Romain Gojon
email: gojon@kth.se

Florian Baier
email: baierfn@mail.uc.edu

Ephraim Gutmark
email: gutmarej@ucmail.uc.edu

Mihai Mihaescu

Temperature effects on the aerodynamic and acoustic fields of a rectangular supersonic jet

In the first part of the paper, a modified artificial dissipation mechanism permitting to perform Large-Eddy Simulations of highly compressible flows is proposed. This dissipation mechanism is evaluated using one linear 2-D test case and two non-linear 2-D test cases. In the second part, the flow and acoustic near-field of rectangular supersonic jets are explored using compressible LES based on this modified artificial dissipation mechanism. At the exit of a converging diverging rectangular nozzle of aspect ratio 2 and of design Mach number 1.5, the jets are overexpanded. Four simulations with four different temperature ratios ranging from 1 to 3 are performed in order to characterize the effect of the temperature on the aerodynamic and aeroacoustic fields of the jets. The geometry of the nozzle and the exit conditions are chosen in order to match those in the experimental study conducted at the University of Cincinnati. It is shown that the total number of cells in the shock cell structure decreases with the increase of the temperature ratio. However, the temperature does not influence the size of the first shock cell and the linear decrease of the shock cell size in the downstream direction. The Overall Sound Pressure Levels are then plotted along the minor and major axis. It is seen that the intensity of the screech feedback mechanism increases with the Temperature Ratio. Moreover, for JetTR25 and JetTR3, the strong flapping motion of the jet along the minor axis due to the screech feedback mechanism seems to yield to an asymmetric organization of the Mach wave radiation. The convection velocity of the turbulent structures in the jet shear layers along the minor axis is then studied. Once normalized by the jet exit velocity, the convection velocity is shown to decrease with the jet temperature ratio. In the last part of the paper, the near-and farfield acoustic are studied. In the near-field, screech tones which frequencies are consistent with both experimental data and a theoretical model are observed. In the far-field, four acoustic components typical of non-ideally supersonic jets are observed, namely the screech noise, the broadband shock-associated noise, the mixing noise and the Mach wave noise. Their directivities and frequencies are in agreement with experimental results and models.

I. Introduction

In future high-speed aircraft, rectangular propulsion systems closely integrated to the fuselage are considered. Supersonic non-ideally expanded jets exiting from such geometries can lead to very intense acoustic noise. Several acoustic components can be observed including screech noise, mixing noise, broadband shockassociated noise and Mach wave noise. The screech noise is due to an aeroacoustic feedback mechanism establishing between acoustic waves propagating upstream and turbulent structures propagating downstream. This mechanism was described by Powell [START_REF] Powell | On the mechanism of choked jet noise[END_REF] and by Raman 2 and consists of two step. First, the turbulent structures developing in the jet shear layers and propagating in the downstream direction interact with the quasi-periodic shock cell structure of the jet, creating upstream propagating acoustic waves. Then, the resonant loop is closed at the nozzle trailing edge where the jet shear layers are excited when the upstream propagating acoustic waves impinge on the nozzle lips. Mixing noise is observed in both subsonic [START_REF] Bogey | Investigation of downstream and sideline subsonic jet noise using large eddy simulation[END_REF] and supersonic [START_REF] Berland | Numerical study of screech generation in a planar supersonic jet[END_REF] jets. The directivity of this noise component is well marked around angles of 160 degrees with respect to the upstream direction and its dominant Strouhal number defined on the nozzle exit lengthscale and the jet ideally expanded velocity is around 0.2. This component is mainly generated at the end of the potential core [START_REF] Bogey | An analysis of the correlations between the turbulent flow and the sound pressure fields of subsonic jets[END_REF][START_REF] Sandham | Nonlinear interaction model of subsonic jet noise[END_REF][START_REF] Tam | Mach wave radiation from high-speed jets[END_REF] . For subsonic jets, Bogey and Bailly 5 proposed that this acoustic component is due to the intermittent intrusion of turbulent structures in the potential core. The broadband shock-associated noise (BBSAN) is produced by the interactions between the turbulent structures in the jet shear layers and the shock cell structure. In his pioneering work, Martlew 8 identified this noise. Several other experiments were conducted [START_REF] Tam | Shock associated noise of supersonic jets from convergent-divergent nozzles[END_REF][START_REF] André | Etude expérimentale de l'effet du vol sur le bruit de choc de jets supersoniques sous-détendus[END_REF][START_REF] Tam | Proposed relationship between broadband shock associated noise and screech tones[END_REF] and the central frequency of this noise component was found to vary with the angle in the far field. Harper-Bourne and Fisher [START_REF] Harper-Bourne | The noise from shock waves in supersonic jets[END_REF] built a model which permits to predict the central frequency of this noise component as a function of the observation angle. Finally, the Mach wave radiation can be seen when the turbulent structures in the jet shear layers are convected at a supersonic speed. The Mach waves can simply be seen as shocks attached to a supersonic traveling object. Its directivity is given by the model of Oertel [START_REF] Oertel Sen | Kinematics of Mach waves inside and outside supersonic jets[END_REF] . For the specific case of rectangular non-ideally expanded jets, a specific asymmetric shock pattern is observed [START_REF] Gutmark | Near acoustic field and shock structure of rectangular supersonic jets[END_REF] but the main acoustic sources are the same as the ones for round jets [START_REF] Berland | Numerical study of screech generation in a planar supersonic jet[END_REF] .

In the present paper, compressible jets exiting from a rectangular converging-diverging nozzle are simulated in order to study the influence of the temperature on the aerodynamic and acoustic fields of the jet. An experimental study of a similar jet is conducted at the University of Cincinnati [START_REF] Mora | Acoustics from a rectangular c-d nozzle exhausting over a flat surface[END_REF] . The spectral and hydrodynamic properties of the jet are described and compared with experimental data and models. A modified artificial dissipation suitable for large-eddy simulations of highly compressible flows is proposed and assessed in section II. The jet parameters and the numerical methods used for the simulations are presented in section III. The aerodynamic and acoustic results are presented in section IV and V, respectively.

II. The flow solver

II.A. Numerical methods

The compressible flow solver [START_REF] Eliasson | EDGE: A Navier-Stokes solver for unstructured grids[END_REF] has already been used in previous studies by Semlitsch et al. [START_REF] Semlitsch | Analyzing the impact of the inlet temperature on the acoustic noise production form a supersonic jet using large eddy simulations[END_REF][START_REF] Semlitsch | Large eddy simulation of fluidic injection into a supersonic convergent-divergent duct[END_REF] to perform Large Eddy Simulations of round non-ideally expanded supersonic jets. The simulations are carried out by using a finite volume method and by solving unsteady compressible Navier-Stokes equations on structured meshes. An explicit standard four-stage Runge-Kutta algorithm is used for time integration and a second order central difference scheme is used for spatial discretization. In order to improve the aeroacoustic capabilities of the solver, at the end of each time step, a modified artificial dissipation is added to the inviscid flows in order to remove grid-to-grid oscillations, to avoid Gibbs oscillations near shock and to relax subgrid-scale turbulent energy. This artificial dissipation has been designed on the same way as the one proposed by Jameson et al. [START_REF] Jameson | Numerical solutions of the euler equations by finite volume methods using runge-kutta time-stepping schemes[END_REF] Moreover, in order to separate turbulent structures and shocks, a modification similar to the one proposed by Ducros et al. [START_REF] Ducros | Large-eddy simulation of the shock/turbulence interaction[END_REF] has been used. Finally, the dissipation added to inviscid flows between the node 0 an the node 1 writes

D 01 = ǫ (2) 01 (U 1 -U 0 ) -ǫ (4) 01 (∆U 1 -∆U 0 ) φ 01 r 01 (1)
where U 0 is the value of the inviscid flow considered at the node 0, ǫ

01 and ǫ

(4)
01 are the dissipation functions, ∆ is the Laplacian operator, φ 01 is the spectral radius and r 01 is a function depending on the grid stretching.

