Stochastic uncertainty quantification of the eddy current in human body by using polynomial chaos decomposition Roman Gaignaire ♣ , Riccardo Scorretti ♥ , Ruth V. Sabariego ♣ and Christophe Geuzaine ♣

The finite element method can be used to compute the electromagnetic fields induced in the human body by environmental extremely low frequency (ELF) fields. However, the electric properties of tissues are not precisely known and may vary depending on the individual, his/her age and other physiological parameters. In this paper, we account for the uncertainties on the conductivities of the brain tissues and spread them out to the induced fields by means of a non-intrusive approach based on the chaos Hermite polynomial with the finite element method as a black box [START_REF] Gaignaire | Stochastic uncertainty quantification of the conductivity in eeg source analysis by using polynomial chaos decomposition[END_REF], [2]. After showing the convergence of the method, we compute the probability to be over the thresholds define by the norms.

Index Terms-Non-intrusive methods, polynomial chaos decomposition, stochastic methods

I. INTRODUCTION

T He uncertainty in electrical parameters, as well as the vari- ability of the human body between different individuals, is a major problem in numerical dosimetry. Determining the conductivities of the human tissues is since years subject of research [START_REF] Geddes | The specific resistance of biological materiala compendium of data for the biomedical engineer and physiologist[END_REF]. Tissues are highly heterogeneous "materials"and possibly anisotropic [START_REF] Sachse | Extension of anatomical models of the human body: Three-dimensional interpolation of muscle fiber orientation based on restrictions[END_REF]. The properties of tissues change rather quickly after death [START_REF] Haemmerich | Changes in electrical resistivity of swine liver after occlusion and postmortem[END_REF], so that measurements performed in vitro on excised tissues may not be representative. Besides for ethical reasons, most of in vivo measurements have been performed on animals. Moreover, the age [START_REF] Gabriel | Dielectric properties of biological tissue: variation with age[END_REF] and the physiological condition [START_REF] Peratta | Modelling the human body exposure to elf electric fields[END_REF] may also significantly alter these properties. Obtaining data for characterising a foetus is even harder [START_REF] Cech | Fetal exposure to low frequency electric and magnetic fields[END_REF]. At ELF frequencies, the measurements are performed either by identifying an equivalent RC circuit, or by a four-point measurement. At those frequencies, electrode polarization is (another) major source of errors. Some tentatives have been performed [START_REF] Peratta | Modelling the human body exposure to elf electric fields[END_REF], [START_REF] Smye | Modelling the electrical properties of tissue as a porous medium[END_REF] for numerically estimating the conductivity and permittivity by modeling tissues as porous media, with a limit success. Gabriel et al have done a huge work [START_REF] Gabriel | The dielectric properties of biological tissues: I. Literature survey[END_REF]- [START_REF]The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues[END_REF] by collecting most of the existing data, and building a tissue database, which is currently the reference for dosimetric computations [START_REF] Andreuccetti | An internet resource for the calculation of the dielectric properties of body tissues[END_REF]. In a recent work [START_REF] Gabriel | Electrical conductivity of tissue at frequencies below 1 MHz[END_REF], new measurements have shown large differences with respect to previously published data: the new values of conductivities are often higher and muscle-type tissues are found to be much less anisotropic. More recently, several techniques based on the magnetic resonance (MR) have been proposed. MR electric impedance tomography (MREIT) methods are based on the detection of the shielding effect of electrical currents on the magnetic field; these electrical currents may be either induced by the MR device itself, by a third external coil [START_REF] Ueno | Impedance magnetic resonance imaging: a method for imaging of impedance distributions based on magnetic resonance imaging[END_REF], or injected into the body through electrodes [START_REF] Kim | In vivo high-resolution conductivity imaging of the human leg using MREIT: the first human experiment[END_REF]. Diffusion tensor MREIT methods are based on the proportionality between water diffusion tensor and electrical conductivity. These last techniques are particularly interesting when dealing with brain [START_REF] Sekino | Magnetic resonance imaging of electrical conductivity in the human brain[END_REF], because the diffusion signal in this organ is strong enough, while other MREIT techniques are not applicable due to the high resistivity of the skull. Moreover, this technique provides information about the anisotropy of tissues. Using this technique, Sekino et al Geuzaine (e-mail: rgaignaire@gmail.com, cgeuzaine@ulg.ac.be).

found higher conductivity values [START_REF] Sekino | Low-frequency conductivity tensor of rat brain tissues inferred from diffusion mri[END_REF] than those reported by Gabriel in [START_REF] Gabriel | The dielectric properties of biological tissues: I. Literature survey[END_REF]. All MRI based techniques share the enormous advantage to be applicable in vivo in humans beings. In spite of the amount of work accomplished, the dielectric properties of human tissues are still poorly known: e.g. at 50 Hz the conductivities of the white and grey matter in [START_REF] Gabriel | The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz[END_REF] and [START_REF] Latikka | Conductivity of living intracranial tissues[END_REF] span, respectively, within [0.0753 ; 0.5155] and [0.0533 ; 0.302] S/m. Therefore, it is crucial to quantify the effect of this uncertainty on the electromagnetic fields computed in the human body. One could use a extremely time-consuming Monte-Carlo (MC) simulation to statistically characterise the induced fields (some weeks in our model).

In this paper, we use a chaos polynomial (CP) approach, the so-called non intrusive probabilistic algorithms, which assumes that the variances of conductivities are finite [START_REF] Gaignaire | Stochastic uncertainty quantification of the conductivity in eeg source analysis by using polynomial chaos decomposition[END_REF], [2] and allows to completely characterise the induced field in the probabilistic dimension with a much lower computational cost (a few hours).

II. INCORPORATION OF STOCHASTIC UNCERTAINTY A. Deterministic framework

The fields induced in the human body are computed by the finite element (FE) method using a φa formulation. Details of this formulation and the phantom used (Fig. 1) can be found in [START_REF] Scorretti | Computation of induced fields into the human body by dual finite element formulations[END_REF]. We simulate the exposure to the field generated by an infinite cable (current I = 1000 A at 50 Hz) placed at a few centimeters from the left side of the head. In the literature on protection against ELF fields, three global quantities are generally computed for each organ from the current density j: the (spatial) average j Avg , the maximum value j max , and the 99% percentile j 99-perc [START_REF] Dimbylow | Induced current densities from low-frequency magnetic fields in a 2 mm resolution, anatomically realistic model of the body[END_REF]. Similar definitions exist for the electric field e. The basic restriction of the 1998 edition of ICNIRP's recommendations [START_REF] Icnirp | Guidelines for limiting exposure to time-varying electric, magnetic, and electromagnetic fields (up to 300 ghz)[END_REF] is based on j max , while the 2010 edition [START_REF]Guidelines for limiting exposure to time-varying electric and magnetic fields (1 Hz to 100 kHz)[END_REF] focus on e 99-perc . In particular, for a 50 Hz occupational exposure the threshold values in the central nervous system (CNS) recommended by the ICNIRP are j max < 10 mA/m 2 and e 99-perc < 100 mV/m.

