Maxim Goryachev

Serge Galliou

Michael E Tobar

Observation of Low Temperature Magneto-Mechanic Effects in Crystalline Resonant Phonon Cavities

We observe magnetic effects in ultra-high quality factor crystalline quartz Bulk Acoustic Wave resonators at milli-Kelvin temperature. The study reveals existence of hysteresis loops, jumps and memory effects of acoustical resonance frequencies. These loops arise as a response to the external magnetic field and span over few Hertz range for modes with linewidths of about 25mHz, which constitute a frequency shift of order 60 linewidths. The effects are broadband but get stronger towards higher frequencies where both nonlinear effects and losses are limited by two level systems. This suggests that the observed effects are due to ferromagnet-like phase of a spin ensemble coupled to mechanical modes. The observed coupling between mechanical and spin degrees of freedom in the ultra low loss regime brings new possibilities for the emerging class of quantum hybrid systems.

INTRODUCTION

Magnetic effects in mechanical resonators have become an interesting subject of research with many potential applications. In particular the combination of magnetic properties of solids and high quality factor mechanical resonators has set a new direction in the sensing technology [1][2][START_REF] Weber | Nano Letters[END_REF][4], such as cantilever magnetometry [START_REF] Weber | Nano Letters[END_REF][5][6][7]. Moreover, careful engineering of coupling between spin and mechanical degrees of freedom [8] opens new possibilities in hybrid quantum systems [9] where one can combine advantages of optomechanics and Quantum Electrodynamics with two level systems [10,11] or magnons [12,13]. Such research leads to better understanding of magnetic impurities and their influence on mechanical properties of solids that, in its turn, creates fundamental knowledge for a new generation of ultra-high purity materials with reduced susceptibility to environmental changes, ageing and other unwanted effects.

To observe such effects in a high purity acoustic system, the most important property is to attain very high quality factors of mechanical resonances. The resulting very narrow linewidths become a very sensitive tool in the discovery of low concentrations of impurities whose effects would not be seen otherwise. Among all mechanical systems, quartz Bulk Acoustic Wave (BAW) resonators cooled to cryogenic temperatures provide the narrowest possible linewidths for many overtone (OT) resonances in the 5 -700 MHz frequency range [14][15][16], and more recently similar performance at GHz frequencies [17], allowing strong coupling to photons and superconducting qubits [18][START_REF] Kharel | Multimode strong coupling in cavity optomechanics[END_REF][START_REF] Chu | [END_REF]. In this work, we study response of these mechanical cavities to external magnetic fields for temperatures near 20 mK. For these measurements we choose an SC-cut [START_REF] Kusters | Ultrasonics Symposium[END_REF] plano-convex electrodeless (BVA type [START_REF] Besson | 31st Annual Symposium on Frequency Control[END_REF]) resonator enclosed in a copper vacuum chamber. While the planoconvex structure of the resonator plate works as a potential trap for the acoustic phonons minimising losses due to supporting mechanisms [START_REF] Tiersten | [END_REF][START_REF] Stevens | The Journal of the Acoustical Society of America[END_REF], the electrodeless (electrodes are deposited on the supporting structure) technology helps to exclude all losses associated with deposition of other materials. It has been discovered previously that the longitudinal (A-type) thickness mode exhibits the superior quality factor compared to shear modes reaching very high overtone numbers. For this type of mode that is used in this work, the mechanical displacement is mostly collinear with acoustical wave vector that is normal to the plate surface. On the other hand, for the shear (B-and C-type) modes used later in the work, the displacement lies mostly in the resonator plane.

In order to investigate response of the acoustic modes to external magnetic field, the quartz BAW cavity is enclosed into a block of Oxygen Free Copper attached to the 20 mK stage of a dilution refrigerator. Because the crystal is enclosed in an individual metallic package inside a copper block, any effects of spurious electric fields, caused by the magnet or other external sources will be shielded. The structure is assembled in such a way that the device under test is placed in the middle of a 7 T superconducting magnet. The resonator plate could be oriented either parallel or normal to the external magnetic field that are further referred as B p and B n respectively (Fig. 1). The acoustic mode frequency (more precisely, frequency deviation from fixed values) is used as a convenient observable of the induced magnetic effects, as it is the most sensitive parameter of a resonant system.

Acoustic resonances are excited and detected using the piezoelectric effects with golden electrodes deposited on the structure supporting the cavity plate. RF fields are supplied via low loss microwave cables split with cryogenic DC blocks. Since the investigated BAW resonator is a single port device, all possible measurements are made in signal reflection where one uses an impedance or vector network analyser to probe the frequency response of the device via the single port. Although, in the case of ultra high-Q acoustic cavities, internal attenuation of typical network analysers is not enough to achieve the linear (small signal) regime. Indeed, it has been observed previously that for a number of ultra-high Quality factor overtones, one needs to greatly reduce the incident power to achieve the small-signal regime. In this case, the feeding and detection lines are split with a circulator that allows to introduce additional attenuation and amplification in the two port measurements as shown in Fig. 1. At low frequencies (< 1 GHz), circulators become lossy at low temperatures due to ferrite properties, its operation is limited to the room temperature. Moreover, comparatively narrow operation widths of these devices limits their operation to a couple of overtones near its central frequency.

To characterise the acoustic resonance in the magnetic fields, we employ the 37th overtone of the longitudinal mode of the plate resonator with the resonance frequency of 116.1642 MHz. For the small signal characterisation, the incident signal was reduced and distributed by a room temperature attenuator and circulator as depicted in Fig. 1. Together with additional low noise amplification, this setup allows to achieve the incident power levels of about -135 dBm, or tenths of a femto-Watt, at the BAW crystal port. This value is estimated based on the specified losses of the connecting cryogenic superconducting cables. The given value of incident power (-135 dBm) refers to the input of the BAW crystal. We take into ac-count losses of cryogenic (superconducting) connecting cables and DC blocks. Such low amounts of power are possible due to very narrow bandwidths (long sweeping rates) of the measurements. Indeed, a system response for a particular value of the magnetic field is taken with the lowest available bandwidth of 1Hz totalling about 26 minutes of sweep time. Such long measurement times are required to avoid ringing effects. The measurement setup is frequency locked to a hydrogen maser for ultimate frequency stability allowing us to resolve sub milli-Hertz frequency shifts at very long averaging times.

Parallel Orientation

The response of the resonant frequency of the 37th OT to the magnetic field oriented in parallel to the resonator plate is shown in Fig. 2. The experiment is repeated by sweeping the magnetic field from negative fields to positive and vice a versa. The results show strong hysteresis of the response together with the memory effect. Two results are almost symmetric around the zero field. The frequency deviation gets saturated at relative large values of the field. Such behaviour including memory and hysteresis is typical for ferromagnetic system [25][26][27].