The dissipation functions write

ǫ (2) 01 = max(ǫ (2) 0 , ǫ (2)
1 ) ǫ

(4) 01 = (1/2 + 1/2min(max(Φ 0 , Φ 1 ), max(Θ 0 , Θ 1 )))max(0, C 4 -ǫ (2) 01 )s 4 (2) with ǫ (2) 0 = C 2 Φ 0 | p k -p 0 | p k + p 0 s 2 , Φ 0 = (∇.u 0 ) 2 (∇.u 0 ) 2 + ω 2 and Θ 0 = ω 2 (∇.u 0 ) 2 + ω 2
where u 0 is the velocity at the node 0, ω is the vorticity, C 2 and C 4 are constants to define, p 0 is the pressure in the node 0 and s 2 and s 4 = s 2 2 /4 are scaling factors which permit to take into account the number of neighbors. Φ 0 varies between 0 for weakly compressible regions to about 1 in shock regions. The capabilities of this dissipation mechanism were assessed for three aeroacoustic test cases, namely an acoustic pulse, a shock propagation and a shock-vortex interaction. Based on this validation, the constants values were set to (C 2 , C 4 ) = (1.5, 0.04). The results for those values are given in the following section.

II.B. 2-D test cases

For the three considered test cases, the variables are nondimensionalized. The first test case is a linear problem and the two others correspond to non-linear problems in order to assess the aeroacoustic capabilities of the solver in highly compressible flows.

II.B.1. Acoustic pulse

The first test case proposed is a Gaussian acoustic pulse placed in a uniform flow. This 2-D test case have been proposed in the first ICASE/LaRC Workshop 21 . The mean density and the mean pressure are equal to ρ = 1 and p = 1/γ where γ = 1.4 is the heat capacity ratio. The mean velocity field is equal to (u, v) = (M, 0) where M = 0.5. The initial perturbation, consisting of a Gaussian acoustic pulse centered at (x, y) = (0, 0), can be written

         p ′ (x, y) = ǫ exp(-ln(2)/b 2 (x 2 + y 2 )) ρ ′ (x, y) = ǫ exp(-ln(2)/b 2 (x 2 + y 2 )) u ′ (x, y) = 0 v ′ (x, y) = 0 (3)
where ǫ = 0.001 is the amplitude of the pulse and b = 3 is the Gaussian half-width.

This problem is solved on a uniform grid with ∆x = ∆y = 0.2 with a temporal discretization of ∆t = 2/30 in order to have a Courant number CF L = (u + c)∆t/∆x = 0.5. The results at three time intervals are represented in Figure 1. For this canonical linear flow scenario, an analytical solution can be found from the linearized Euler equations. The 2-D solution can be found in Tam and Webb [START_REF] Tam | Dispersion-relation-preserving finite difference schemes for computational acoustics[END_REF] and the 3-D solution can be found in Bogey and Bailly [START_REF] Bogey | Three-dimensional non-reflective boundary conditions for acoustic simulations : far field formulation and validation test cases[END_REF] . In 2-D, the analytical solution writes

p ′ (x, y, z) = ρ ′ (x, y, t) = ǫ 2α ∞ 0 exp(-ξ 2 /4α) cos(ξt)J 0 (ξη)ξdξ (4)
where α = ln(2)/b 2 , η = (x -M t) 2 + y 2 1/2 and J 0 is the Bessel function of the first kind and of order 0. The analytical solution has been added in Figure 1 for the three time instances considered. A perfect match is visible between the analytical solution and the simulation results.

II.B.2. Shock propagation

The second 2-D test case is a non-linear propagation of a shock. This problem is also proposed in the first ICASE/LaRC Workshop 21 . The mean density and the mean pressure are equal to ρ = 1 and p = 1/γ. The mean velocity field is equal to (u, v) = (0, 0). The initial field is given by

                       u(x, y) = 0.5 exp -ln(2)(x/5) 2 v(x, y) = 0 p(x, y) = 1/γ 1 + γ -1 2 u 2γ γ -1 ρ(x, y) = 1 + γ -1 2 u 2 γ -1 (5)
An uniform grid with ∆x = ∆y = 1 is used to solve this problem. A temporal discretization of ∆t = 1/3 is chosen, yielding a Courant number CF L = (u + c)∆t/∆x = 0.5. The results along the line y = 0 at three time instances are represented in Figure 2. The numerical results from Bogey et al. [START_REF] Bogey | A shock-capturing methodology based on adaptative spatial filtering for high-order non-linear computations[END_REF] are added for comparison. A very good match is found with the position, the amplitude and the shape being in agreement. An oscillatory behavior can however be seen in the present simulation.

II.B.3. Shock-vortex interaction

The third 2-D test case is the non-linear interaction between a vortex and a shock. The mean field is a normal shock at x = 0 characterized by an upstream Mach number M 1 = u 1 /c 0 = 1.2. The mean flow upstream and downstream from the shock are given by

         u 1 = 1.2 v 1 = 0 ρ 1 = 1 p 1 = 1/γ                          u 2 = u 1 (γ -1)M 2 1 + 2 (γ + 1)M 2 1 v 2 = 0 ρ 2 = ρ 1 u 1 u 2 p 2 = p 1 ρ 2 ρ 1 (γ + 1) -(γ -1) (γ + 1) - ρ 2 ρ 1 (γ -1) (6)
where the subscripts 1 and 2 denote the upstream and downstream parts of the normal shock. The initial perturbation, consisting of a single vortex centered at (x, y) = (-2, 0), is characterized by the velocity distributions of a Taylor vortex. Its velocity components in the cylindrical coordinate system centered at (x, y) = (-2, 0) can be written

   u r (r) ′ = 0 u θ (r) ′ = M v r exp 1 -r 2 2 (7)
The density and pressure in the vortex region can then be written

         ρ(r) = 1 + γ -1 2 M 2 v exp(1 -r 2 ) 1 γ -1 p(r) = 1 γ ρ(r) γ (8)
where M v = 0.25.

The simulation has been performed on a uniform grid with a mesh spacing equal to ∆x = ∆y = 0.2 with a temporal discretization of ∆ = 0.05, yielding CF L = (u 1 + c)∆t/∆x = 0.55. Snapshots of the pressure field are represented in Figure 3 for four different times. In Figure 3(a), at t = 0, the Taylor vortex is located just upstream from the normal shock, at (x, y) = (-2, 0). The vortex then passes through the normal shock, creating an acoustic wave in the process, visible in Figure 3(b,c,d), at t = 10, t = 20 and t = 30.