B. Uncertainties

In this paper, we focus on the fields induced in the brain. The conductivities of the white matter σ W (ω) and of the grey matter σ G (ω) are modeled within a probabilistic framework1 , by assuming that they are random variables defined on a probabilistic space Θ, F, P . Therefore j Avg , j max , j 99-perc , e Avg , e max and e 99-perc turn out to be random as well. In particular, by using the maximum entropy principle [START_REF] Jaynes | Information theory and statistical mechanics[END_REF] we model (arbitrarily) σ G (ω) and σ W (ω) as independent random variables, uniformly distributed:

σ G (ω) ∼ U([0.0753 ; 0.5155]) (S/m) (1) σ W (ω) ∼ U([0.0533 ; 0.3020]) (S/m) (2)

C. The non intrusive approach

As the conductivities of the brain and the cerebellum are two independant random variables of finite variance, we can expand them as a truncated series of order p in in the bidimensional Hermite polynomials of a random gaussian vector ξ(ω) = (ξ 1 (ω), ξ 2 (ω)), known as Hermite chaos polynomials [2]:

σ G (ω) ≈ Pin i=0 σ Gi ψ i (ξ(ω)) (3)
σ W (ω) ≈ Pin i=0 σ W i ψ i (ξ(ω)) (4)
where σ Gi and σ W i are scalar values that depend on the probabilistic law of the conductivities, P in = C pin 2+pin is the number of bi-dimensional polynomials of order less than p in , and ψ i is the i th bi-dimensional polynomial of Hermite. To solve the stochastic problem, we use an approach based on a CP decomposition of both the conductivity and the induced fields [2]. We assume the conductivities to be of finite variance, with no assumption on the shape of the probabilistic distribution.

The values of those induced fields -the average current density in the brain j Avg (ω) = j Avg (ξ(ω))are computed by the FEM from any couple of values (σ G (ξ(ω)), σ W (ξ(ω))). The average density belongs to a space that can be spanned by the polynomials ψ(ξ(ω)) and thus written as a truncated series to an order p out :

j Avg (ω) = Pout m=0 j Avg m ψ m (ξ(ω)). (5)
To compute the value of the unknown real coefficients j Avg m , we use the orthogonality properties of the Hermite polynomials:

j Avg m = E[j Avg (ω)ψ m (ξ(ω))] E[ψ m (ξ(ω)) 2 ] , (6)
where E[•] is the mathematical expectation. The denominator can be computed analytically. The integral in the numerator is computed by means of a Hermite Gauss integration scheme with d integration points [2]:

E[j Avg (ω)ψ m (ξ(ω))] ≈ d i=1 ... d j=1 w i,j (j Avg ((t 1 , t 2 ) i,j ))ψ m ((t 1 , t 2 ) i,j ), (7)
with (t 1 , t 2 ) i,j the i, j-th Gauss point and w i,j the associated weight in the bi-dimensional Cartesian rule. The deterministic problem must thus be computed d 2 times, with the conductivity evaluated through (3) and (ξ 1 (ω), ξ 2 (ω)) = (t 1 , = t 2 ) i,j , i, j = 1, . . . , d.

III. RESULTS AND DISCUSSION

The non intrusive method is governed by three parameters: p in , p out and d; p in is linked to the precision on the approximation made on the input random variables σ G (ω) and σ W (ω), p out is the order of truncation of the studied global quantities (j Avg , j max , j 99-perc and the corresponding quantities for the electric field) and d is the number of quadrature points. Herein, we have chosen p in = 16, while p out and d vary. For the sake of conciseness, we deal with the white and grey matter (though the method could handle other tissues during the same computation as well).

A. Influence of the input parameters

The probabilistic density (PD) of e 99-perc in the grey matter obtained with p out = 8, p in = 16 and different values of d is shown in Fig. 2 The PD of j max in the white matter obtained with d = 16, p in = 16 and different values of p out are plotted in Fig. 3. Again, one observes that convergence is achieved as p out increases. The value of p out has a minor influence on the central dispersion parameters (mean and variance): the mean is constant and equal to 13.1 mA/m, and the standard deviation converges with p out ≥ 3 to 3.4 mA/m. It can be observed that the support of the PD is bounded to 21 mA/m for p out ≥ 4 (that is, most likely j max < 21 mA/m): conversely, p out = 2 would lead to the wrong conclusion that j max may exceed 21 mA/m with a non negligible probability.

B. Analysis of the results

The PD of j Avg , j 99-perc and j max linked to the induced current density in the grey matter are represented in Fig. 4 for p out = 4 and p out = 8. For p out = 6, the curves are very close to those for p out = 8 so the method has converged. One observes that while j Avg is always under to 10 mA/m 2 , this does not hold for j 99-perc and J max , which are likely to exceed this hazardous limit -the exact probability is computed hereafter.

The PD of e Avg , e 99-perc and e max related to the electric field in the white matter are depicted in Fig. 5 for p out = 4 and p out = 8. The curves for p out = 4 and p out = 8 are much more similar than in the previous case. These PDs are more peaked than those corresponding to the current density, i.e. they are less dispersed around their means. Moreover, the area under these curves for ||e|| > 100 mV/m equals 0 for e Avg and e 99-perc and nearly 0 for e max . In order to avoid health hazards, ICNIRP recommends that in the central nervous system j max ≤ 10 mA/m 2 [START_REF] Icnirp | Guidelines for limiting exposure to time-varying electric, magnetic, and electromagnetic fields (up to 300 ghz)[END_REF] or e 99-perc ≤ 100 mV/m [START_REF]Guidelines for limiting exposure to time-varying electric and magnetic fields (1 Hz to 100 kHz)[END_REF]. As these global quantities are available as a polynomial expansion like (5), we can estimate the probability -p-that these recommendations are not fulfilled -thus, in the case of [START_REF] Icnirp | Guidelines for limiting exposure to time-varying electric, magnetic, and electromagnetic fields (up to 300 ghz)[END_REF], p is: P{j max ≤ 10 mA/m 2 }. To this aim, a large number n of couples of independent values following a normal variable (ξ 1i , ξ 2i ) 1≤i≤10 4 are sampled. The polynomial expansion (5) is evaluated for each pair of values and the number of occurrences (i.e. the number of pairs) for which the basic restriction is exceeded are counted. The probability p and the confident interval CI are estimated by means of the central limit theorem as:

p = Number of occurences n (8)
CI = z .05/2 p(1 -p) n (9)
where z .05/2 is the confidence interval of 97.5% of a normal random variable. The cost of this computation is negligible with regard to the coefficient evaluation via [START_REF] Gabriel | Dielectric properties of biological tissue: variation with age[END_REF].

The probability that each of the global quantities exceeds the basic restriction for the simulated exposure has been computed with n = 10 5 and the obtained values are reported in Table I. For instance, the third line in Table I reads: the probability for e max > 100 mV/m is of 80% ± 0.27% with a risk of 5% 2 . According to most recent recommendations [START_REF]Guidelines for limiting exposure to time-varying electric and magnetic fields (1 Hz to 100 kHz)[END_REF], the basic

Field

Probability in Grey matter in white matter eavg 0 ± 0 0 ± 0 e 99-perc 0 ± 0

0 ± 0 emax 0.8 ± 2.7 × 10 -3 0.14 ± 2.2 × 10 -3 javg 0 ± 0 0 ± 0 j 99-perc 0.73 ± 2.8 × 10 -3 0.16 ± 2.3 × 10 -3 jmax 1 ± 0 0.80 ± 2.5 × 10 -3
restriction e 99-perc ≤ 100 mV/m is most likely fulfilledit would not have been the case if we consider e max instead of e 99-perc . Concerning the 1998 recommendations [START_REF] Icnirp | Guidelines for limiting exposure to time-varying electric, magnetic, and electromagnetic fields (up to 300 ghz)[END_REF], the restriction j max ≤ 10 mA/m 2 is clearly not fulfilled for the grey matter (see also Fig. 4), while for the white matter the probability for j max to exceed 10 mA/m 2 is of 80%. These probabilities decrease respectively to 73% and 16% when dealing with j 99-perc instead of j max .

IV. CONCLUSION

The proposed recommendations for avoiding health issues due to over-exposure to ELF radiations may require to evaluate the induced fields in the human body by dosimetric methods.

Unfortunately, these computations are largely affected by the uncertainty on the conductivities of human tissues. Indeed, an arbitrary security factor 3 is considered in [START_REF]Guidelines for limiting exposure to time-varying electric and magnetic fields (1 Hz to 100 kHz)[END_REF] in order to account for "dosimetric uncertainties".