(Hz) FIG. 2. Frequency deviation as a function of external magnetic field parallel to the crystal plate for sweeping magnetic field up and down for the 37th OT of the longitudinal mode.

Normal Orientation

Similar type of behavior is observed for the 33rd and 37th OTs in the normal orientation of the magnetic field as shown in Fig. 3. In these results, the field was swept down (green ↓ curve) and then up (blue ↑ curve). For this field orientation the observed hysteresis width in terms of the applied magnetic field is considerably larger. The corresponding frequency deviation is of the same order. The observed difference with the parallel orientation of the field suggest anisotropy of the magnetic system responsible for the mechanical frequency shifts. It is worth noting that two measurements (↓ and ↑) in Fig. 3 (A) and (B) have different frequencies at zero field as well as they are not symmetric under the rotation around zero field, thus, measurement prehistory, which suggests strong memory effects. Despite the fact that the observed frequency shifts are on the order of few Hertz, given extremely narrow linewidths in studied BAW resonators (25 mHz for the 37th OT at 116.164MHz [15]), the observed effects appear to be very significant constituting about 60 linewidths. This makes the BAW cavities unique for study of such weak magneto-acoustic effects.

STRONG SIGNAL MEASUREMENTS

In order to investigate the dependence of the magnetic hysteresis on absolute values of the frequency, the quartz resonator is probed directly with the Network Vector Analyser. Such simplified setup avoids circulators that allows to measure overtones of acoustic vibration in a very broad frequency range. As a result of impossibility to decrease the Network Analyser output signal strength below a certain limit, the device incident power is relatively strong on the order of -45 dBm or tens of nanoWatt. Such level of incident power results in ob-servable nonlinearity of the Duffing type. Results of frequency response to magnetic field B n sweeps for various resonant modes are shown in Fig. 4. This figure shows that the spread of the hysteresis loop increases with increasing frequency from tens of milli-Hertz for less than 10MHz modes to a couple of Hertz for high frequency modes. This fact also implies a broadband nature of the effect which is different to the linear interaction of impurity ion ensembles with photonic modes leading to narrow band avoided level crossings [28]. It is also observed that all modes demonstrate saturation at high magnetic fields. Finally, the response of 33rd and 37th OTs of the longitudinal mode in the strong signal regime is quite different to that in the weak signal setup shown in Fig. 3 suggesting an important role of nonlinearity in the interaction between magnetic and acoustic subsystems. The observed hysteresis and memory effects have strong temperature dependence. They are specifically milli-Kelvin phenomena that vanishes at temperatures above 1K, so that no considerable frequency shift has been observed at 4K.

DISCUSSION

The response of quartz BAW resonators to the magnetic field at room temperature have been observed previously [START_REF] Brendel | Proceedings of IEEE 48th Annual Symposium on Frequency Control, Proceedings of IEEE 48th Annual Symposium on Frequency Control[END_REF]. It was attributed to ferromagnetic behaviour of supporting springs holding the resonator plate. The same mechanism cannot be responsible for the magnetic effects observed in this study for three reasons. Firstly, all magnetic effects are specific to temperatures below 1K: no observable frequency shifts are visible at 4K where as clamping mechanism impact is observed even at room temperature. Secondly, the resonator used in the current work is built using a different clamping technology that avoids magnetic elements. Thirdly, the magnetic field effects are accompanied with strong nonlinear response to the excitation signals that should have no effect on supporting mechanism. Finally, increase in the OT number usually leads to improvement of the phonon trapping [START_REF] Goryachev | [END_REF] that means that a larger fraction of acoustic energy is focused in the centre of the resonator place. This effect typically leads to increase in Q factor with frequency due to uncoupling of acoustic vibration from support structure. This fact also implies that the supporting mechanism has less effect on resonance frequency for higher OTs, that contradicts the observed results (Fig. 4).

Coupling between mechanical and spin degrees of freedom may be accounted for several effects such as magnetostriction, Einstein-de Haas, Wiedemann and related effects. Although all of these phenomena require ferromagnetic materials, the class of materials quartz does not belong to. The evidence strongly suggests that the magnetic effects observed in this work originates in crys-tal impurities themselves. Traces of magnetic impurities have been observed in pure synthetic quartz crystals using whispering gallery microwave modes [28], and has been shown to be the mechanism for the largest magnetic response of the crystal material. Furthermore, the temperature dependence of mechanical Quality factors of quartz resonators showed a relaxation phenomenon due to the presence of Na + , Al 3+ ions substituting some Silicon ions in the lattice [START_REF] Mason | Effect of impurities and phonon process on the ultrasonic attenuation of germanium, crystal quartz and silicon[END_REF][START_REF] Fraser | Impurities and anelasticity in crystalline quartz[END_REF][START_REF] Martin | [END_REF][34][START_REF] Halliburton | Precision Frequency Control[END_REF][START_REF] Stevels | [END_REF]. These and other impurities in quartz have been a subject of extensive studies in literature [37,38]. Moreover, two level systems were found to be responsible for strong nonlinearities in acoustic resonators made of unswept (non purified) quartz [39], jump chaotic response of such devices at low temperatures [40], and extra losses at higher frequencies [15]. Additionally, interaction of high frequency acoustic phonons with magnetic spins has been studied within the research field of paramagnetic acoustic resonance [START_REF] Bolef | Interaction of acoustic waves with nuclear spins in solids[END_REF][42][43].

Although, the ferromagnetic-like behaviour was not observed in quartz studies with Whispering Gallery Mode (WGM) resonators [28], there are a good reasons why this is the case. Firstly, concentration of defects and impurities depends on the crystal manufacturer, and the crystals used for WGM resonators and BAW devices came from different sources. In particular, crystal treatment plays a crucial role for impurity behaviour. For example, it has been experimentally observed that annealing of sapphire crystals can convert Fe 2+ ions into Fe 3+ which caused a number of significant new effects unobserved in "as-grown" crystals [44]. Similarly, quartz crystals used for BAW resonators undergo the so-called sweeping procedure, whereas the quartz used in the WGM resonators had no such treatment. During the sweeping procedure quartz samples are heated to a few hundreds degrees and high voltage is applied to remove certain cations [START_REF] Martin | [END_REF][START_REF] Gualtieri | Proceedings[END_REF]. Secondly, the BAW devices are highly sensitive, exhibiting tens of milli-Hertz acoustic linewidths, which is many orders of magnitude narrower than that of the dielectric linewidths of WGM resonators. While the former approach detected effects on the Hz scale, the WGM modes in quartz [28] have typical linewidths of tens of kHz leaving any effect of the order of 1Hz, like ones observed in this work, very hard to observe.