III. The rectangular supersonic jet

III.A. Jet parameters

Four Large Eddy Simulations of supersonic rectangular jets are performed, corresponding to four Temperature Ratios (TR) of 1, 2, 2.5 and 3. The jets will be referred as JetTR1, JetTR2, JetTR25 and JetTR3 respectively. The different cases are presented in Table 1. The jets originate from a rectangular convergingdiverging nozzle of aspect ratio (AR) of 2 : 1, which has a height of h = 12.95mm in the minor axis plane. The design Nozzle Pressure Ratio of this nozzle is 3.67, yielding a Mach design of M d = 1.5. In this study, the jets are overexpanded with a Nozzle Pressure Ratio (NPR) of 3. The geometry and the operating conditions are similar to those in the experimental study of Mora et al. [START_REF] Mora | Acoustics from a rectangular c-d nozzle exhausting over a flat surface[END_REF] N P R M j T R u j (m.s -1 ) JetTR1

III.B. Convergence study

The Large Eddy Simulations are carried out using the compressible flow solver [START_REF] Eliasson | EDGE: A Navier-Stokes solver for unstructured grids[END_REF] with the modifications presented in section II.A. First, a convergence study have been carried out for the cold jet JetTR1. Three structured meshes consisting of 64 blocks were designed with 40, 80 and 160 millions of nodes respectively. A visualization of the converging diverging nozzle along the minor axis plane is proposed in Figure 5(a) picturing half the nozzle [START_REF] Mora | Acoustics from a rectangular c-d nozzle exhausting over a flat surface[END_REF] . In Figure 5(b), the intermediate structured mesh along the same plane is represented.

It can be noted that the finest mesh has been designed in order to have a mesh size of y + ∼ 1 in the wall normal direction and of x + < 10 in the wall parallel directions in the diverging part of the nozzle. Moreover, the stretching of the mesh is kept below 5% in the domain of interest to preserve numerical accuracy. Total temperature and total pressure are imposed at the inlet of the nozzle. Characteristic boundary conditions are applied on all the other boundaries. Those conditions combined with the implementation of sponge zones permit avoiding spurious reflections in the physical domain. Finally, adiabatic no-slip conditions are used at the nozzle walls. Visulatizations of the flow and near-field acoustics as calculated with two different grid resolutions, namely 40 and 160 millions of nodes are represented in Figure 6. The pressure fields along the major axis are also shown. Qualitatively, when the number of nodes increases, smaller structures are observed in the jet shear-layers and higher frequencies are propagated in the surrounding near acoustic field. Data obtained with LES for the three different grid resolutions are compared in Figure 7 with experimental results along the jet axis and along the minor/major axis on the plane z = 2h. The shock-cell structure and the levels obtained in the LES and in the experiment are overall in good agreement. However, when increasing the number of nodes, the length of the shock cells follows better the experimental results. The mesh with 160 millions of nodes will be used in this study, as it permits to follow better the shock cell structure observed experimentally and to propagate higher frequencies in the near acoustic field. Please note that the 160 millions of nodes mesh has been designed in order to allow acoustic waves with Strouhal numbers up to St = f D eq /u j = 3 to be well propagated, where f is the frequency, D eq is the equivalent diameter of the jet, and u j is the ideally expanded jet velocity.

IV. Aerodynamics results

The four LES simulations (see Table 1) are performed on the 160 millions of nodes mesh. A total of 200, 000 time steps of ∆t = 0.002h/u j are computed in each case after the transient period, permitting a simulation time of 400h/u j .

IV.A. Instantaneous features

In order to visualize simultaneously the jet flow and the near acoustic field, three-dimensional snapshots are displayed in Figure 8 for JetTR1, JetTR2 and JetTR3. The jet shear-layers are shown using isosurfaces of density and the near acoustic fields of the jets along the minor axis and along the major axis are represented using the instantaneous pressure.

In Figure 8(a,b), for the cold jet (T R = 1), three features are observed in the near acoustic field. First, acoustic waves propagating in the upstream direction in the vicinity of the nozzle are visible. These waves are linked to the screech noise. This noise component have been observed in various studies, notably in Westley and Woolley 26 , Tam and Tanna 9 , Panda et al. [START_REF] Panda | Underexpanded screeching jets from circular, rectangular and elliptic nozzles[END_REF] , and André et al. [START_REF] André | Investigation of the mixing layer of underexpanded supersonic jets by particle image velocimetry[END_REF] The second acoustic contribution that can be seen consists of circular acoustic waves coming from the jet shear layers at different axial positions. These acoustic waves are due to the interactions between the turbulent structures and the shock cell structure in the jet shear-layers. Those interactions lead to the observation of broadband shock-associated noise in the far-field [START_REF] Tam | Shock associated noise of supersonic jets from convergent-divergent nozzles[END_REF][START_REF] Tam | Proposed relationship between broadband shock associated noise and screech tones[END_REF][START_REF] Dahl | Predictions of supersonic jet mixing and shock-associated noise compared with measured far-field data[END_REF] . The third acoustic component visible propagates in the downstream direction and is associated to a low frequency. This acoustic contribution corresponds to the mixing noise, as observed by Bogey and Bailly 5 , Sandham and Salgado 6 and Tam 7 for instance.

In Figure 8(c-f), for the higher temperature ratios investigated (T R = 2 and T R = 3) it can be observed a temperature effect on the amplitude of the upstream propagating waves. One have to note that the amplitude of the upstream propagating waves is stronger along the minor axis plane than along the major axis one. Moreover, these waves are visibly antisymmetrically organised with respect to the jet axis. These two observations suggest that the screech mechanism in those jets is associated with a flapping motion of the jets along the minor axis. Moreover, it seems that the amplitude of the screech noise is increasing with the temperature. Secondly, when the Temperature Ratio increases, a new acoustic component arises in the sideline direction with long and straight acoustic waves organized along a clear direction. This acoustic contribution is linked to Mach wave radiation. This specific noise component appears when the convection velocity of the turbulent structures in the jet shear-layers becomes supersonic. This acoustic component have a marked directivity and a broadband spectrum. It has been studied experimentally [START_REF] Panda | Experimental investigation of density fluctuations in high-speed jets and correlation with generated noise[END_REF] and numerically [START_REF] Nichols | Large eddy simulation of crackle noise in hot supersonic jets[END_REF] .

IV.B. Flow field statistics

The mean axial velocity fields obtained in the present LES for the four Temperature Ratios are represented in Figure 9. For JetTR1, JetTR2 and JetTR3, they are compared to experimental PIV results in Figure 9(a,b,c,e). Despite of the unavoidabale mismatch in the boundary conditions between the simulations and experiments there is an overall good agreement between the two sets of data. Moreover, as noted by Hsia et al. [START_REF] Hsia | Effects of mach number on the development of a subsonic rectangular jet[END_REF] for subsonic jets and by Zaman [START_REF] Zaman | Spreading characteristics of compressible jets from nozzles of various geometries[END_REF] for supersonic jets, a turbulent rectangular jet spreads faster along its minor axis than along its major axis, evolving from a rectangular cross section to a circular cross section. This evolution is clearly visible in Figure 9 where the jets appear circular at z = 15h. In Figure 9, it can be observed that the length of the shock cells decreases in the downstream direction. This trend is due to the growth of the shear layer and to the dissipation of the shock-cell structure by the turbulence [START_REF] Tam | A multiple-scales model of the shock-cell structure of imperfectly expanded supersonic jets[END_REF] . For rectangular jets, Tam [START_REF] Tam | The shock-cell structures and screech tone frequencies of rectangular and non-axisymmetric supersonic jets[END_REF] built a model by considering the corresponding rectangular ideally expanded jet to predict the length of the first shock cell. The model gives