We propose an effective method for quantifying the uncertainty on some relevant quantities (notably j max and e 99-perc ), provided that a characterisation of the statistical distribution of the conductivities is available. Our simulations suggest that latest 2010 ICNIRP recommendations are more permissive than those in the former edition. However, it has to be pointed out that in our computations we use a quite pessimistic statistical law for the conductivities, and also that other sources of uncertainty (posture, physiognomy...) are disregarded.

Manuscript received July 1 ,

 1 2011. Corresponding authors: R. Gaignaire, C.

Fig. 1 .

 1 Fig. 1. a) Mesh of whole head; b) grey matter; c) and white matter.

  . The curves of the PD obtained with d = 10 and d = 16 are nearly superposed, what proofs the convergence of the method with increasing values of d. Concerning dispersion parameters as the mean and the standard deviation, the convergence is reached as soon as d ≥ 4 (mean: 0.0487V /M , standard deviation: 0.07 when d = 2 to 0.0044 as soon as d = 6).

Fig. 2 .Fig. 3 .

 23 Fig. 2. Probabilistic density of e 99-perc in the grey matter for pout = 8, p in = 16 and d = 6 (blue), d = 10 (red), d = 16 (green).

Fig. 4 .

 4 Fig. 4. Density of probability (in A/m 2 ) of j Avg (blue), j 99-perc (red) and jmax (green) in the grey matter (p in = 16, d = 16).

Fig. 5 .

 5 Fig. 5. Density of probability (in V/m) of e Avg (blue), e 99-perc (red) and emax (green) in the whie matter (p in = 16, = . 16).

TABLE I PROBABILITY

 I TO EXCEED THE THRESHOLD DEFINED BY THE NORMS.

The notation: x(ω) means that x is a random variable.

ACKNOWLEDGMENT

This work has been supported by the Belgian French Community (ARC 09/14-02) and the Belgian Science Policy (IAP P6/21).


Loïc Deyra
email: loic.deyra@institutoptique.fr

Igor Martial

Julien Didierjean

François Balembois

Patrick Georges

W, 300 µJ, 25ns pulsed 473nm blue laser based on actively Q-switched Nd:YAG single-crystal fiber oscillator at 946 nm

HAL is

The realization of high average power, millijoule Qswitched 946 nm laser based on Nd:YAG has several applications in spectroscopy, material processing and LIDAR applications. The second harmonic at 473 nm can be used for underwater communication [1], optical storage or spectroscopy. The fourth harmonic of 946 nm at 236 nm allows the LIDAR detection of dangerous compounds [2].

So far the reported powers and energies at 946 and 473 nm were limited. At 946 nm, energies superior to the millijoule have only been obtained at repetition rates of tens of hertz [3] [4]. At tens of kilohertz, energies up to 430 µJ have been demonstrated [5] [6]. At 473 nm, up to 1.5 W of average power and 150 µJ have been reported [6].

Indeed the main limitations of those lasers comes from the spectroscopic properties of the Nd:YAG 946 nm quasi-three level laser line. 946 nm laser operation in Nd:YAG suffers from reabsorption losses (0.7% of thermal population in the lower level 4 I9/2) and low emission cross section that limit its overall efficiency (σem946=5x10 -20 cm²). It requires crystals with high doping concentration to achieve high power emission, but then suffers from strong thermal lens. On the opposite, a low doping value keep the thermal effects at a manageable level, but reduces the absorption over the crystal length. Single-crystal fiber (SCF) overcomes this problem by using low-doped (0.2 %) and long crystals (50 mm), with an excellent thermal management and pump confinement. They already have demonstrated a significant output power at 946 nm, with more than 30 W of output power, but in CW regime, with a spatially multimode and unpolarised beam [7].

In this paper we propose to use single-crystal fiber technology to realize a high power, high energy, polarized oscillator with good beam quality adapted for efficient nonlinear frequency conversion. We will first study how to polarize efficiently a Nd:YAG oscillator. Then, we will detail how the laser parameters can be optimized for a Q:switched regime delivering an energy at the millijoule level with a significant average power. Finally, we will study a simple set-up of extracavity second harmonic generation in a BiB3O6 (BiBO) crystal.

The first difficulty arises if the laser needs to be polarized for nonlinear frequency conversion. Nd:YAG, being a naturally non-birefringent material, suffers from strong thermal-stress induced depolarization losses when a polarizing element is inserted inside the cavity, the maximum losses being located in the crystal at 45° of the imposed polarization [8]. In this work we will use a technique proposed by Clarkson & al in 1999 [9] : a quarter-wave plate is inserted between the input mirror and the laser crystal to reduce the depolarization losses.. An intracavity thin film polarizer (TFP) is used as a polarizing element because of its higher extinction ratio compared to the commonly used Brewsterangle glass plate.

The experimental set-up is detailed in figure 1. The pump source is a 120 W, 200 µm, NA 0.22 fiber coupled laser diode at 808 nm. It is focused inside the single-crystal fiber (SCF) at a pump spot diameter of 480 µm. The laser crystal is a commercial Nd:YAG single-crystal fiber module (Taranis model, Fibercryst). It consists of a 50 mm long, 1 mm diameter, 0.2 % doped Nd:YAG, AR coated for 808 and 946 nm rod embedded in a cooling system, set at 12 °C. The pump absorption efficiency without laser operation was measured to be 96%. Thanks to the pump guiding in crystal fibers, a significant part of the pump power is still present at the bottom end. As demonstrated before [10] we can consider that about 50% of the unabsorbed pump power is located on the output face in a diameter equivalent to the pump spot diameter at the focus. This corresponds to an intensity of about 1200 W/cm², compared with the transparency intensity of about 500 W/cm². [11] The laser cavity is designed with two mirrors. The input plan mirror M1 is coated for high reflectivity (HR) at the 946 nm laser line, and high transmission (HT) at 808 nm and 1064 nm. A plano-concave mirror M2 of curvature radius 200 mm closes the cavity: different transmission values are used depending on the measurements. The cavity length is 270 mm, and the laser fundamental mode diameter at maximum pump power is 400 µm, taking into account the 40 mm focal length of the thermal lens induced in the single crystal fiber. This thermal lens value was estimated with a theoretical calculation for a pump power of 100 W, and confirmed experimentally by studying the stability of a plano/plano cavity. In order to polarize the laser beam, an intracavity thin-film polarizer (TFP) coated for high reflectivity at 946 nm with an extinction ratio of 200:1 is placed between the SCF output face and M2. A quarter-waveplate (QWP) is inserted between the input mirror M1 and the SCF to compensate for the depolarization losses.

A first set of characterization is realized in continuous wave (CW) regime, where the M2 mirror is now an output coupler with a reflectivity of 98% to serve as a reference output for the determination of the depolarization losses. The measurements of those depolarization losses and the influence of the QWP are presented in Figure 2. Three output beams are simultaneously monitored (see figure 1). P1 gives the depolarization losses, P2 is the reference output and P3 is the 946 nm laser output. Those three measurements are used to calculate the round trip depolarization losses. We can see that until 50 W of pump power, the depolarization losses are reduced from 5 to 2%. Then, without the QWP, the depolarization losses raise to reach a maximum of 20% at 120 W of pump power. With the QWP, the round trip depolarization losses are partially compensated until 120 W of pump power where the QWP effect becomes negligible. This behavior was expected, as the quarter waveplate only compensates the depolarization at low powers [9]. Above 100 W of pump power, the beam quality starts to degrade. Therefore we limit our pump power to a 100 W, where the depolarization losses are 10%.

Coupled optimization of energy and average power in Qswitched require an accurate control of the laser parameters. Here we decided to act on two elements: the output coupling and the repetition rate.