In summary, the magnetic effects observed in this work have only become apparent due to the phonon trapped acoustic modes reaching extremely high quality factors of a few billion. Indeed, with other technologies providing quality factors of less than one million or linewidths of a few hundreds of Hertz, such hysteresis loops observed in this work could not have been seen. These high acoustic Q-factors makes the phonon trapped BAW resonators a new tool to for studying solid state physics [16], such as the ferromagnetic-like behaviour discovered in this work as well as a new way of realising a magneto-mechanical system in the ultra-low loss regimes. It fits the new framework of magneto-mechanical systems that have re-cently emerged bringing together advantages of both spin and mechanical degrees of freedom. For example, such systems have been recently used for applications such as magnetometry [START_REF] Li | [END_REF] and optomechanical transduction [47].

This work was supported by the Australian Research Council Grant No. CE170100009.

FIG. 1 .

 1 FIG.1. Experimental setup for measuring frequency response of BAW cavity with acoustic modes standing along k as a function of external magnetic field that is parallel ( Bp) or normal ( Bn) to the resonator plate.

FIG. 3 .

 3 FIG.3. Frequency deviation as a function of external magnetic field normal to the crystal plate for sweeping magnetic field up and down for the 33st (A) and 37th OTs (B) of the mode.

FIG. 4 .

 4 FIG.4. Frequency deviation as a function of external magnetic field normal to the crystal plate in the strong signal regime. Indices in boxes denote OT numbers and types of vibration: A -longitudinal, B -fast shear, C -slow shear.


K Z Djelloul

K Boughrara

R Ibtiouen

F Dubas

NonLinear Analytical Calculation of Magnetic Field and Torque of Switched Reluctance Machines

Keywords: Harmonic modeling, subdomain method, magnetic field, electromagnetic performances, finite elements method, switched reluctance machine

The paper deals with the nonlinear analytical harmonic modeling (HM) of 3-phases, and 6-slots/4-poles switched reluctance machine (SRM). The proposed model consists in computing magnetic field distribution and static torque of a conventional winding distribution SRM. Because a SRM has inherently nonlinear characteristics, the analytical subdomain model which does not take into account the saturation has a limited accuracy; this is due to the assumption of infinite tooth permeability. The new analytical subdomain model based on the HM technique, favorable to beyond this limitation, is presented with the consideration of the local magnetic saturation on the teeth. The results obtained with the nonlinear analytical model (NAM) are compared with the analytical subdomain model (without the saturation effect) and nonlinear finite elements method solutions.

INTRODUCTION

SRMs have attracted increasing attention of electrical machine researchers and industrials due to their simplicity, robust performance, capable of handling high temperatures, reduction in the cost of power electronics devices and low cost maintenance as well as high torque density [1]- [START_REF] Ahn | Design and Analysis of High Speed 4/2 SRMs for an air-blower[END_REF]. They have been successfully applied in electric vehicles, aircraft, mining machinery, autonomous generator,… [8]- [START_REF] Cheng | Design indicators and structure optimisation of switched reluctance machine for electric vehicles[END_REF]. Besides the previous advantages, some of their drawbacks are also evident such as high torque ripple and high acoustic noise. In order to decrease the noise emission and the vibration, many studies have been performed and some methods have been proposed

[10]- [START_REF] Zhu | Analytical Model for Predicting Maximum Reduction Levels of Vibration and Noise in Switched Reluctance Machine by Active Vibration Cancellation[END_REF]. Because a SRM has inherently nonlinear magnetic characteristics and a doubly salient pole structure, a finite element method (FEM) is often adopted to obtain accurate magnetic field distribution. However, the solution time can be large if the mesh is detailed and/or many simulations are required. For rapid solution, some analytical models of SRM were presented in [15]- [START_REF] Schinnerl | Novel Analytical Calculation Method for the Non-Linear Ψ-I-Characteristic of Switched Reluctance Machines in the Aligned Rotor Position[END_REF]. In those papers, the ferromagnetic materials are considered with infinity permeability and/or only aligned position is considered.

In this paper, which takes the solution of magnetic field in the slots/teeth into account

II. MOTOR GEOMETRY AND ASSUMPTIONS

The geometric representation of a 3-phases, and 6-slots/4poles conventional winding distribution SRM is shown in Fig. 1 and 2. In the Fig. 2, Region I represents the rotor slots/teeth, Region II the air-gap and Region III stator

, , = ∞ = ∞ , , , , , , , slots/teeth
. The angular position of the th stator slot-opening and th rotor slot-opening are defined respectively by

= 2 - (1) = 2 - + (2)
with 1 ≤ ≤ , 1 ≤ ≤ , and the position of the rotor.

The model is formulated in 2-D polar coordinate system with the following assumptions:

• Eddy current effects are neglected.

• The current density has only one component along the z-axis. • The stator and rotor slots/teeth have radial sides.

• The axial length of the machine is considered infinite and invariant, end effects are neglected. • The magnetic permeability in the stator and rotor (slots/teeth) is invariant in the radial direction, and variant in the tangential direction. • The cores of the outer and inner part of the stator and rotor respectively are assumed infinitely permeable, for simplification.

III. ARMATURE CURRENT DENSITY AND PERMEABILITY SLOTS/TEETH DISTRIBUTION

A. Stator Current Density Source

The stator slot contains two coils as shown in Fig. 2. The current density and the complex Fourier series expansion ( ) of this current density is given by

= ( ) (3)
( ) = , (4)
with ( ) = -1

The complex Fourier coefficients of , are calculated as

, = 1 2 ,
( -1)
+ , ( 1 - ) (5)
where the stator current densities in the slots for double layer armature winding are defined as

, = C ( ) (6)
, = C ( ) (7)
where = d( -) 2 ⁄ is the surface of the stator slot coil, and C ( ) & C ( ) are the transpose of the connecting matrix between the 3-phases current and the stator slots, that represent the distribution of stator windings in the slots of the 6/4 SRM with double layer winding is given by

C ( ) = -1 0 0 0 1 0 0 0 -1 1 0 0 0 -1 0 0 0 1 (8)
C ( ) = 0 0 -1 -1 0 0 0 1 0 0 0 1 1 0 0 0 -1 0 (9)

B. Stator and Rotor Permeability Distribution

For the slots/teeth in the rotor (i.e., Region I) and stator (i.e., Region III) of 6/4 SRM, the representation of permeability is given by

( ) = ̂ (10)
( ) = ̂ (11)
The coefficients of ̂ and ̂ for the Region I and III, given in the Appendix, are calculated with the distribution of permeability defined as following

( ) = , ∈ -2 , + 2 , , ∈ -2 -, -2 (12)
( ) = , ∈ -2 , + 2 , , ∈ -2 -, -2 (13)
where

= (2 / ) - & = ( 2
/ )are respectively the stator and rotor tooth opening.