L model = 2(M 2 j -1) 1/2 h j (1 + h 2 j /b 2 j ) 1/2 (9)
where h j and b j are the dimensions of the corresponding ideally expanded rectangular jet. For our aspect ratio 2 rectangular nozzle, their expressions 35 can be reduced to

h j h = [(A j /A d ) -1] * 2/3 + 1 b j 2h = [(A j /A d ) -1] * 1/3 + 1
where A j is the nozzle exit area of the fully expanded equivalent jet and A d = h * 2h is the nozzle exit area. A j is given by

A j A d 2 = M 2 d M 2 j (1 + γ -1 2 M 2 j )/(1 + γ -1 2 M 2 d ) (γ+1)/(γ-1)
where M d = 1.5 is the design Mach number of the nozzle. Equation ( 9) permits to find a value L model = 1.57h. Values of L s = 0.99L model , L s = 0.97L model , L s = 0.97L model and L s = 0.95L model are found for the size of the first shock cell of JetTR1, JetTR2, JetTR25 and JetTR3, respectively. The size of the cell decreases by few percent with the jet temperature ratio but overall, the model gives a very good approximation for the length of the first shock cell. The normalized shock cell size L s /L model are reported in Figure 10 for all the visible cells in the mean fields of all investigated temperature ratios. The total number of shock cells visible decreases with the temperature ratio, from 10 for T R = 1 to 4 for T R = 3. However, the variation of the shock-cell size appears to behave linearly for the four jets. Such evolution was already observed by Harper-Bourne and Fisher 12 for round supersonic jets. They proposed a relation for the size of the n-th shock cell which writes

L n = L s -(n -1)∆L ( 10
)
where ∆L is the variation of the cell size from one cell to another. For the present jets, a mean value of ∆L/L s = 5.5% is obtained. For round underexpanded supersonic jets, André et al. [START_REF] André | Investigation of the mixing layer of underexpanded supersonic jets by particle image velocimetry[END_REF] and Harper-Bourne and Fisher 12 found ∆L/L s = 3% and ∆L/L s = 6%, respectively. It is worth noting that the lower value found by André et al. [START_REF] André | Investigation of the mixing layer of underexpanded supersonic jets by particle image velocimetry[END_REF] can be attributed to the presence of a secondary flow characterized by a Mach number of 0.05. To conclude, for our rectangular aspect ratio 2 jet, a linear evolution of the shock cell size similar to the one in round jets is observed. Overall Sound Pressure Levels (OASPL) obtained along the minor and the major axis are represented for the four jets in Figure 11. Isocontours are added in order to improve the readability. The OASPL is computed from the rms pressure fields. Several acoustic components are visible. The first acoustic component is visible in Figure 11(a,b,c,d) in the upstream direction along the minor axis plane. This component is linked to acoustic waves propagating in the upstream direction, permitting to close the aeroacoustic feedback loop leading to the screech noise. This acoustic component is only visible along the minor axis and its amplitude increases with the Temperature Ratio, confirming the observations made in Figure 8. Moreover, the cell structures obtained in the jets shear-layers in the amplitude fields of figure 11(c,d) for Temperatures Ratio of 2.5 and 3 are due to the presence of hydrodynamic-acoustic standing waves creating by the screech feedback loop, explaining their presence only along the minor axis plane. Such structures were previously observed by Panda et al. [START_REF] Panda | Underexpanded screeching jets from circular, rectangular and elliptic nozzles[END_REF] for screeching supersonic jets and by Gojon et al. [START_REF] Gojon | Investigation of tone generation in ideally expanded supersonic planar impinging jets using large-eddy simulation[END_REF] for ideally expanded planar impinging jets. The second acoustic components visible is linked to the Mach wave noise, whose intensity increases with the Temperature Ratio. Moreover, it can be seen that this component is organized in a axisymmetric manner for JetTR2 in Figure 11(b,f) but in a asymmetric manner for JetTR25 and JetTR3 in Figure 11(c,d,g,h) with a lot stronger Mach wave radiation along the minor axis compared to the major axis. The strong flapping motion of the jet along the minor axis due to the screech feedback mechanism seems to lead to this asymmetric organization of the Mach wave radiation. (a,b,c,d) the minor axis and (e,f,g,h) the major axis. The nozzle is in black.

IV.C. Temperature effect on convection velocity

In order to assess the presence of the Mach wave radiation in the hot cases, the convection velocity of turbulent structures in the jet shear layers is computed. It is calculated in the jet shear layers, where the turbulence kinetic energy is maximum, from cross-correlations of axial velocity between 70 neighbors points equidistantly spaced at 0.25h from each other. For the four jets, the normalized convection velocity and the convective Mach number along the jet shear layer in the minor axis are represented in Figure 12. For all the jets, the convection velocity is not constant but varies according to the shock cell structure, as already observed for round jets experimentally 10 and numerically [START_REF] Gojon | Large-eddy simulation of underexpanded round jets impinging on a flat plate 4 to 9 radii downstream from the nozzle[END_REF] . The mean values of the convection velocity and of the convective Mach number between z = 1 and z = 15 along the minor axis are reported in Table 2. Overall, the mean normalized convection velocity < u c > of the turbulent structures decreases with the increase of the Temperature Ratio, from 0.80u j to 0.64u j . The same observation has been made numerically in a recent study by Liu et al. [START_REF] Liu | Effects of temperature on noise generation in supersonic jets[END_REF] Indeed, for round supersonic underexpanded jets, they found a convection velocity decreasing from 0.68u j for a Temperature Ratio of 1 to 0.56u j for a Temperature Ratio of 7. In Figure 12(b), the convective Mach number remains below 1 for JetTR1, yielding the absence of Mach waves in Figure 8(a,b). For JetTR2, JetTR25 and JetTR3, a convective Mach number of about M c = 1.25 is found. This supersonic Mach number yields the creation of the Mach waves observed in Figure 8(c-f). JetTR25, and JetLTR3.

u j (m.s -1 ) < u c /u j > M c =< u c /c 0 >

V. Acoustic results

V.A. Near-field acoustics

The pressure spectra obtained in the vicinity of the nozzle at (x, y, z) = (0, -2h, -2h) are represented in Figure 13 as functions of the Strouhal number St = f D eq /u j . There is a change in the dominant frequency as the jet's temperature is changing, with dimensionless frequencies ranging between St = 0.29 and St = 0.36. Those frequencies, visible in the upstream direction in the vicinity of the jet, correspond to screech components, see for instance in Westley and Woolley 26 , Tam and Tanna 9 , Panda et al. [START_REF] Panda | Underexpanded screeching jets from circular, rectangular and elliptic nozzles[END_REF] , and André et al. [START_REF] André | Investigation of the mixing layer of underexpanded supersonic jets by particle image velocimetry[END_REF] Two other secondary tones can be seen in the spectra in Figure 13; a low frequency ranging between St = 0.16 and St = 0.21 and the first harmonic of the screech frequency.