In our experiment, we implement a variable output coupler based on a half-wave plate (HWP) and the TFP to have a complete overview of the laser behavior with an increase of cavity losses. By adjusting the HWP, the output coupling can be varied from 0 to 100%. In Figure 3, the "useful" CW output power (P2+P3, meaning the output on the 2% coupler plus the output from the variable output coupler) is plotted versus the output coupling (value coming from the HWP plus 2%) for 40, 60, 80 and 100 W of pump power. For every pump power value, the optimum output coupling is between 10 and 20 % of transmission. At 100 W, as much as 11.6 W of average power can be extracted for an output coupling value of 12%. This optimum output coupling is close to the one reported in other papers [3][7].

This CW characterization gives an order of magnitude of what performances will be available in Q-switched operation. The lifetime in Nd:YAG is 230 µs : therefore, one can expect the average power to drop in Q-switched operation for repetition rates between 5 and 15 kHz. Since we obtained 11.6 W in CW with an output coupler of 12%, we should be capable of operating the Q-switched laser around 10 kHz and reach an energy in order of the millijoule. However based on the SCF coating damage threshold of 3 J/cm² and the estimated beam diameter inside the SCF, we calculate the maximum output pulse energy cannot overcome 300 µJ with an output coupling value of 12%,.

Here comes the interest of the variable output coupler: it can be used to increase gradually the output coupling, in order to maximize the output energy while keeping a satisfying average output power and an intracavity energy under the components damage threshold. As can be seen on Fig. 3, an output power of 10 W in CW can still be extracted with an output coupling of 30%. It means that mJ level pulses can be safely generated by this laser in Qswitched regime with average power in the multiwatt level, assuming that around 10 kHz the average power only drops by a little amount.

To obtain pulsed operation, we used acousto-optical Qswitching for the simplicity, and the large range of repetition rate available. A compact Gooch&Housego acousto-optic modulator (AOM) is placed between the SCF output and the TFP. The AOM is 35 mm long, coated for low reflectivity at 946 nm, and has a diffraction efficiency for a polarized beam of 60%. The cavity length was minimized in order to obtain the shortest possible pulse duration. As M2 mirror, we now use a high-reflectivity mirror for 946 nm instead of the 2% output coupler since the coupling is achieved by the thin film polarizer (output P3 on Fig 1). The output coupling is set to a transmission of 30% for the reasons described above. The passive losses in the laser cavity were about 3% without adding the depolarization losses.

In Q-switched operation we monitored at the same time the average output power, the repetition rate and associated pulse duration. The results are presented in figure 4. The maximum output energy of 1 mJ is obtained for a repetition rate of 7 kHz, with a pulse duration of 23 ns. As the average power starts to decrease severely under 15 kHz, we will operate the laser at 10 kHz, where the average power is 8 W, with a pulse duration of 30 ns (displayed in figure 5). This configuration, while producing a slightly lower energy, provides a better compromise between average power and energy. The beam profile at 946 nm is displayed in figure 6. It has a slightly elliptical Gaussian intensity profile, and we measured its beam quality to be M²x=1.32 and M²y=1.08. We found that the elliptical profile only appears at high pump powers. Therefore, we attributed it to the very strong thermal lens that will make the cavity very sensitive to a slight pump misalignment.

The laser output is stable with a measured fluctuation of 0.4% over one hour, calculated by dividing the standard deviation by the root mean square value (stability plot is displayed in figure 7.

The nonlinear crystal used for the type-I SHG of 946 nm to 473 nm is a 10 mm long BiB3O5 [12] crystal cut at θ=161.6° and φ=90°. For these cut angles, BiBO has a nonlinear coefficient deff =3.34 pm/V, a walk-off angle of 40 mrad for the fundamental beam, and an acceptance angle of 1.28 mrad.cm. The BiBO crystal is kept inside an oven at a stabilized temperature of 50°C. The second harmonic generation is realized in a single-pass, extracavity set-up. In this experiment we try to preserve the fundamental beam quality as much as possible: we use a loose focusing with a beam diameter of 190 µm, which results in a beam divergence of 1.5 mrad, only slightly higher than the BiBO acceptance angle. The resulting blue laser is filtered from the fundamental with a dichroic mirror. The conversion efficiency is displayed in figure 8: a maximum conversion efficiency of 37.5 % is obtained for an input power of 8 W, which results in an output 473 nm power of 3 W and energy of 300 µJ. The dotted line corresponds to the simulation made with the simulation software SNLO . The blue beam profile is displayed in figure 6, and the beam quality is only slightly degraded with a measured beam quality of M²x=1.38 and M²y=1.12. The pulse duration is reduced to 25 ns (see figure 5). The blue beam is stable with a calculated RMS stability of 1.8% (displayed in Fig 7). The increased of the blue power was expected, as SHG is a nonlinear process and dependent on the square of intensity.

In conclusion, we demonstrated a Q-switched 473 nm laser based on Nd:YAG single-crystal fiber. We first studied the polarization of Nd:YAG in CW regime, and extracted more than 10 W of average power for a polarized beam at 946 nm. Then, we used a variable output coupler to operate the laser in Q-switched mode at an energy level more than two times above the previously reported results. Finally, we realized an extracavity SHG in a BiBO crystal, and obtained an average output power of 3 W at 10 kHz, which is two times the previously reported results in both average power and energy. The pulse duration was 25 ns, and the beam quality was measured to be M²x=1.38 and M²y=1.12. Many progress can still be made to increase the oscillator performances. While a straight increase in pump power will cause too severe thermal problems, alternative solutions can be found. In order to lower the thermal lens inside the crystal, pumping directly into the emitting level of Nd:YAG at 885 nm seems a promising solution [13]. the nonlinear frequency conversion could also be improved by using more elaborate set-ups such as double-pass SHG [6] or more efficient nonlinear crystals such as quasi-phase matched crystals [14].

Aknowledgments

Loïc Deyra thanks the DGA for the funding of his PhD This work has been partially funded by the ANR through the program "UV Challenge"

Figure 1 :Figure 2 :

 12 Figure 1: Experimental set-up

Figure 3 :Figure 4 :

 34 Figure 3: Output CW power at 946 nm in function of the halfwave plate induced polarizer transmission for different pump power values

Figure 5 :Figure 6 :

 56 Figure 5: Pulse shape at 946 nm and 473 nm

Figure 7 :Figure 8 :

 78 Figure 7 : Output power stability for 8 W of 946 nm(red) and 3 W of 473 nm (blue)


Loïc Deyra
email: loic.deyra@institutoptique.fr

Xavier Délen

G Mennerat

Phillipe Villeval

François Balembois

Patrick Georges

Nonlinear properties of RTP for second harmonic generation at 1030nm

We demonstrate that around 1030 nm, RbTiOPO4 (RTP) realizes the second harmonic generation of Yb-doped lasers in non-critically phase-matched configuration at ambient temperature, with efficiencies comparable to LBO and wide thermal acceptance.

Frequency doubled lasers are widely used in today's industry, from micromachining to solid state pumping or spectrometry applications. While it is a nonlinear process fully mastered today, with conversion efficiencies routinely reaching more than 60%, it is still a technique sensitive to many outside parameters such as angular misalignments or temperature operation. In order to increase the robustness of a frequency conversion scheme, one needs to reduce the dephasing between the fundamental and the second harmonic cause by either an angle misalignment (angular sensitivity) or a temperature change (thermal sensitivity). RbTiOPO4 (RTP) is an isomorph of KTP, used mainly for high repetition rate electro-optic modulators [START_REF] Albrecht | Study of RTP crystal used as Electro-Optic modulator[END_REF]. Since its discovery, its optical and nonlinear properties have been widely studied : among them, one very interesting feature is its large thermal acceptance centered at 100°C for SHG at 1064 nm [START_REF] Mangin | Thermal expansion, normalized thermo-optic coefficients, and condition for second harmonic generation of a Nd:YAG laser with wide temperature bandwith in RbTiOPO4[END_REF], and simultaneous temperature non-sensitive and angular non-critical SHG at 1030 nm at room-temperature [START_REF] Mennerat | Simultaneous temperature and angular non-critical frequency doubling at 1030 nm in RTP[END_REF]. In addition to its non-linear properties comparable to KTP, it should be a good candidate for highly stable, frequency doubled lasers.