The field vectors and are coupled by

= ( 14
)
In the case, that the magnetic permeability is invariant in the tangential direction, (14) becomes = and = . In our case, the magnetic permeability is variant in the tangential direction, (14) is given by [23]

= (15) = ( 16
)
where and are respectively the radial and tangential magnetic permeability.

By using the complex Fourier and the Cauchy's product theorem in (15) and (16), explained in more detail in [START_REF] Sprangers | Magnetic Saturation in Semi-Analytical Harmonic Modeling for Electric Machine Analysis[END_REF], the coefficients constitutive relation of and are given by

̂ , = ̂ (17)
̂ , = ̂ (18)
The permeability convolution matrices for the radial and the tangential of these coefficients of (17) and (18) is given by

, = ̂ ⋯ ̂ ⋮ ⋱ ⋮ ̂ ⋯ ̂ ( 19
) , = , = ̂ ⋯ ̂ ⋮ ⋱ ⋮ ̂ ⋯ ̂ (20)
IV. MAGNETIC FIELD IN 6/4 SRM By using the complex Fourier series decomposition and the method of separation of variables in and , we can find the solution to Poisson's and Laplace's equation in all regions, for that, all coefficient , of the vector potential is gathering in one column vectors .

A. Armature Slots/Teeth (Region III) Magnetic field calculation in the stator slots with double layer winding and teeth region is calculated with solving the following Poisson's matrix equations

+ 1 - 1 = -, (21) where = , ,
, and is the diagonal matrix of n, given by

= - ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ (22)
The expression for the general solution of (21) is formulated as

= + + (23)
where and are respectively the diagonal eigenvalue and the eigenvector matrix of ( ) , & are the column vectors of the constants unknown coefficients, and the term represented the particular solution of (21) with given by

= ( -4 ) , ( 24
)
where I is a diagonal identity matrix with same size as , and is the column vectors of the all coefficients , .

B. Rotor Slots/Teeth (Region I)

The Laplace matrix equation in the rotor slots/teeth, which is an annular domain delimited by the radii and , is given by

+ 1 - 1 = 0 (25)
where = , , .

The general solution can be obtained directly from (23)

with = 0 = + (26)
where and are respectively the diagonal eigenvalue and the eigenvector matrix of ( ) .

C. Air-Gap (Region II)

In the air-gap region, we have to solve Laplace's matrix equation

+ 1 - = 0 (27)
The general solution of (27) can be written as The system of six boundary conditions matrix equations (33) to (38) permits to determine the coefficients of the vector potentials in the three regions of 6/4 SRM.

= + (28)

V. ALGORITHM FOR NONLINEAR SOLVING

The magnetic saturation effects are calculated only in the stator and rotor teeth of the SRM. The back iron has been chosen in order to amplify a bit this effects are shown in Fig. 3, with the characteristic ( ) [see Fig. 3(b)] which is obtained directly by = /( ) form the curve ( ) [see Fig. 3(a)].

The algorithm presented in Fig. 5 is used to estimate the level of saturation in the SRM by an iterative procedure, with is the maximum allowable error to achieve convergence of solution, if the error less than , the analysis is finished. The update permeability values is assured by the algorithm shown in Fig. 4, in first step the relative permeability equal the maximum value given by the curve ( ), the next update determined through by the difference between and

, in all teeth one by one, is by adding half the previous value either a positive or negative value only in the case that the error greater than .

VI. ELECTROMAGNETIC TORQUE CALCULATION

According to Maxwell stress tensor method, electromagnetic torque is computed by

= , , (39)
where is the radius of a circle placed at the middle of the air-gap, and the axial length of the motor.

VII. RESULTS AND VALIDATION

The NAM is used to determine the magnetic field distribution and the electromagnetic performances of a 6/4 SRM. The main dimensions and parameters of the studied machine are given in Table I. The NAM results are verified by nonlinear FEM, and compared with the solutions of linear AM. The description of AM without the saturation effect of a SRM can be found in [START_REF] Boughrara | General Subdomain Model for Predicting Magnetic Field in Internal and External Rotor Multiphase Flux-Switching Machines Topologies[END_REF].

The magnetic field distribution is done for one phase fed by a constant current with a current density per coil equal to 3 / in aligned and unaligned position corresponds to = 0° and 45° rototation of rotor position respectivly. Figs. 6 and8 show the relative permeability distrubition in the stator and rotor slots/teeth of 6/4 SRM. Figs. 7 and9 show a comparison between analytical, nonlinear analytical and FEM radial and tangential flux densities of 6/4 SRM. In the unaligned position where the flux density in the iron is situated in the linear part of the curve ( ), is confirmed by Fig. 6, the relative permeability vary between = 1 in slots and , = 1200 in all teeth. The results of NAM is very accurate and in excellent agreement with FEM [see Fig. 7], which confirm the accuracy of this model for magnetic field prediction. However, by comparised between analytical solution and the other solution in the zoom zone [see Fig. 7(a)], we find little difference, as a magnetic short circuit exists, following to the presence of a convergence between some of the stator and rotor teeth, is gives zero magnetic field in the airgap, due to the permeability in AM are assumed to be infinitely ( = ∞) in the iron, but the NAM take the solution of this problem into account.

In the aligned position, the effect of nonlinear magnetic characteristic is important for the current density of 3 / confirmed by Fig. 9, the estimation of the saturation level is obtained with the relative permeability in 1 st and 3 nd tooth of rotor pole stay in 225 for nonlinear analytical and 250 for nonlinear FEM, and in 1 st and 4 nd tooth of stator stay in 430 and 455 for nonlinear analytical and FEM respectivly, in other teeth stay in the linear part with , = 1200.

The results shown in Fig. 9, note that the nonlinear analytical and FEM study results are in very good agreement. In Fig. 9(a), the radial flux density obtained with nonlinear analytical and FEM is lower to the values obtained by AM, the NAM is very accurate and in excellent agreement with nonlinear FEM is confirmed than NAM beyond the limitation of AM.

For the conventional type of winding distribution SRM, in Fig. 10, the static electromagnetic torque due to phase A given by the nonlinear analytical agree very well with nonlinear FEM, on the other side, in the result of the AM a considerable difference in amplitude between this model and other models can be shown especialy when the position of rotor is aproaching the aligned position. The NAM is accurate for this type of machine in aligned and unaligned position.