For the four jets, the amplitude and the frequency of the low frequency and screech components are given in Table 3. The screech frequencies found experimentally 15 are also reported. For the low frequency The Strouhal numbers are in excellent agreement with the experimental results from the University of Cincinnati [START_REF] Mora | Acoustics from a rectangular c-d nozzle exhausting over a flat surface[END_REF] . Please note that those frequencies have been reported using a far-field microphone array organized along the minor axis. However, Mora et al. [START_REF] Mora | Acoustics from a rectangular c-d nozzle exhausting over a flat surface[END_REF] observed an overall decrease of the amplitude of the screech component with an increase of the Temperature Ratio, in contradiction with the present increase observed. It is believed that this behavior is due to the lack of turbulence at the nozzle exit in the simulations. For a rectangular jet, Tam [START_REF] Tam | The shock-cell structures and screech tone frequencies of rectangular and non-axisymmetric supersonic jets[END_REF] built a model in order to predict the screech frequency. In the present scenario (2:1 AR rectangular jet), the model writes

St T am = D eq u c /u j 2h(1 + u c /a 0 )(M 2 j -1) 1/2 (h j /b j ) 2 + 1        1 + γ -1 2 M 2 j 1 + γ -1 2 M 2 d    (γ+1)/(2(γ-1)) M d M j -1     2 3 + 1 (11)
The values found by evaluating equation ( 11) using the mean convection velocity and mean convective Mach number from Table 2 are reported in Table 3. For JetTR2, JetTR25, and JetTR3, a very good agreement is found between the simulation results, the experimental results, and the value found using equation (11). For JetTR1, equation (11) yields to an overestimation of about 10% of the screech frequency. Overall, the decrease of the frequency of the screech component with the Temperature Ratio is recovered using equation (11).

V.B. Far-field acoustics

The Ffowcs Williams-Hawkings (FW-H) surface used to compute the far-field acoustics is an ellipsoid represented in Figure 14 along the minor axis for the case JetTR2. It is placed at the end of the first acoustic zone in the structured mesh, corresponding to a Strouhal number resolved of St ∼ 6. The surface consists of about 1 million nodes and the flow variables are stored at a sampling frequency corresponding to St ∼ 10. Please note that the surface is placed comparatively to other study quite far from the jet for two main reasons. First, the jet being rectangular, the structured grid in the jet region is rectangular and a rectangular FW-H surface is not ideal. Second, it was too expensive in terms of storage to use the method with several outflow disks of Shur et al. [START_REF] Shur | Noise prediction for increasingly complex jets. part i: Methods and tests[END_REF] , used for example by Brès et al. [START_REF] Brès | Nozzle wall modeling in unstructured large eddy simulations for hot supersonic jet predictions[END_REF] and in the case of an open surface, a "loose" FW-H surface is then preferable [START_REF] Shur | Noise prediction for increasingly complex jets. part i: Methods and tests[END_REF] . Indeed, this surface located quite far from the jet permits to avoid the creation of "pseudo-sound" [START_REF] Crighton | Noise Source Mechanisms[END_REF] . This unphysical sound results from the convection through the surface of slow turbulent structures that are not canceled because of the absence of the outflow disk. The FW-H surface is used to compute the fluctuating pressure in the far-field at a distance of 40D eq along the minor axis. The OASPL directivities as predicted in the far-field of the jet in the minor-axis plane are represented for the four jets in Figure 15 and compared to the experimental results for JetTR1, JetTR2 and JetTR3. Overall, a good agreement is reached.

The acoustic spectra in the far-field are represented in Figure 16 as functions of the Strouhal number and the directivity angle, with 20 degrees being the most upstream location and 160 degrees angle corresponding to the most downstream side angle. Several acoustic components typical of non-ideally expanded supersonic jets can seen in Figure 16. In the upstream direction, for θ < 60 degrees, the screech tone frequencies dominate, as already pointed out in the pressure spectra of Figure 13. In the downstream direction, θ > 140 degrees, the mixing noise generated by large scale turbulent structures appears around a Strouhal number of 0.25. The direction and frequency of this acoustic component are in good agreement with the numerical results of Berland et al. [START_REF] Berland | Numerical study of screech generation in a planar supersonic jet[END_REF] for a planar supersonic jet with M j = 1.55. A third acoustic component is visible for 60 < θ < 160 degrees. Its central frequency is varying with the angle of observation. This component is associated with broadband shock-associated noise, as observed in experimental 9-11 and numerical 4,29 studies. A mechanism was proposed by Harper-Bourne and Fisher 12 for this acoustic component. In this mechanism, the broadband shock-associated noise is generated by the interactions between the turbulent structures propagating downstream in the jet shear layers and the shocks of the shock cell structure. Each interaction is considered as an acoustic source. The directivity of constructive interference is then determined. This model yields a central frequency

f shock = N u c L s (1 + M c cos(θ)) ( 12
)
where N is the mode number, L s is a length scale related to the shock cell size, and M c is the convective Mach number. As the cell length varies with the axial direction, it is difficult to chose a value of L s . A value of L s = 0.8L model is used for our study. The corresponding central frequency of the broadband shockassociated noise is represented with dashed line in Figure 16 for N = 1 and using the values in Table 2 for u c and M c . A good overall agreement is found. It is worth noting that at angles around θ = 80 degrees, for JetTR2, JetTR25 and JetTR3, the first harmonics of the screech component coincide with the BBSAN component and becomes dominant. The same observation has been made in the experimental study [START_REF] Mora | Acoustics from a rectangular c-d nozzle exhausting over a flat surface[END_REF] . One can finally note that in the upstream direction, the relation (12) tends to the values of St = 0.50, St = 0.48, St = 0.45 and St = 0.44 for JetTR1, JetTR2, JetTR25 and JetTR3, respectively. Those tones are visible in the spectra of figure 13, between the screech tones and their harmonics. Finally, for JetTR2, JetTR25 and JetTR3, a fourth acoustic component is visible at an angle of θ = 140 degrees. This broadband component corresponds to the Mach wave radiation. Its directivity is given by the model of Oertel 13 , which writes

θ = 180 -arccos 1 M c ( 13
)
where φ is the angle with respect with the jet axis and M c is the convective Mach number. Using a convective Mach number of M c = 1.25, relation (13) gives an angle of θ = 143 degrees. Please note that this angle is also in agreement with the peak values visible in Figure 15(b-d) for JetTR2, JetTR25 and JetTR3.

VI. Conclusions

An artificial dissipation mechanism derived from the one of Ducros et al. [START_REF] Ducros | Large-eddy simulation of the shock/turbulence interaction[END_REF] is proposed in the first part of this paper. This mechanism is assessed using linear and non-linear 2-D test cases. The results are presented and demonstrate the ability of the modified solver to perform large-eddy simulations of highly compressible flows. The solver is then used to simulate rectangular supersonic overexpanded jets. The geometry of the nozzle and the exit conditions are similar to those in the experimental study carried out at the University of Cincinnati. A convergence study is first performed. Mean fields are found to be in good agreement with experimental results. Four simulations with different temperature ratios are then presented in order to characterize the effect of the temperature on the aerodynamic and aeroacoustic fields of the jets. First, it is shown that the total number of cells in the shock cell structure decreases with the increase of the temperature ratio. However, the temperature ratio does not influence the size of the first shock cell and the linear decrease of the shock cell size in the downstream direction. In case of the shock cell structure, results are found to be similar to those for round non-ideally expanded supersonic jets. The Overall Sound Pressure Levels are then represented along the minor and major axis. It is seen that the intensity of the screech feedback mechanism increases with the Temperature Ratio. For JetTR25 and JetTR3, the strong flapping motion of the jet along the minor axis due to the screech feedback mechanism is believed to lead to an asymmetric organization of the Mach wave radiation. Consequently, the convection velocity of the turbulent structures in the jet shear layers along the minor axis is studied. Once normalized by the jet exit velocity, the convection velocity is shown to decrease with the jet temperature ratio, a similar behavior as observed in round jets. In the last part of the paper, the near-and far-field acoustic fields are studied. In the vicinity of the nozzle, a screech frequency is observed for each jet. This frequency is consistent with both experimental data and a theoretical model. The far-field acoustic fields are finally computed using a Ffowcs Williams-Hawkings based approach. Four acoustic components typical of non-ideally supersonic jets are observed, namely the screech noise, the broadband shock-associated noise, the mixing noise and the Mach wave noise. Their directivities and frequencies are in agreement with experimental results and models.