The refractive indexes of RTP have been precisely characterized in previous works [START_REF] Mennerat | Simultaneous temperature and angular non-critical frequency doubling at 1030 nm in RTP[END_REF], and show that type-II non-critical phase matching (NCPM) at room temperature is possible in the y-z plane for a wavelength around 1030 nm. Its deff in this direction has been previously determined to be deff = 2.04 pm/V [START_REF] Oseledchik | Nonlinear optical properties of the flux grown RbTiOPO4 crystal[END_REF]. Usually with standard frequency doubling crystals, the NCPM wavelength can be adjusted with temperature: as an example, the NCPM wavelength of the well-known LBO crystal can easily tuned from 1 µm to 1.25 µm by adjusting the operating temperature from 200°C to -20 °C [START_REF] Ukachi | Measurements of non-critically phase-matched second-harmonic generation in a LBO crystal[END_REF]. For RTP crystals, this is a bit more complicated: as RTP shows very wide thermal acceptance, the NCPM wavelength cannot be tuned with temperature. Thus, in RTP, the laser wavelength needs to be tuned to be exactly at (or slightly above) the NCPM wavelength. To verify the NCPM wavelength of RTP at room temperature, we first used a broadband fluorescence source based on Amplified Spontaneous Emission (ASE) from an Yb-doped fiber amplifier that delivered an output of 5 W at 30 kHz with pulse width of 15ns with a spectrum ranging from 1020 to 1045 nm. This source was focused in the RTP crystal and the green output was monitored with a spectrometer that had a resolution of 0.2 nm (See Figure 1). The random distribution of the ASE power and the low resolution of our spectrometer can't give an exact value of the SHG efficiency or the exact phase matching wavelength, but it is enough to know at what wavelength the fundamental laser should be tuned. The green spectrum was centered on 1031.6 nm.

Having the accurate NCPM wavelength for SHG in RTP, the frequency doubling experimental setup is very straightforward, and is detailed in Figure 2. The fundamental laser is a Master Oscillator Power Amplifier (MOPA) [START_REF] Délen | Hybrid master oscillator power amplifier nanosecond laser source at 257 nm[END_REF] emitting at a wavelength centered at 1031,6 nm by tuning the seed diode wavelength. It emits 4W of average power at a repetition rate of 30 kHz, with pulse width of 15 ns and a diffraction-limited beam profile (M²<1.1). The output spectral linewidth is Fourier-transform limited (Δν=40MHz). Then the laser is focused into the nonlinear crystal at a focal spot size of 120 µm. The nonlinear crystal is a Y-cut RTP of dimensions 4x4x10mm. It is AR coated for 1030 and 515 nm. The AR coatings on the RTP crystal had an unsually low damage threshold so loose focusing had to be used. To show the advantage of a large thermal acceptance, no thermal regulation system is implemented. The performances of the RTP crystal is compared with a 3x3x20 mm3 LBO cut for Type-I SHG at 1030nm, with AR coatings at 1030 and 515 nm, and operated in NCPM configuration at 188°C. As shown in Figure 3, in the same focusing conditions both crystals yield comparable results: we obtained 1.4W of green average power with LBO and 1.35W with the RTP crystal. At the highest pump power we were very close of the damage threshold of the RTP AR coating (evaluated at about 2.5 J/cm²), so we lowered the output green power to about 1.2 W. After operating the RTP for 1h, no grey-tracking damages were observed and the output power was very stable with calculated RMS fluctuations of 2%. (Figure 4) We then characterized the two interesting properties of SHG in the RTP crystal, namely the NCPM behavior and wide thermal acceptance. Inset in Fig. 2a is the angular acceptance measurement, done by detuning the crystal until significant power drop was observed.

An acceptance angle of about 300 mrad was measured, showing the NCPM behavior (by comparison, the angular tolerance for a standard KTP in critical phase matching and with the same length is about 8 mrad) (See Figure 5).

The thermal acceptance was measured by heating the crystal from 20 to 90°C: the power dropped to half of its initial value when the temperature was raised by 60°C, leading to an estimated thermal acceptance of 120°C. (Fig 6) A more detailed characterization of the thermal acceptance can be found in [START_REF] Mennerat | Simultaneous temperature and angular non-critical frequency doubling at 1030 nm in RTP[END_REF]. In summary, we designed a frequency doubled 515 nm source based on a 1030 nm ns pulsed Yb-based laser and a 10 mm RTP crystal. At 1030 nm, RTP combines for SHG a large deff, NCPM behavior at room temperature and wide thermal acceptance, a combination never demonstrated before to the best of our knowledge. For these reasons, it will be an excellent solution for applications requiring a simple and robust frequency doubling scheme such as intracavity frequency doubling or military laser systems.

Loïc Deyra acknowledge the DGA for the funding of his PhD

Figure 1 :

 1 Figure 1: Frequency doubled wavelength by YZ-cut RTP at room temperature and normal incidence

Figure 2 :

 2 Figure 2: Experimental setup

Figure 3 :Figure 4 :

 34 Figure 3: Output green average power with LBO (green squares) and RTP (red circles). Inset beam profile and measured beam quality with RTP

Figure 5 :

 5 Figure 5: Angular acceptance of SHG in RTP crystal Figure 6: Thermal acceptance of SHG in RTP crystal


Michel Gondran
email: michel.gondran@polytechnique.org

Alexandre Gondran
email: alexandre.gondran@enac.fr

Energy ow lines and the spot of Poisson-Arago

We show how energy ow lines answer the question about diraction phenomena presented in 1818 by the French Academy: "deduce by mathematical induction, the movements of the rays during their crossing near the bodies". This provides a complementary answer to Fresnel's wave theory of light. A numerical simulation of these energy ow lines proves that they can reach the bright spot of Poisson-Arago in the shadow center of a circular opaque disc. For a monochromatic wave in vacuum, these energy ow lines correspond to the diracted rays of Newton's Opticks. 1

INTRODUCTION

The answer Fresnel provided in 1818 in response to the French Academy's competition marks the beginning of the refutation of Newton's corpuscular theory of light and the rehabilitation of the Huygens wave theory. The competition topic was presented as follows: "...diraction phenomena have been a subject of research for many physicists...but research has not yet suciently determined the movement of the rays near the body where the change occurs...it important...to further study...the physical manner in which rays are inected and separated into dierent bands ... As a result the Academy is proposing this research...to be presented as follows: 1. Determine all the eects of ray diraction ...direct and reected when they ... pass near the extremities of a body... 2. Deduce from these experiments, by mathematical induction, the movements of the rays during their crossing near the bodies." Fresnel's essay develops a mathematical wave theory which seems be in conict with the corpuscular theory. It describes an impressive number of diraction experiments all explained by the same principle: the fringes are due to interference waves issued by each of the screen points. Fresnel's principle generalizes Huygens' principle.

Poissson carefully studied Fresnel's theory and deduced "that the center of the shadow of a circular opaque disc... (should)... be as enlightened as if the disc didn't exist" 1 ; this bright spot of light at the center of the shadow, he claimed, "violated common sense" and hence refuted Fresnel's wave theory. However, Arago almost immediatly veried the existence of the spot experimentally. Fresnel won the competition and this discovery marks the beginning of the acceptance of wave theory and the refutation of corpuscular theory. This spot of light, today known as Poisson's bright spot or spot of Arago, had been observed a century earlier (1723) by Maraldi, who had not published his work. 3,4 This paper proposes to complete Fresnel's answer and to show how the energy ow lines are (in the special case of a monochomatic wave in vacuum) the answer to the French Academy's question about diraction phenomena:"deduce by mathematical induction, the movements of the rays during their crossing near the bodies". We study, by a numerical simulation, the case of diraction by a circular aperture as well as diraction by a circular opaque disc in order to nd the spot of Poisson-Arago. In Section 2, we recall how to calculate bright densities with wave theory. In Section 3, we show that the energy ow lines can correspond to Newtonian rays. Then in section 4, we discuss the interpretation of these energy ow lines.