VIII. CONCLUSION

In this paper, we have proposed an accurate Nonlinear Analytical subdomain Model (NAM) for predicting the magnetic field distribution in any rotor position and the static electromagnetic torque in multiphase SRM. The developed model takes into account the effect of the magnetic saturation by introducing iteratively the curve B(H) of the soft magnetic material. Because the conventional winding distribution SRM presents a nonlinear characteristic, the linear analytical subdomain method (AM) is not accurate compared to NAM. Therefore, the proposed analytical model is considered as a viable to FEM for analysis of SRM.

  [20]-[21], a NonLinear Aanalytical Method (NAM) based on the subdomain Harmonic Model technique is presented. The developed model allows predicting the magnetic field distribution in the aligned and unaligned rotor position and the static electromagnetic torque in multiphase SRM with considering the nonlinear ferromagnetic characteristic. All results obtained with the NAM are then compared to those found by subdomain analytical model (AN) with the assumption of infinite tooth permeability and the nonlinear FEM [22].

Fig. 1 .

 1 Fig. 1. Studied 6/4 SRM.

Fig. 2 .

 2 Fig. 2. Simplified model of SRM.

  Conditions The studied SRM [see Fig. 2] is constituted with three regions, there are two interface continuous boundary conditions at & , and the application of the conditions Neumann boundary to the inner (outer) boundary of Region I (Region III) with infinitely permeable material is assumed,

Fig. 3 .

 3 Fig. 3. Characteristic of the stator/rotor back iron used in nonlinear analytical/FEM simulation: (a) ( ) and (b) ( ).

Fig. 4 .Fig. 5 .

 45 Fig. 4. Organigramme for update step relative permeability values.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Relative permeability distribution due to phase A in the unaligned position in the middle of: (a) Region I, and (b) Region III.

Fig. 8 .

 8 Fig. 8. Relative permeability distrubition due to phase A in the aligned position in the middle of: (a) Region I, and (b) Region III.

Fig. 9 .

 9 Fig. 9. Flux density due to phase A in the aligned position: (a) radial, and (b) tangential.

Fig. 10 .

 10 Fig. 10. Static electromagnetic torque due to phase A.

TABLE I .

 I PARAMETERS OF 6/4 SRM.

	SYMBOL	Parameter	Value and unit
	Number of stator slots	6
	Number of rotor poles	4
	Internal radius of rotor slot	17.3 mm
	External radius of stator slot	36 mm
	Radius of the external stator surface	45 mm
	Radius of the stator internal surface	25.7 mm
	Radius of the rotor surface	25.5 mm
	Air-gap length		0.2 mm
	Stack length		60 mm
	Radius of the shaft	10 mm
	Rotor slot opening	60°
	Stator slot opening	36°
	Opening of a slot coil	17°
	Phase current		15 A
	Number of conductor of slot coil	20

APPENDIX

The coefficients of ̂ and ̂ for the Region I and III is given by


Guillaume Alombert-Goget
email: guillaume.alombert-goget@univ-lyon1.fr

Yannick Guyot

Abdeldjelil Nehari

Omar Benamara

Nicholas Blanchard

Alain Brenier

Nicolas Barthalay

Kheirreddine Lebbou

Scattering defect in large diameter titanium-doped sapphire crystals grown by the Kyropoulos technique

Keywords: Ti-doped sapphire; Kyropoulos method, defects, laser emission

I. Introduction

The excellent mechanical, thermal and optical properties of titanium doped sapphire (Ti 3+ : Al 2 O 3 ) allow the development of laser systems with very high average and peak powers. Doping sapphire with Ti 3+ , confers it broad emission band to allow tunable laser of wavelength about 660-1180 nm, and therefore produce laser of short pulses which are less than 10 fs [1][2]. This amplifier material is currently the most widely used in the PetaWatt fs laser chains based Chirped pulse amplification(CPA) technique [START_REF] Alombert-Goget | Large Ti-doped sapphire bulk crystal for high power laser applications[END_REF]. The development of these high-power laser systems opens the way to applications based on table-top relativistic laserelectron accelerators [START_REF] Esarey | Physics of laser-driven plasma-based electron accelerators[END_REF] or on laser-driven ion sources [START_REF] Macchi | Ion acceleration by superintense laser-plasma interaction[END_REF].

New academic projects aimed to obtain powers of the order of tens and hundred Petawatt are under development. Such objectives necessarily require crystals with large diameter size, ultra-high quality, almost without defects, with relatively high concentrations of dopants.

It is therefore significant to develop titanium-doped sapphire with a diameter greater than 100 mm. These crystals must have a good resistance to damage, a good homogeneity of the Ti 3+ ions distribution controlled longitudinally and transversely in all the parts. This development requires the control and the well understanding of the various phenomena that occurred during crystallization.

The largest Al 2 O 3 : Ti 3+ crystals existing today in the world have a diameter of the order of 180 mm with great difficulties of reproducibility. They are obtained by the growth method called HEM (Heat Exchange Method) [START_REF] Joyce | Progress in the growth of large scale Ti:sapphire crystals by the heat exchanger method (HEM) for petawatt class lasers[END_REF]. But the other techniques such as Kyropoulos method are promising for obtaining large-diameter sapphire crystals able to meet the specifications [START_REF] Alombert-Goget | Large Ti-doped sapphire bulk crystal for high power laser applications[END_REF] [START_REF] Nehari | Ti-Doped Sapphire (Al 2 O 3 ) Single Crystals Grown by the Kyropoulos Technique and Optical Characterizations[END_REF].

The current limitations of Ti 3+ : Al 2 O 3 crystals are caused by several reasons. Firstly, reproducing the same growth behaviors with Ti 3+ dopants is difficult, the dopants usually show a strong longitudinal and transverse segregation, which degrades the optical quality of the crystal for laser applications [3][8]. Then the segregation results in the propagation of bubbles [START_REF] Ghezal | Effect of Pulling Rate on Bubbles Distribution in Sapphire Crystals Grown by the Micropulling Down (μ-PD) Technique[END_REF] [START_REF] Li | Qualitative and quantitative bubbles defects analysis in undoped and Ti-doped sapphire crystals grown by Czochralski technique[END_REF], and finally will lead to the thermal stresses which cause great optical damage.