Figure 1 .

 1 Figure 1. (a,b,c) Snapshots of the fluctuating pressure field and (d,e,f ) cut along the line y = 0 at times (a,d) t = 0.5, (b,e) t = 20 and (c,f ) t = 40; • analytical solution and present simulation.

Figure 2 .

 2 Figure 2. (a,b,c) cut along the line y = 0 of the fluctuating pressure γp -1 at times (a) t = 0.6, (b) t = 100 and (c) t = 200; • solution found by Bogey and Bailly 24 and present simulation.

Figure 3 .

 3 Figure 3. Snapshots of the pressure field at times (a) t = 0, (b) t = 10, (c) t = 20 and (d) t = 30.The results are then compared to the Direct Numerical Simulation results obtained by Inoue[START_REF] Inoue | Propagation of sound generated by weak shock-vortex interaction[END_REF] along the line starting from the center of the vortex and making an angle of -45 degrees compared to the downstream direction. The line is represented in Figure3(b) for clarity. The comparison is proposed in Figure4. An excellent agreement is visible. The acoustic wave created by the interaction between the shock and he vortex has the same amplitude, the same propagation speed and the same shape as the one in the Direct Numerical Simulation.

Figure 4 .

 4 Figure 4. Radial distribution of the fluctuating pressure starting from the center of the vortex along the line making an angle of -45 degrees compared to the downstream direction at (a) t = 10, (b) t = 20, (c) t = 30 and (d) t = 40; • Direct Numerical Simulation results obtained by Inoue 25 and present simulation.

Figure 5 .

 5 Figure 5. (a) Picture of half the nozzle showing the converging diverging geometry along the minor axis and (b) 80 millions of nodes structured mesh along the same plane.

Figure 6 .

 6 Figure 6. Isosurfaces of density for the cold jet JetTR1 with a (left) 40M nodes and (right) 160M nodes meshes. The isosurfaces for 1.2 kg.m -3 are represented colored by the Mach number. The pressure field along the major axis is also shown. The nozzle is in black.

Figure 7 .

 7 Figure 7. Mean axial velocity field for the cold jet JetTR1 along (a) the jet axis, (b) the minor axis at z = 2h and (c) the major axis at z = 2h; • PIV results and 40M, 80M and 160M LES results.

Figure 8 .

 8 Figure 8. Isosurfaces of density colored by the Mach number for (a,b) JetTR1, (c,d) JetTR2 and (e,f ) JetTR3. The pressure field along (a,c,e) the minor axis and (b,d,f ) the major axis are also shown. The nozzle is in black.

Figure 9 .

 9 Figure 9. Mean axial velocity field for (a,e) JetTR1, (b,f ) JetTR2, (c,g) JetTR25 and (d,h) JetTR3 along (a,b,c,d) the minor axis and (e,f,g,h) the major axis. The PIV results are added in the black rectangles. The nozzle is in black.

Figure 10 .

 10 Figure 10. Normalized lengths of the shock cells obtained for -• -JetTR1, -• -JetTR2, -• -JetTR25, and -• -JetLTR3.

Figure 11 .

 11 Figure 11. Overall Sound Pressure Levels for (a,e) JetTR1, (b,f ) JetTR2 (c,g) JetTR25 and (d,h) JetTR3 along(a,b,c,d) the minor axis and (e,f,g,h) the major axis. The nozzle is in black.

Figure 12 .

 12 Figure 12. (a) Convection velocity and (b) convective Mach number Mc of the turbulent structures along the jet shear-layer in the minor axis plane as functions of the axial position for JetTR1, JetTR2, .

Figure 13 .

 13 Figure 13. Pressure spectra at (x, y, z) = (0, -2h, -2h) as functions of the Strouhal number for (a) JetTR1, (b) JetTR2, (c) JetTR25 and (d) JetTR3.

Figure 14 .

 14 Figure 14. Representation of the contour of the FW-H surface in black over a 2-D snapshot along the minor axis representing the Mach number in the jet and of the fluctuating pressure around for JetTR2. The nozzle is in black.

Figure 15 .

 15 Figure 15. OASPL obtained on the circle of radius R = 40Deq in the minor axis plane as functions of the angle θ with respect to the upstream direction for (a) JetTR1, (b) JetTR2, (c) JetTR25 and (d) JetTR3; • experimental results and current results (LES & FW-H).

Figure 16 .

 16 Figure 16. Far-field SPL as predicted based on the FW-H approach R = 40Deq as functions of the Strouhal number and of the angle θ with respect to the upstream direction; data in the minor-axis plane of the jet for (a) JetTR1, (b) JetTR2, (c) JetTR25 and (d) JetTR3. The color scale ranges from 110 and 140 dB/St.

Table 1

 1

		3	1.36	1.	398
	JetTR2	3	1.36	2.	563
	JetTR25	3	1.36 2.5	629
	JetTR3	3	1.36	3	689

. Jet parameters: Nozzle Pressure Ratio N P R, ideally expanded Mach number Mj , Temperature Ratio T R and ideally-expanded jet velocity uj .

Table 2 .

 2 Convection velocity along the minor axis: ideally expanded velocity uj , mean convection velocity < uc/uj > and mean convective Mach number Mc =< uc/c0 >.

  St low dB low St screech dB screech St screech exp. St T am

	JetTR1	0.21	135	0.36	150	0.37	0.41
	JetTR2	0.18	140	0.31	159	0.31	0.33
	JetTR25 0.17	139	0.30	171	-	0.30
	JetTR3	0.16	142	0.29	174	0.28	0.28

Table 3 .

 3 Frequencies Stlow and Stscreech and amplitudes dBlow and dBscreech of the two tone frequencies emerging in Figure13for each jet; screech frequency Stscreechexp. found in the experimental study[START_REF] Mora | Acoustics from a rectangular c-d nozzle exhausting over a flat surface[END_REF] ; screech frequency StT am found using the model of Tam[START_REF] Tam | The shock-cell structures and screech tone frequencies of rectangular and non-axisymmetric supersonic jets[END_REF] .

Acknowledgments

The computations were performed using HPC resources provided by the Swedish National Infrastructure for Computing (SNIC) at the PDC center. Jing Gong at PDC is acknowledged for assistance concerning technical and implementational aspects.


Capital-labour substitution in defence provision

Ron Smith, Humm Anthony, Jacques Fontanel Defence Security and Development S. Deger and R. West, Frances Pinter, London, 1987. The choice between labour and capital arises for military expenditures necessary for national security. Economic constraints play a major role, because the military sector is highly technological and aims for the highest performance for its equipment, which has become very expensive. Should conscription be maintained? Is a highly capital-intensive defence function preferable? The countries that spend the most in military terms have higher per capita spending than other types of defence and the army is traditionally more "labour intensive".