II. INTENSITY DISTRIBUTIONS

Let us consider a monochromatic plane wave of light whose wavefronts lie in the plane of a circular aperture (resp. an opaque disc) placed in the xy plane, and a detector in a parallel plane at distance z. Set (x M , y M ) the coordinates of a point M in the diracting plane and (x, y) the coordinates of the observation point P on the detector.

First, if we neglect the polarisation of light and if we suppose that the incident wave is of the form A 0 e ikz on the aperture, the amplitude A(P ) for z > 0, which veries the Helmholtz equation, is given by the Rayleigh-Sommerfeld formula: 5,6

A(P ) = - iA 0 λ ∫ S e ikr r (1 - 1 ikr ) cos θdx M dy M (1) where r = √ (x -x M ) 2 + (y -y M ) 2 + z 2 , cos θ = z r , k = 2π
λ and where the integration is taken on the surface S of the aperture. Notice that the formula gives a more exact solution, in particular for a very small distance of the aperture thanks to -1/ikr; see Fig. 2 of Gillen and Guha. 7 A.

Intensity distributions for a circular aperture Far from the aperture, the classical Fraunhofer diraction appears, and the emerging beam has a well-dened angular dispersion, in the order of ∆θ ∼ λ R ; this is the Airy disc. Near the aperture, we get the Fresnel diraction. We note a succession of bright and dark areas on the axis.

Because the incident wave is a plane wave, the intensity value is calculated by the Babinet's principle 8,9 in taking the square of

A(P ) = A 0 (e ikz + i λ ∫ S e ikr r (1 - 1 ikr ) cos θdx M dy M );
(2) numerical integration is taken on the surface of the opaque disc S. We see clearly the area corresponding to the geometric shadow, but also the bright spot of Poisson-Arago in the center of the shadow; with the choice of λ = R 10 , this bright spot assumes a great importance.

Newton, who carried out the experiment of the opaque disk using a coin, does not report the presence of fringes within the shadow in his Opticks. 2 With λ = 0.5 µm and a coin (R = 1 cm), it is dicult to see the bright spot of Arago on a detector placed at 5 m [see Fig. 4]. The radius of the spot is 0.1 mm and just visible to the naked eye. This oversight was to have an unfortunate consequence a century later!

III. ENERGY FLOW LINES FOR A MONOCHROMATIC WAVE

Let us consider a monochromatic electromagnetic eld {-(H, t), *(H, t)} which is the real part of the complex monochromatic electromagnetic eld {E(H, t) = -0 (H)e -iωt , B(H, t) = * 0 (H)e -iωt }.

The Poynting vector 5 = 1 µ 0 -× * is the instantaneous rate of energy ow per unit area at a point

; u = 1 2 (ǫ 0 -2 + 1 µ 0 * 2 )
is the instantaneous electromagnetic energy density.

Since the optical frequencies are very large (ω is of order 10 15 s -1 ), one cannot observe the instantaneous values of any of the rapidly oscillating quantities, but only their time average taken over a time interval which is large compared to the fundamental period T = 2π ω . 11,12 The energy ow, which is interpreted as the time-averaged Poynting vector, 11,12 is determined from the real part of the complex Poynting vector S = 1 µ 0 E × B * and from the energy density of the complex eld U = 1 2 (ε 0 EE * + 1 µ 0 BB * ). The time-averaged ux of energy and the time-averaged energy density are given by 5

= 1 2µ 0 Re[-0 × * * 0 ], u = 1 4 (ε 0 -0 - * 0 + 1 µ 0 * 0 * * 0 ).
(3)

The energy ow lines are obtained by the equation

dH dt = 5 u .
(4)

For diraction problems, these energy ow lines have been discussed at length. In 1952, they were calculated numerically for two-dimensional diraction on a half-plane by Braunbeck and Laukien 10 and recalled in Born and Wolf's textbook 11 p. 575-577. In 1976, Prosser 13 proposed an interpretation of diraction and interference with electromagnetic elds in terms of energy ow lines. These lines are recently demonstrated and discussed in distributions of incoherent light for various two-dimensional situations by Wünscher et al. " Their interpretation will be discussed in the following section.

To solve the circular aperture and opaque disc problems, we look for an electromagnetic eld (-0 , * 0 ), written in cylindrical coordinates (ρ, ϕ, z), in the form -0 = {e ρ , 0, e z } and * 0 = {0, b ϕ , 0}, where e ρ , e z and b ϕ are functions of (ρ, z).

From Maxwell equation

curlB = ε 0 µ 0 ∂E ∂t we deduce -0 = ic k [-∂ ∂z b ϕ , 0, ∂ ∂ρ b ϕ ] and 5 = 1 2µ 0 cλ 2π Im(b * ϕ ∇b ϕ ).
(5) From Faraday's law curlE = -∂B ∂t , we show that b ϕ veries the Helmholz equation

∆b ϕ (ρ, z) + k 2 b ϕ (ρ, z) = 0. ( 6
)
If we take as boundary conditions b ϕ = A 0 e ikz on the aperture, then the solution found

(-0 , * 0 ) is solution of Maxwell equations.
Taking into account the time-averaged energy conservation law ∇ • 5 = 0, we deduce We notice that after a disturbance in the Fresnel zone, lines gradually become straight in the Fraunhofer area, in agreement with the diracted rays proposed by Newton in Figure 1 for a circular aperture.

ε 0 -0 - * 0 = 1 µ 0 * 0 * *

B. Spot of Poisson-Arago and energy ow lines for a circular opaque disk

Figure 6 shows energy ow lines where the initial positions are drawn at random outside the circular opaque disk.

0 10 R 20 R 30 R -3 R -2 R -R 0 R 2 R 3 R
FIG. 6: Energy ow lines behind the circular opaque disk.

Using the presence of energy ow lines behind the opaque disc, we propose an explanation of the bright spots of Poisson-Arago in the next section.

C. Energy ow lines for Young's double slit experiment We now complete these numerical simulations by determining the energy ow lines for the Young's double slit experiment. Carried out in 1802 by Thomas Young, some years before Fresnel's theory, this well-know experiment is the rst that clearly demonstrates the wave nature of light. # Let us consider a monochromatic plane wave of light (λ = 0.5 µm) propagating perpendicular to two slits placed in the xy plane, and a detector in a parallel plan at distance z.

The slits have a width d = 5 µm along x and innitely long along y; 2 d is the distance between slits, center to center.

The electromagnetic eld (-0 , * 0 ) is function of (x, z) and can be written *

0 = [0, 0, b z ], -0 = ic k [ ∂ ∂y b z , -∂ ∂x b z , 0].
The energy ow lines after the slits are then dened by

dH dt = cλ 2π
Im(b * z ∇b z ) b z b * z . ( 8
)
If the incident wave b z is of the form A 0 e ikz on the slits, b z is, after the slits, given by the Fresnel-Kirchho solution:

b z (P ) = A 0 √ λz e -i π 4 e ikz ∫ S e ik(x-x M ) 2 2z dx M (9)
where the integration is taken on the length S of the two slits.

Figure 7 shows 20 energy ow lines where the initial position are drawn at random in the two slits. IV.

INTERPRETATION OF MONOCHROMATIC ENERGY FLOW LINES

For monochromatic waves in the vacuum, we can establish a correspondence between these energy ow lines and the diracted rays proposed by Newton in Principia $ (1687):"Moreover, the rays of light that are in our air (as lately was discovered by Grimaldi, by the admission of light into a dark room through a small hole, which I have also tried) in their passage near the angles of bodies, whether transparent or opaque (such as the circular and rectangular edges of gold, silver and brass coins, or of knives, or broken pieces of stone or glass), are bent or inected round those bodies as if they were attracted to them."