The defects of single crystals obtained by different drawing methods depend on the growth conditions. Indeed, temperature gradients, convections and surface tensions have much effect on crystal growth. The sapphire melts at 2050 °C, despite its congruent behavior (no change in composition or phase transition during solidification), it is difficult to grow crystals in a stationary state of drawing. The sapphire has a very particular defect: microbubbles of gas in the form of a diameter varying from one micrometer to a few tens of micrometers [START_REF] Ghezal | Effect of Pulling Rate on Bubbles Distribution in Sapphire Crystals Grown by the Micropulling Down (μ-PD) Technique[END_REF][10] [START_REF] Li | Bubbles defects distribution in sapphire bulk crystals grown by Czochralski technique[END_REF][12] [START_REF] Fukuda | Growth of bubblefree Ti-doped Al2O3 single crystal by the czochralski method[END_REF]. Microbubbles degrade optical properties by decreasing transparency [START_REF] Ghezal | Effect of Pulling Rate on Bubbles Distribution in Sapphire Crystals Grown by the Micropulling Down (μ-PD) Technique[END_REF][10] [START_REF] Li | Bubbles defects distribution in sapphire bulk crystals grown by Czochralski technique[END_REF]. The presence of colored centers (F, F + ) is also detrimental for laser application [START_REF] Alombert-Goget | Luminescence and coloration of undoped and Ti-doped sapphire crystals grown by Czochralski technique[END_REF] [START_REF] Zhou | Growth and spectroscopic properties of Ti-doped sapphire single-crystal fibers[END_REF]. FOM (figure of merit) is one of the values most commonly used to evaluate the laser potential for titanium-doped sapphires, which is the ratio of its attenuation coefficients at the laser pump wavelength to the wavelength of the laser emission. To get high FOM values, it is essential to minimize the residual absorption at 800 nm (center of the laser emission wavelength range) which is attributed to the presence of Ti 3+ -Ti 4+ pairs [START_REF] Aggarwal | Residual infrared absorption in As-grown and annealed crystals of Ti:Al2O3[END_REF]. Thus, the difficulties in controlling the valence of titanium (Ti 3+ , Ti 4+ ) constitute the major problem for using titanium-doped sapphire as a laser material [16][17][18] [START_REF] Rapoport | Titanium sapphire laser characteristics[END_REF]. After annealing the crystal under a reducing atmosphere, it is possible to achieve FOM values higher than 200 of on small diameter components (<1 cm). It is commonly accepted that sapphires used in a laser cavity must have a FOM greater than 100 [START_REF] Pishchik | Sapphire[END_REF]. However, for sapphires containing a large quantity of titanium (± 532nm close to or greater than 1 cm -1 ), it is very difficult to achieve FOM values close to or higher than 100 [START_REF] Aggarwal | Residual infrared absorption in As-grown and annealed crystals of Ti:Al2O3[END_REF]. On the large diameter crystals (> 50 mm) obtained by HEM [START_REF] Rapoport | Titanium sapphire laser characteristics[END_REF] and Czochralski [START_REF] Uecker | Czochralski growth of Ti:Sapphire laser crystals[END_REF] without post-treatment, FOM values of 100 or slightly higher were measured locally but not homogeneously. Compare to the results published in the literature growing by the other techniques, we have shown that the 100 mm diameter sapphires obtained by the Kyropoulos method have equivalent FOM values [START_REF] Alombert-Goget | Large Tidoped sapphire single crystals grown by the kyropoulos technique for petawatt power laser application[END_REF].

In this paper, we observe and discuss an optical defect responsible for light scattering in titanium-doped sapphires and peculiar to Kyropoulos grown crystal. There are very few references in the literature about this defect. This paper presents its specificity and discuss its the possible nature.

II. EXPERIMENTAL PROCEDURES

II.1. Sample preparation

Ti-doped sapphire single crystals have been grown by the Kyropoulos [START_REF] Nehari | Ti-Doped Sapphire (Al 2 O 3 ) Single Crystals Grown by the Kyropoulos Technique and Optical Characterizations[END_REF]. The Ti: sapphire crystals used in this study were both grown along a-axis [112 � 0] and m-axis [1100] in argon atmosphere. All the starting titanium concentration in raw material is 0.10 atom%. The details of the growth process have been published in a previous paper [START_REF] Nehari | Ti-Doped Sapphire (Al 2 O 3 ) Single Crystals Grown by the Kyropoulos Technique and Optical Characterizations[END_REF]. Fig. 1 shows the grown crystals obtained by Kyropoulos.

II.3. X-ray rocking curve measurements

The equipment used is a SmartLab X-ray diffractometer from the Japanese company RIGAKU. It is equipped with a rotating anode of copper, used at 9kW (45kV, 200mA). The optics used, before sample, are a 1D parabolic mirror, a Germanium oriented (2 2 0) monochromator with 4 reflections, a 5 mm mask and a 1 mm slit. The optics used, after sample, are two 1 mm x 10 mm slits, 2,5 ° Sollers slits, possibly a Germanium oriented (2, 2, 0) analyzer with 2 reflections and a scintillation detector. The sample support plate is installed on a double tilts table at 90 ° RxRy. A motor placed under the RxRy table, allows the sample to be rotated. With the double tilts table placed above, the axis coincides with the normal to the planes of type (1, 1, 0).

II.4. transmittance measurements

The absorption coefficient profiles were obtained after transmittance measurements by the propagation of laser diode emissions at 532 nm and 808 nm through the sample. The laser beam diameter on the surface of the sample is about 1 mm. The polarization of the laser beam was linear and setting parallel to the c-axis of the crystal (À polarization) by a Glan-Taylor polarizer. Sample positioning (horizontal and vertical) was performed by linear translation stages with micrometric precision.

II.5. Laser measurements

The laser cavity was constituted with a plan dichroic input mirror highly transparent near 532 nm and highly reflective at 800 nm. The output mirror had 20% transmission near 800 nm.

The cavity length was 46 mm. The pump beam from the second harmonic of Quantel Brillant Eazy Nd-YAG laser, at 532 nn wavelength, was focused with a 30 cm focal length at 1cm before the sample. Over than 95% of the pump power was absorbed by the sample; the remaining pump beam was eliminated after the output mirror with a 750 nm long pass filter.

II.6. luminescence measurements

The measurements were performed using an EQ-99 broadband Laser-Driven Light Source (LDLS) from Energetic coupled with a Jobin Yvon Gemini 180 monochromator as excitation source. The luminescence is collected with a UV fiber to a Jobin YvonTriax 320 monochromator and detected by a Jobin Yvon 3000 V CCD camera. This setup allows obtaining excitation emission amplitude map of the sample luminescence.

