments with 15 W at 1.064 µm, the power generated at 3.4 µm was only a few milliwatts. But breaking records in terms of power was not the goal of the DFG CBC experiment previously presented. OPO CBC is much more interesting as placing a parametric amplifier in a cavity can significantly increase the efficiency of the parametric process.

OPOs have higher efficiencies than DFGs (10 -20 % efficiency can be achieved in OPOs) and demonstrating experimentaly that CBC of OPO is feasible could offer an interesting option to power scale frequency converters. However, an OPO places the parametric amplifier inside a cavity that adds some phase constraints to the process and indirect phase control of an OPO is not as simple as for the DFG modules. There are two main architectures for a continuous wave OPO, as can be seen in figure 3: ring and linear cavities can be used. A ring cavity generally offers lower threshold and the first challenge of developing cw OPOs for CBC was to lower as much as possible their threshold.

Lowering the threshold too much is not suitable either, as a cw OPO becomes unstable in terms of emitted power when pumped more than 3 times its threshold. Analytical estimates of continuous wave OPOs are available in the literature [START_REF] Sutherland | Handbook of nonlinear optics[END_REF].

Optimizing the design of the cavity in terms of pump, signal and idler beams overlap and minimizing the threshold simultaneously, different configurations with a theoretical threshold around 10 W were chosen and tested. Another configuration of nested cavity doubly resonant OPO (NesCOPO [START_REF] Hardy | Compact, single-frequency, doubly resonant optical parametric oscillator pumped in an achromatic phase-adapted double-pass geometry[END_REF]) was also optimized with a threshold that could be as low as a few hundred milliwatts and fine wavelength tuning capability.

Combining these OPOs could lead to interesting sources that could be power scaled by increasing the number of OPOs combined, and benefit from the fine tuning capability of one of them.

The last challenge to overcome is the coupling between two feedback loops that is required by OPO CBC. As can be seen in figure 4, the phase control feedback loop has of course to be operated to perform efficient OPO CBC. But simultaneously, one has to guarantee that the idler wavelength amplified in both OPO cavities is the same. That can only be achieved by adding a second feedback loop controlling the cavity length. Operating both feedback loops simultaneously is the main challenge of OPO CBC.

OPO CBC EXPERIMENTS

We set up an OPO CBC experiment and we are testing the double feedback loop control to check if OPO CBC is practically feasible. The experimental setup is similar to the one presented in figure 4: two singly resonant OPOs with low threshold pumped by two 15 -20 W emitting Yb:fiber amplifiers are combined. Simultaneous control of the phase of one of the pumps and of the cavity length of one of the OPOs is tested. First results of these experiments will be presented during the conference.

CONCLUSION AND FUTURE WORK

In this paper, the challenges of applying coherent combining by active phase control to much more powerful frequency converters, optical parametric oscillators (OPO), is presented. Optimizing the OPO threshold and coupling two different feedbacks are the main challenges in designing an optimal experiment. We also described the experimental demonstration of OPO CBC we are now testing.

The first results of this experiment will be presented during the conference.

Figure 1 .

 1 Figure 1. Schematics of the experimental setup for demonstrating CBC of two 3.4-µm difference frequency generators by phase-control of the 1.064-µm pump wave.

Figure 3 .

 3 Figure 3. Cavity architectures of continuous wave OPOs.

Figure 4 .

 4 Figure 4. Schematics of the basic architecture required to combine two OPOs.As can be seen in figure4, the phase control feedback loop has of course to be operated to perform efficient OPO CBC. But simultaneously, one has to guarantee that the idler wavelength amplified in both OPO cavities is the same. That can only be achieved by adding a second feedback loop controlling the cavity length. Operating both feedback loops simultaneously is the main challenge of OPO CBC.

ACKNOWLEDGEMENTS

This work was partially funded by Delegation Generale pour l'Armement.