Can these energy ow lines be interpreted as light rays?

When light is incoherent, we must reject this interpretation as recalled by Wünscher et al. " Indeed, when the light is not monochromatic, it must be regarded as a mixture of monochromatic waves as Newton showed in his famous experiments of light decomposition. Each monochromatic wave of white light gives energy ow lines which depend on its

wavelength.

The answer is more complex for a monochromatic wave. These energy ow lines are a generalization of the rays of the geometrical optics. Indeed, if we increase the frequency of the light wave towards innity, the energy ow lines converge towards the straight rays of the geometrical optics. This is shown in Fig. 8

  This announcement was made by a jury of great scientists: Pierre-Simon Laplace, Jean B. Biot, Simeon D. Poisson, Joseph L. Gay-Lussac -all Newtonians -as well as Dominique F. Arago, who was the only one who believed in wave theory.

FIG. 1 :

 1 FIG. 1: Newton's rays diracted by a hair and a small circular aperture (1704).

Figure 2

 2 Figure2shows, in a plane (z,x) containing the optical axis, intensity behind a circular aperture with a radius R = 5µm of a monochromatic plane wave of light with wavelength λ = R 10 = 0.5µm.

FIG. 2 :

 2 FIG. 2: Calculated intensity distributions behind the circular aperture in the plane (z, x).

FIG. 3 :

 3 FIG. 3: Calculated intensity distributions behind the circular opaque disk in the plane (z, x).

Figure 3

 3 Figure 3 represents the intensity behind an opaque disc of radius R = 5µm of a monochromatic wave of light with wavelength λ = R 10 = 0.5 µm.

  FIG. 4: Spot of Poisson-Arago: Intensity distribution behind a coin (R = 1 cm) on a detector placed at 5m.

0

  Figure5shows 26 energy ow lines where the initial positions are drawn at random in the circular aperture.

  FIG.7: 20 energy ow lines behind the two slits.

  FIG. 8: Evolution of the energy ow lines as the frequency increases: λ = 0.5 µm; λ = 50 nm; λ = 5 nm.

Am. J. Phys., Vol. 78, No. 6, June 2010 http://aapt.org/ajp


Z J Ajaltouni
email: ziad@clermont.in2p3.fr

Symmetry and Relativity : From Classical Mechanics to Modern Particle Physics

Keywords: Symmetry, Relativity, Classical Physics, Particle Physics

The aim of this review is highlighting the common aspects between Symmetry in Physics and the Relativity Theory, particularly Special Relativity. After a brief historical introduction, emphasis is put on the physical foundations of Relativity Theory and its essential role in the clarification of many issues related to fundamental symmetries. Several examples from classical mechanics to modern particle physics will be given and some open questions will be raised.

Introduction

Physical sciences rely essentially on two pillars : experimental facts, on one side, and their translation into a coherent mathematical formalism on another side. In both the two approaches, a physical insight is used, symmetry. Does the physical process under analysis present symmetrical properties, or does its formulation contain some symmetry elements ? This kind of intellectual attitude has been adopted in Physical Sciences since the famous paper of P.Curie written at the end of the 19 th century in which the author stresses on symmetry properties of some ectromagnetic phenomena, notably those ones occurring in crystal bodies [1]. Quoting H.Weyl's sentence summarized by :"Symmetry is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection..." ([2]), one can add that symmetry represents a methodology followed by Modern Physics in order to build coherent and successful models whose aim is understanding the fundamental physical laws at all scales, from the microscopic world to the macroscopic Universe.

Let us come to Relativity Theory and how it is related to Symmetry in Physics. Firstly, one has to remind that Relativity is born from the unsufficiencies of Classical Physics, both Newton Mechanics and Maxwell Electromagnetism, in which are postulated the absolute character of time and the existence of an hypothetic medium, the aether, which fills the vacuum and serves as a support for the propagation of electromagnetic waves, among them the visible light. After the publication of A.Einstein historical paper [START_REF] Einstein | On the Electrodynamics of Moving Bodies[END_REF], time like space components becomes an ordinary component of a four-dimensionnal structure, the Minkowski spacetime manifold. Thus, a complete symmetry arises between space and time, and this new feature leads to important physical consequences as it will be shown in the next sections. This review is organized around three main topics :

• Contribution of Relativity Theory to the different symmetries in Classical Physics (section 2).

• Relativistic Symmetries in Quantum Mechanics (section 3).

• Role of the Relativity Theory in the discovery of new symmetries (and asymmetries) in Elementary Particle Physics (section 4). The conclusion will be devoted to some open issues in the common field of Relativity and Symmetry.

2

Relativity and Symmetries in Classical Physics

Classical Mechanics

One of the main difficulties which appeared in Classical Mechanics was its unconsistency with the electromagnetic phenomena, notably the problem of magnetic induction in a moving circuit. This unconsistency has been amplified by the negative result of the Michelson-Morley experiment whose aim was measuring the translation velocity of the Earth in the aether. In order to cure these problems, A.Einstein succeeded in establishing new basis for Mechanics by setting three new postulates : ⋆ Time is a relative notion and each observer possesses its own time.

⋆ The speed of light is isotropic and does not depend on the state of motion of the observer. ⋆ (R ′ ) being an inertial frame moving with a constant velocity v according to another frame (R) , the coordinate transformation between these two inertial frames is given by the Lorentz formula [START_REF] Lorentz | Electromagnetic Phenomena[END_REF].

A straightforward consequence of these new principles was the fusion of space and time in one entity called "Space-Time", which was performed by H. Minkowski ([5]). In this new 4-space, time and space components are completely symmetric, as it is shown by the following series of equalities relating space-time components of the same event between the two frames (R) and (R ′ ) :

x = x ′ , y = y ′ , z = γ(z ′ + vt ′ ) , t = γ(t ′ + vz ′ /c 2 ) (I) and x ′ = x , y ′ = y , z ′ = γ(z -vt) , t ′ = γ(t -vz/c 2 ) (II) where γ = 1 √ 1-v 2 /c
2 is the Lorentz factor, and the signappears in equations (II) because the velocity of (R) according to (R ′ ) is -v.

Physical interpretation of these relations becomes more obvious by adopting the point of view of Special Relativity : Analysis of physical phenomena does not depend on the motion of the observer(s). Observer O in frame (R) and observer O' in frame (R ′ ) have completely a symmetric role . It is worth stressing that Einstein's postulates and Minkowski's 4-space lead to a new physical concept, the Relativistic Invariance one, which have important consequences in the foundations of Modern Physics. Thanks to a new mathematical entity, the 4-vector introduced by Minkowski, one can set many fundamental invariants, among them : ⋆ The square of the space-time interval, ds2 = dx µ dx µ = c 2 dt 2 -dx 2 -dy 2 -dz 2 , from which one can deduce automatically the relation of clock retardation [START_REF] Landau-Lifschitz | The Classical Theory of Fields[END_REF]. ⋆ The square of the energy-momentum 4-vector,

P µ P µ = E 2 /c 2 -p 2 = m 2 c 2 .
Applying this last relation to a massive particle at rest ( p = 0), the Einstein breakthrough formula emerges, E (0) = mc 2 , which represents a new symmetry, the equivalence between Energy and Mass [START_REF] Einstein | Does the Inertia of a body depend upon its Energy content[END_REF] ; symmetry which does not exist in Newton Mechanics.