III. Results

III.1. Light scattering

Large diameter titanium sapphire grown by Kyropoulos technique presents a current defect: a significant scattering area. This scattering area starts from some cm under the seed and passes through almost all the crystal, just stops at the end of the crystal (see figure 1.b and figure 1.c). Observed by the cross section of the crystal, this scattering area is extended on c-plan, it crosses the section by its diameter and has a width of a few mm (figure 1.d). For the crystal grown along a-axis, the scattering area crosses the crystal following a diameter corresponding to m-axis (see figure 1.b). For the crystal grown along m-axis the scattering area staying on cplan (and crosses the crystal following a diameter corresponding to a-axis). This scattering area is not homogeneous, it looks like a cloud in the material with some zones presenting different level of density. The scattering intensity was depended on the polarization of the incident light and the orientation of the crystal. In figure 4.a, the scattering effect of incident light polarized parallel to the c-axis (À polarization) of the crystal is stronger than in the case of an incident light polarized perpendicular to the c-axis (Ã polarization) presented in figure 4b. Moreover, the scattering is not isotropic and its intensity is different in function of direction of crystallographic axes of the crystal. The scattering is stronger in the direction of the c-axis (see on figure 4c) than in the perpendicular direction (see on figure 4d). This effect of the polarization of the light and the crystallographic aces suggest that the scattering elements in the material are not spherical and present a specific orientation following the crystallographic axes. These losses in transmission measurement have a strong impact on the measured value of the figure of merit. The transmittance at 532 nm and 808 nm were measured on five points of the Ti-Sapphire wafer (see the schema in figure 5.b) and reported on table 1. By comparison between the points 2 and 3 in the scattering defect and the points 4 and 5 in the free defect area, the losses by scattering corresponding are evaluated. So, if the scattering can induce losses up to 0.1 cm -1 at 532 nm, at 808 nm the losses achieve 0.025 cm -1 . These losses correspond at 532 nm at 12% of the absorption of the material but in 808 nm the losses by scattering is 4 to 5 time more important than the absorption. At 808 nm, the scattering has a very harmful impact on the FOM values. The figure of merit value is defined as:

𝐹𝐹𝐹 = 𝛼 532 𝑛𝑛 𝛼 808 𝑛𝑛
± 532 nm is the attenuation coefficient at 532 nm, ± 808 nm is the attenuation coefficient at 808 nm.

And, the attenuation coefficient is defined as:

𝛼 = 𝛼 𝑎 + 𝛼 𝑠
± a is the absorption coefficient, ± s is the scattering coefficient.

At 808 nm with a scattering coefficient 5 times more important than the absorption coefficient, the value of FOM in scattering area does not depend of the absorption coefficient but depends of the scattering coefficient at 808 nm. Thus in these areas the measured values of the figure of merit are 5 times lower than in other parts of the crystal.

III.2. Crystal quality

The sapphires are perfectly monocrystalline, between two cross polarizer the crystals didn't present any mark of defect (see figure 2) [21] [22] . The scattering area cannot be explained by crystal twinning, grain boundary or any crystalline disorientation. The birefringence proprieties of the crystal are homogenous on all the crystal, so the "inclusion" responsible for the scattering effect must be isotropic. The high crystalline quality of the sapphires grown by Kyropoulos method was confirmed by X-ray diffraction measurement. Figure 3 shows the results of Rocking curve measurements obtained for the plane (1,1,0). The FWHM values for the measurements taken without analyzing are 12.6 arcsec in the area without scattering and 11.9 arcsec in scattering area, respectively. With the analysis FWHM values of the curves for the two areas are 9.7 arcsec.

The measured FWHM values are small [START_REF] Jafari | Rocking curve imaging of high quality sapphire crystals in backscattering geometry[END_REF] [24] both in the defect-free zone and in the defective zone. Thus the crystal quality of the sapphire is extremely good in the areas presenting the scattering defects. The quality of the crystal seems even better in areas with a scattering effect. These results are consistent with the measurement of dislocation density realized by Sen et al [START_REF] Sen | Synchrotron X-ray diffraction imaging studies of dislocations in Kyropoulos grown Ti doped sapphire crystal[END_REF]. They show a lowest level of dislocation density in the central part of the crystal where the scattering defect is present. Due to this high level of crystalline quality, the presence of this scattering area in the crystal can't be explained by a crystallographic defect in the sapphire.

III.3. Laser efficiency

Laser efficiency measurements were realized in order to estimate the possibility to use a sample with this scattering area in laser cavity (see on figure 6). The slope of the laser efficiency on clear area of the sample was estimate at 48.6 %. The laser measurement realized on scattering area presented a slope of 38 %. The scattering area induces a loss of 22% of the laser efficiency compared to the free scattering area. The scattering loss induced on the energy of the pump and on the laser emission of the cavity. So a loss of 22% is not a surprise for an oscillator if you considering the high number of passes inside of the crystal during the laser emission. These measurements confirm that a sapphire containing this scattering defect cannot be used in a laser cavity. As we know, the titanium sapphire crystals of this dimension are not grown to be used in oscillator part but in amplification part of a CPA laser. The losses induced by the scattering area are in energy not so high in absolute value (0.1cm -1 at 532 nm and 0.025cm -1 at 800 nm).

With a limited number of pass in the crystal the spurious effect could by less important in amplificatory part than in oscillator part. So considering only the losses in energy, the crystal presenting this type of defect could be used in amplification part of a CPA laser. However, the presence of scattering effect in amplification part of a CPA laser can induce a reduction of the temporal contrast of the laser pulse and considering the optical energies involved t may be the source of crystal damage.

III.3. Photoluminescence

The photoluminescence intensity is not affected by the diffusion of the material. The emission amplitude maps realized on scattering area do not change compare to those realized on the area defect free (see figure 7). They all present the blue emission with the same excitation and emission range. The luminescence properties seem homogenous on the samples. The maps don't show an increase of the F or F + center emission in particular. If the emission amplitude maps allow a comparison of the spectroscopic properties, it didn't allow a comparison of emission intensity. The blue emission of the Ti-doped sapphire is less intense in the scattering area (see on figure 7e). The defects responsible to this scattering have an impact on the mechanism of the blue emission which is linking with color center and Ti 4+ ions [START_REF] Alombert-Goget | Luminescence and coloration of undoped and Ti-doped sapphire crystals grown by Czochralski technique[END_REF] [START_REF] Zhou | Growth and spectroscopic properties of Ti-doped sapphire single-crystal fibers[END_REF][29] [START_REF] Mikhailik | Studies of concentration dependences in the luminescence of Ti-doped Al2O3[END_REF]. So the amounts of color center or Ti 4+ are reducing in the scattering area, which may contradict the explanation of an oxygen deficiency in the sapphire. If the stoichiometry of the sapphire is a hypothesis that cannot be dismiss, other possibilities can be evaluated by the comparison between the techniques Kyropoulos and Czochralski as we show below.