Résumé : Le choix entre le travail et le capital se pose pour les dépenses militaires nécessaires à la sécurité nationale. Les contraintes économiques interviennent fortement, car le secteur militaire est hautement technologique et qu'il vise les plus hautes performances pour ses matériels devenus alors fort onéreux. Faut-il maintenir une conscription ? Peut-on préférer une fonction de défense fortement intensive en capital ? Les pays les plus dépensiers en termes militaires ont des dépenses par personne supérieures aux autres types de défense et l'armée de terre est traditionnellement plus « travail intensive ».

Dépenses militaires, structures des dépenses militaires, capital militaire, travail militaire, conscription. Military expenditure, military expenditure structures, military capital, military labour, conscription.


Nacera Bouldja
email: nacera.bouldja@centralesupelec.fr

Marc Sciamanna

Delphine Wolfersberger

Slow light using photorefractive nonlinear optics

Slow and fast light are achieved in two experiments using SPS photorefractive crystal at room temperature. We report that the photorefractive gain achieved by the two wave mixing (TWM) method can control the group velocity of the transmitted light pulses at visible wavelength. It is shown theoretically and experimentally that the time delay and the shape of the output pulse change as a function of the photorefractive gain. The beam fanning has also been used to control the velocity propagation of a single light pulse in the same crystal at λ = 1064 nm. Depending on the orientation of the polar axis, it is possible to accelerate or decelerate a short pulse with duration of order of µs in the crystal.

Slow light, photorefractive crystal, two wave mixing, photoréfractive gain, beam fanning, deplition factor

INTRODUCTION

Deceleration of light group velocity has some potential for future applications such as optical delay lines and buffer memories in optical communication networks [START_REF] Tucker | Slow-light optical buffers: capabilities and fundamental limitations[END_REF]. Slow light is characterized by a very small group velocity (v g << c) which can be achieved in high dispersion nonlinear materials such as optical fibers [START_REF] Thévenaz | Slow and fast light in optical fibres[END_REF], photonic [START_REF] Baba | Slow light in photonic crystals[END_REF][START_REF] Li | Systematic design of flat band slow light in photonic crystal waveguides[END_REF] or photorefractive crystals [START_REF] Sturman | Slowdown and speedup of light pulses using the self-compensating photorefractive response[END_REF][START_REF] Bouldja | Improved slow light performances using photorefractive two-wave mixing[END_REF][START_REF] Podivilov | Light pulse slowing down up to 0.025 cm/s by photorefractive two-wave coupling[END_REF][START_REF] Shumelyuk | Slowing down of light in photorefractive crystals with beam intensity coupling reduced to zero[END_REF][START_REF] Bouldja | Slow light with photorefractive beam fanning[END_REF]. Recent studies have shown that the two waves mixing (TWM) method can be used to strongly improve the photorefractive crystal dispersion and reduce light propagation velocity at room temperature [START_REF] Bouldja | Improved slow light performances using photorefractive two-wave mixing[END_REF]. The TWM consists in the coupling of strong pump beam and probe signal to modulate the refractive index and leads to the output pulse amplification, resulting in the generation of a photorefractive gain Γ. Using this method, group velocities lower than 0.025 cm/s and 0.9 cm/s were respectively achieved in BaTiO 3 [START_REF] Podivilov | Light pulse slowing down up to 0.025 cm/s by photorefractive two-wave coupling[END_REF] and Sn 2 P 2 S 6 (SPS) [START_REF] Bouldja | Improved slow light performances using photorefractive two-wave mixing[END_REF] crystals.

The delay is usually accompanied by the measurement of the achievable bandwidth of the output signal, in order to present a more complete evaluation of the slow light performances. The photorefractive crystal suffer from high dispersion when the light pulse group velocity is low which causes the widening and distortion of the output pulse and limits its data rates [START_REF] Deng | Time-bandwidth problem in room temperature slow light[END_REF]. Usually, this parameter is characterized by the fractional delay FD which is the ratio between the time delay and the output pulse full-width at half maximum (FWHM). The optimization of two-wave mixing in the SPS crystal was used to provide a solution to the problem of pulse widening. Then FD of the order of 0.79 [START_REF] Bouldja | Improved slow light performances using photorefractive two-wave mixing[END_REF] was measured for a pulse duration of the order of 100 ms.

In this work, we present two experimental methods for slowing down or speeding up an optical pulse in SPS photorefractive crystal at room temperature. Unlike the BaTiO 3 , this crystal is known for its fast response time of the order of 10 ms, which makes it possible to process shorter pulses and its large gain in the visible and infrared domains [START_REF] Odoulov | Photorefraction in tin hypothiodiphosphate in the near infrared[END_REF]. First, using the TWM method, the SPS crystal can slow down the light Gaussian pulses with durations of order of 14 ms close to the SPS response time. Both experimental and theoretical studies show that the time delay and the shape of the transmitted pulse change as a function of the photorefractive gain. In the second part, the beam fanning in the SPS photorefractive crystal has also been used to slow-down a single light pulse. The beam fanning occurs when a single optical passes through the crystal and scatters from the crystal imperfections. The coupling between the fanning and the input beam gives rise to noisy refractive index gratings and causes the depletion and slowing down the output beam. Here, a group velocity deceleration and acceleration have been observed for pulse durations of order of µs and ms. We show that both the delay and the advancement depend on the crystal orientation and the input pulse durations.

Slowdown of pulse with duration close to SPS response time

In this section, we investigate the performance of slow light in photorefractive TWM configuration. The setup of Fig. 1(a) makes use of a 0.5 cm-thick z-cut Sn 2 P 2 S 6 (SPS) sample and a Crystal laser LC Model DL at 632 nm. The laser beam is split into two parts, a pump and signal. The signal beam is modulated by a Newfocus Model 4002-Visible Phase Modulator (EOM) to be in the Gaussian shape. The pulse gets amplified inside the photorefractive crystal because of its interference with the pump with different intensities. The intensity of the input and of the output pulses is detected with two amplified photodetectors D 1,2 and analyzed by an oscilloscope.

By analyzing the intensity evolution measured by D 1,2 , we observe that the transmitted pulse is delayed by comparison to the reference pulse. As shown in Fig. 1(b), the maximum of the output pulse duration is about 10 ms (red line) and is shifted in time with ∆τ = 9.4 ms compared to the input one (black line). As we show in [START_REF] Bouldja | Improved slow light performances using photorefractive two-wave mixing[END_REF], the obtained experimental delay depends on the coupling strength between the pump and the signal beams or the photorefractive gain. For the characterization of the delay as a function of the photorefractive TWM gain Γ, we fix the input pulse duration to 14 ms close to the crystal response time and we analyze the evolution of the transmitted pulse for different values of Γd, with d is thickness of the crystal. This included studies of the delay, the full output pulse width at half maximum (FWHM) and the fractional delay FD. The measurements results are compared with the theoretical ones calculated by using the following equation [START_REF] Sturman | Slowdown and speedup of light pulses using the self-compensating photorefractive response[END_REF] The output pulse amplitude A(d, t) of Eq. ( 1) depend on the time t, the photorefractive gain Γd, the 1/e half-width of the pulse intensity T 0 and the input amplitude A 0 . The numerical simulations of the delay and the duration of the output pulses as a function of the gain are presented in Figs. 2(a 1 ) and 2(b 1 ). The delay is found to increase with increasing the photorefractive gain . Furthermore, for sufficiently large values of Γ, the delay can be higher than the input pulse duration. Also, the output pulse width is modified compared to the input one because of its broadening which increases with Γd. From Fig. 2(a 1 ), the input pulse widens by almost three times its duration for Γd = 4.7. The resulting fractional delay (FD) is shown in Fig. 2(a 1 ). The value of theoretical FD increases with Γd and its maximum value is about 0.5.