IV Discussion

There is very few mention of similar defect in the publications. Some scattering centers that look like small bubbles are mentioned by Lacovara et al and attributed to oxygen vacancy [START_REF] Lacovara | Growth, spectroscopy, and lasing of titaniumdoped sapphire[END_REF]. To produce large diameter titanium sapphire crystal the kyropoulos technique used a large diameter molybdenum crucible. In order to avoid the oxidation of the crucible, the furnace integrates an important graphite thermal insulation. The presence of this huge quantity of carbon in the furnace induces a strongly reductive environment [START_REF] Bruni | Will Czochralski Growth of Sapphire Once Again Prevail?[END_REF]. According to Vishnevski-et al [START_REF] Vishnevski- | Specific features of the formation of submicron inclusion in Ti:sapphire[END_REF] [START_REF] Vyshnevskiy | Foreign phase inclusions in Ti-sapphire grown in a carbon-containing medium[END_REF], this reductive environment during growing process induce a oxygen deficiency in the melt stoichiometry, which produces inclusions with an excess of aluminum ions 40-50 h after the crystallization starting. They evaluated the size of the inclusions at 70-300 nm. Contrary to what they have presented in their publications, we have not been able in the present work to observe these defects by electron microscopy. On the samples extracted from zones having a high scattering density, our measurements did not show the presence of structure which can be attributed to the defects in this range of sizes.

The oxygen deficiency in the stoichiometry of the sapphire should increase the presence of F center. In absorption spectra, the absorbance attributed to the F center seems increase in the defect area [START_REF] Vishnevski- | Specific features of the formation of submicron inclusion in Ti:sapphire[END_REF]. However the losses by scattering increase the level of the base line in absorption measurement. This is more intense for low wavelength and produces an effect on the F center absorbance band by the increase of its intensity.

One of the main defects in titanium doped sapphires grown by Czochralski technique are the presence of bubbles as shown in figure 8 already well study previously [START_REF] Li | Qualitative and quantitative bubbles defects analysis in undoped and Ti-doped sapphire crystals grown by Czochralski technique[END_REF] [START_REF] Fukuda | Growth of bubblefree Ti-doped Al2O3 single crystal by the czochralski method[END_REF]. But in your knowledge nobody see and report scattering defect in sapphire obtained by Czochralski as presented in this article. They are many differences in the growing process between Czochralski and Kyropoulos. The crystallization takes place in the melt with a highly convex crystallization interface [START_REF] Stelian | Comparison between numerical modeling and experimental measurements of the interface shape in Kyropoulos growth of Tidoped sapphire crystals[END_REF]. This interface shape coupled with the segregation effect induces is responsible to a strong gradient of Ti concentration in the crystal [START_REF] Stelian | Interface effect on titanium distribution during Ti-doped sapphire crystals grown by the Kyropoulos method[END_REF]. The interface shape is driving by the thermal gradient in the melt and the crystal due to the isolated configuration and the rotation speed imposed on the crystal. In Kyropoulos technique, the rotation speed currently of 2 rpm is low compared to the rotation speed used in Czochralski technique [START_REF] Li | Qualitative and quantitative bubbles defects analysis in undoped and Ti-doped sapphire crystals grown by Czochralski technique[END_REF] [START_REF] Li | Bubbles defects distribution in sapphire bulk crystals grown by Czochralski technique[END_REF]. In Kyropoulos technique, due to this rotation speed and the interface shape, it could be possible than different crystallization speeds in function of the crystal axis occur. Thus, the crystallization in direction of the c axis is faster than in other direction. This difference of crystallization speeds in function of the crystal axis can explain the creation of defect localized specifically along of on a diameter of the crystal and oriented on the c-plan of the crystal. Reduce the convexity of the interface should homogenize the growth rates as a function of the crystallographic axes and potentially eliminate this defect.

Increase the rotation speed at a comparable level to the Czochralski technique can reduce the pressure in the melt below the crystallization interface, and reduce the convexity of the interface. In addition to changing the rotation speed, some other adaptation in the furnace could be considered in order to modify the thermal gradient with the target to reduce the convexity of the interface. Thus, some modifications in the isolation configuration or in the shape of the crucible should be considered to achieve this target.

Obtain an interface more flat would improve the quality of the Ti sapphire for CPA laser application. It allows better radial reparation of the dopant and could reduce the formation of this scattering defect.

V. Conclusion

Defects corresponding to the presence of a scattering area in titanium doped sapphire obtained by Kyropoulos technique. This scattering cannot be attributed to the presence of bubbles or induced by a crystallographic defect. In this area a diffusion coefficient of 0.1 cm -1 at 532 nm and 0.025 cm -1 at 808 nm are measured. This scattering area has catastrophic effect on the laser properties of the Titanium doped sapphire. The measurements showed that sapphires with this defect could not be used in a cpa laser.

The measures did not allow us to observe or identify the precise nature of the scattering element in the material. The hypothesis of an oxygen deficiency was discussed. But, the spectroscopic measurements, in particular the F center emissions, do not confirm this explanation.

By comparing with the Czochralski technique (for which the crystals present no defect), the convex interface shape in the kyropoulos could be responsible of this defect. Thus, increase of the rotation speed in the Kyropoulos, allowing a decrease of the convexity of the interface shape can be a way to eliminate this defect and increase the optical quality of the crystal.

Figure 1 :

 1 Figure 1: Picture of the kyropoulos sapphire crystal on intense white light. The scattering area is

Figure 4 :Figure 5 :

 45 Figure 4: a) Ti-doped sapphire illuminated by light in À polarization; b) Ti-doped sapphire illuminated

Figure 2 :

 2 Figure 2: Kyropoulos Ti doped wafer between two cross polarizer. In a) the sapphire is oriented in

Figure 3 :

 3 Figure 3: X-ray rocking curve on the plan (1,1,0) in a clear area (without scattering) and in scattering

Figure 6 :

 6 Figure 6: Laser efficiency measurement. The square dots correspond to measurements realized on

Figure 7 :

 7 Figure 7: Emission amplitude maps in a) and c) on free defect area, in b) and d) on scattering area. e)

Figure 8 :

 8 Figure 8: in a titanium sapphire crystal wafer obtains by Czochralski technique; in b same crystal on

Table 1 :

 1 Attenuation coefficients measured at 532 nm and 808 nm in À polarization on different positions on the sample. The value of the figure of merit (FOM) calculated from the attenuation coefficients are presented for each position of measurements.

		Attenuation coefficient
		(cm -1 )		FOM
		808 nm	532 nm
	1	0,0209	0,7186	34
	2	0,0301	0,9010	30
	3	0,0294	0,9555	32

Acknowledgment

This study was performed in the framework of French national research project FUI TITANSAPHIR with the financial support of the Région Auvergne Rhône-Alpes, project N° 13 023257 01. The author would wish to thank the staff at CMTC, Grenoble and in particularly Hervé ROUSSEL for the help with the rocking curve analysis.