A(d, t) = A 0 T 0 2 √ π ∞ -∞ e -0.5ω 2 T 2 0 + Γd 1-iωt -iωt dω (1)
Now, if we analyze the experimental results, we note that the behaviors of the curves of Fig. 2(a 2 ) is the same as in Fig. 2(a 1 ). On the other hand, for the same value of Γd, the measured values of delay and widening of the output pulse are less important than those calculated with Eq. ( 1). The output pulse amplitudeA(d,t) of Eq. ( 1) depend on the timet, the photorefractive gain d, the 1/ehalf-width of the pulse intensityT0and the input amplitudeA0.

The results plotted in the right column in

Slow and fast light using photorefractive beam fanning

The SPS photorefractive crystal has already been used for slow light in the visible range using the two [START_REF] Bouldja | Improved slow light performances using photorefractive two-wave mixing[END_REF][START_REF] Podivilov | Light pulse slowing down up to 0.025 cm/s by photorefractive two-wave coupling[END_REF] or four [START_REF] Shcherbin | Slowing down of light pulses using backward-wave four-wave mixing with local response[END_REF] wave mixing method . In this section, we propose the slow and the fast light propagation scheme using the beam fanning in the SPS photorefractive crystal at 1064 nm wavelength. We experimentally study the slowing down and the speeding up of shorter pulses of order of µs and ms. This process is easily achieved by the scattering of a single light beam in an asymmetrical way [START_REF] Segev | Nonlinear multi two-wave mixing, the fanning process and its bleaching in photorefractive media[END_REF] and it can be observed in all crystal possessing large photorefractive gain Γ. On the other hand, it was reported that the presence of two types of charge carriers in the SPS photorefractive crystal [START_REF] Sturman | Slowdown and speedup of light pulses using the self-compensating photorefractive response[END_REF][START_REF] Bouldja | Improved slow light performances using photorefractive two-wave mixing[END_REF], provide two different response times (slow and fast response) known as "self compensation response" [START_REF] Sturman | Slowdown and speedup of light pulses using the self-compensating photorefractive response[END_REF][START_REF] Bouldja | Improved slow light performances using photorefractive two-wave mixing[END_REF][START_REF] Shumelyuk | Light pulse manipulation in sn2p2s6[END_REF] . The presence of this response in the crystal can control the dispersion and group velocity of the pulse at the output of the crystal. Indeed, by changing the orientation of the polar axis of the crystal, it is possible to slow down or speed up the velocity light propagation simultaneously.

To control the light velocity propagation using the beam fanning, we change the visible laser of the above experience by a Crystal laser LC Model IR-700 at 1064 nm and we turn off the pump beam as shown in Fig. [START_REF] Baba | Slow light in photonic crystals[END_REF]. We choose the incidence angle θ = 45 • between the input beam and the SPS c-axis to get the maximum fanning in the direction of the transmitted pulse. The input pulse power was adjusted by the attenuator A to maintain a constant intensity I 0 = 1.2 W/cm 2 of throughout the experiment. In these conditions, a noisy refractive index grating is created in the crystal via the TWM between the fanning and the input beam. In the last conditions, a noisy refractive index grating is created in the crystal via the TWM between the fanning and the input beam. Also, the interference between the beams causes energy transfer beam coupling between the two beams [START_REF] Segev | Temporal evolution of fanning in photorefractive materials[END_REF] and the depletion of the transmitted pulse. The variation of this depletion allows to measure fanning strength defined by the depletion factor D [START_REF] Grabar | Manipulation of fast light using photorefractive beam fanning[END_REF]. At the same time, the maximum of the output pulse is delayed compared to the reference one. The left column in Fig. (4) represents the measured performance of the slow light achieved for depletion factor of order of 0.54 and different values of the full input pulse width at half maximum t 0 . The delay and the output pulse duration depend on t 0 and from Fig. 4(a 1 ), we note that for large t 0 >> 50 ms, the time delay increases slowly which reduces the performance of slow light. The maximum fractional delay measured in these experimental conditions is of order of 0.2. Now, we rotate the polar axis of the crystal by 180 • to change the direction of energy transfer between the input beam and the fanning or the sign of D. For D = -0.54, the same experiment ensures pulse amplification and its acceleration at the output of the SPS crystal. The performances of the fast light are plotted in Figs. 4(a 2 ) and 4(b 2 ). As shown in Fig. 4(a 2 ), the advancement is observed for short pulse durations of order of µs. the advancement values vary from 0.5 µs to 3 µs for t 0 which vary between 15 µs to 30 µs.

Conclusion

In summary, we have shown the photorefractive gain achieved through TWM can be used to slow down light pulses of durations close photorefractive response to the time of the crystal (here τ = 10 ms) and with FD of max 0.56.

In addition instead of using wave mixing with an additional pump laser, we can efficiently use the photorefractive beam fanning to generate wave mixing and slow down or speed up light pulses.

Figure 1 :

 1 Figure 1: (a) Two Wave Mixing setupin the photorefractive SPS crystal with a time response τ = 10 ms and laser at 638 nm ; A is the attenuator, EOM is the electrooptic modulator, D1,2 are the amplified detectors, M1,2 are mirrors and BS beam splinters. (b) Experimental results of the temporal envelopes of the normalized input (black line) and output (red line) pulses as a function of the time for full input pulse duration at half maximum (FWHM) t0 = 10 ms, ∆τ = 9.7 ms.

Fig. ( 2 )

 2 demonstrate that the fractional delay increases also with the photorefractive gain. The curves in Figs. 2(b 1 ) and 2(b 2 ) show that the values of the experimental fractional delay are almost the same as those obtained theoretically. The maximum FD of order of 0.5 and 0.56 are respectively obtained theoretically and experimentally for Γd = 4.7. The FD values do not exceed the unit due either to the large widening of the output pulse (theoretical case) or to the small values of the delay (experimental case).

Figure 2 :

 2 Figure 2: Slow light performances as function of Γd for a full input pulse duration at half maximum t0 = 14 ms and 0.5 cm-SPS crystal with response time τ = 10 ms. (a1) and (b1) Theoretical results (a2) (b1) experimental result. (ai) Time delay ∆τ and output pulse duration, (bi) fractional delay.

Figure 3 :

 3 Figure 3: Setup of the slow light using the beam fanning in 0.5 SPS crystal with response time τ = 10 ms performed with a laser at 1064 nm. A is the attenuator, EOM is the electrooptic modulator, p1 is the polarizer, BS is the beam splitter D1,2 are the amplified detectors.

(a 2 )(b 2 )Figure 4 :

 224 Figure 4: (a1, b1) Performances of the slow light as function of the input pulse durations for laser beam at λ = 1064 nm, the input intensity I0 = 1.2 W/cm 2 and D = 54. (a2, b2) Performances of the fast light s function of the input pulse durations when the SPS is rotated through 180 • and D = -54. (a1) delay, (a2) advancement, (b1) and (b2) output pulse durations in the case of the slow and fast light respectively.

FUNDING

Chair in Photonics: Region Grand Est; Airbus GDI Simulation; Departement de la Moselle; European Regional Development Fund (ERDF); CentraleSupélec; Fondation Supélec; Metz Metropole. The authors thank A. Grabar from the University of Uzhgorod for the SPS sample.