Classical Electrodynamics

In his seminal paper, A.Einstein was motivated by showing that the two standard views of the induction phenomenon (Lenz-Faraday law) were really the same one. Departing from the Relativity principles, he achieved merging the two different processes and he found again the standard results of the Maxwell's equations : any variable electric field E generates a magnetic field B, and reciprocally. So, a complete symmetry is established between the electric and magnetic fields, despite the absence (temporary ?) of magnetic charges. This physical property is clearly reflected in the relativistic formalism with the introduction of a new entity, the electromagnetic tensor

F µν = ∂ µ A ν -∂ ν A µ 2
, which is an antisymmetric tensor whose six components are respectively those of the electric E and magnetic B fields. Doing some calculations with this new tensor like the following,

∂ α F βγ + ∂ γ F αβ + ∂ β F γα = 0
(indices α, β, γ are varying fro 0 to 3), allows to find again the standard Maxwell-Faraday equations :

-→ rot E = -∂ B/∂t and div B = 0.

Summary

At the classical level, the Relativity Theory introduces new aspects of symmetry in fields where symmetry was not conjectured : Space and Time , Electric and Magnetic fields and the Mass-Energy equivalence. Furthermore, its mathematical formulation includes, in a simple and direct way, the symmetry properties of the physical processes [START_REF] Einstein | The Meaning of Relativity[END_REF].

3 Relativity and the Symmetry Properties of Quantum Mechanics

We will deal succesively with non-relativistic Quantum Mechanics (NRQM), relativistic Wave Mechanics of De Broglie and the relativistic approach to Quantum Mechanics initiated by P.Dirac. By following the historical steps, emphasis is put on the basic physical ideas and the main experimental discoveries in each field. Technical calculations will be avoided.

Symmetry in Non-Relativistic Quantum Mechanics

The importance of Symmetry in NRQM leads to a wide use of Group Theory methods when describing quantum systems, from the simple hydrogen atom to complicated molecular structure [START_REF] Wigner | Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra[END_REF]. This feature is related to the mathematical structure of Quantum Mechanics itself : any physical quantity which can be measured, or observable, is represented by an hermitean operator A (A = A † ) acting on a Hilbert space built from state vectors or kets; the last ones represent the physical states of the system and form an algebraic vector space [START_REF] Dirac | The Principles of Quantum Mechanics[END_REF].

How ordinary QM has been made relativistic ? Before answering this question, it is fair to recall how Special Relativity has been firstly introduced in Quantum Physics.

De Broglie Relativistic Wave Mechanics

In his doctoral thesis (Paris, 1924), the french physicist L.de Broglie generalized the important notion of wave-particle duality (first introduced by A.Einstein for the Photon in 1905 [START_REF] Einstein | Point de vue heuristique concernant la production et la transformation de la Lumière[END_REF]) by extending it to massive relativistic particles [START_REF] Broglie | Recherches sur la Théorie des Quantas[END_REF]. Departing from the hypothesis that each material corpuscle has an internal vibration whose frequency is given by the Planck-Einstein relation, hν = γmc 2 , and supposing that this vibration is in phase with a "phase wave" propagating at the speed V = c 2 /v ( 3 ), De Broglie inferred the expression of the particle wave-length :

λ = V ν = h γmv = h p
which is identical to the photon wave-length given by the Planck-Einstein relation, E = pc = hν =⇒ λ = c/ν = h/p . This famous relation, which has been confirmed by the experimental discovery of the electron diffraction (Davisson and Germer, 1927), links together the two main aspects of matter, wave and/or particle. Thus, a new symmetry at the level of the elementary particles has been revealed, as a consequence of the Einstein-De Broglie relations. Then, the wave nature of any quantum system has been exploited very deeply to set fundamental equations : the non-relativistic Schrodinger equation [START_REF] Messiah | Mécanique Quantique. Tome I[END_REF] , and the relativistic Klein-Gordon one [START_REF] Messiah | Mécanique Quantique. Tome II[END_REF].

Dirac Equation and its Consequences

In the years 1927-1928, the british physicist P.Dirac succeeded in formulating a relativistic wave equation for the electron characterized both by its simplicity and its ingenuity [START_REF] Dirac | Quantum Theory of the Electron I and II[END_REF].

γ µ p µ Ψ = mcΨ
where (p µ ) is the 4-momentum of the particle and γ µ are the Dirac matrices with µ = 0, 1, 2, 3.

The Dirac equation has a linear evolution according to time, like the Schrodinger equation and, furthermore, it reveals a perfect similarity among space and time components. Thus, it has the great advantage to be relativistic invariant. Applied to the known massive particles (at this epoch there were only electrons and protons), Dirac equation leads to fascinating results : ⋆ The value of the spin, s = 1/2 , is deduced automatically with a gyromagnetic factor of the electron, g e = 2, which agrees with the experimental measurements. ⋆ There appear negative-energy states for the electron, E = -p 2 + m 2 , as with the Klein-Gordon equation. But, according to the fermionic nature of the electron, these states are interpreted as an electron with positive electric charge, or an anti-electron called also a positron [START_REF] Dirac | Théorie du Positron[END_REF]. This particle has been discovered by Anderson in 1933 [START_REF] Anderson | The Positive Electron[END_REF], which marks the beginning of the Antimatter Era.

• Thanks to the relativistic Dirac equation a new symmetry is born :

Matter ⇆ Antimatter and it confirmed the prediction of Dirac that "every particle has its own antiparticle,whatever its spin is " . Then, there were the discoveries of the antiproton, the anti-muon, the antineutron and the other ones [START_REF] Cahn | The Experimental Foundations of Particle Physics[END_REF].

• What is really fundamental in this new kind of symmetry is that it firstly appeared in the world of elementary particles, a field governed by the rules of Quantum Mechanics, and not in the macroscopic world where Classical Physics dominates. It is worth saying too that taking account of the importance of the Relativity principles in both the two approaches to Quantum Physics, De Broglie's Wave Mechanics or Standard (Dirac) Quantum Mechanics, have paved the way to the development of a new field of research, the physics of Elementary Particles.

Modern Particle Physics

At the same period than the Dirac equation, Quantum Field Theory (QFT) was developed on relativistic principles [START_REF]Selected Papers on Quantum Electrodynamics[END_REF]. It was essentially devoted to the interactions between quantized electromagnetic fields and charged particles, which is called Quantum Electrodynamics (QED). The formal language of QFT uses intensively Relativity Theory; its exposition lies outside the scope of this review. However, QFT is the best framework for describing interactions among elementary particles; this research field being characterized by new symmetry properties and, sometimes, by asymmetry ones.

Elementary Particles and Fundamental Symmetries

As shown in the preceding sections, Relativity Theory introduces fundamental tools for apprehending new physical processes. These are essentially : ⋆ Relativistic Conservation Laws, like the conservations of the total energy-momentum (including the mass) and of the total angular momentum (including the spin of the particles). ⋆ Relativistic Covariance. It is a new principle which stipulates that the description of any quantum process must have the same analytical form, independently of the reference frame in which the process is observed; the best example being the computation of the transition probabilities between different quantum states.

• According to these new principles, E.Wigner wrote a paper devoted to the representations of the Lorentz Group [START_REF] Wigner | On Unitary Representations of the Inhomogenuous Lorentz Group[END_REF] which constitutes a new breakthrough in Relativistic Quantum Physics (including QFT). He applied techniques of Group Theory to the Hilbert space describing a quantum system obeying Relativity principles; and he succeeded to derive two Relativistic Invariants (RI), independently of the internal dynamics of the physical system. The two RI are respectively the mass of the quantum system (already found since the introduction of the Minkowski 4-space) and its intrinsic angular momentum or its quantized spin. An immediate consequence of these RI is related to the massless particles which move at the speed of light c like photons, gravitons and even neutrinos if their masses are neglected : the projection of the spin of a massless particle along its momentum or helicity, λ = s • p/| p|, has only two values, λ = ±s for s = 1/2, 1, 3/2, .... It could be seen that the massless quantum state with λ = -s is the mirror image of the other state with λ = +s and vice-versa. This intuitive symmetry will be tested experimentally and important features will be deduced.

• It is quite remarkable that Wigner's achievement completes the seminal work of Dirac : by means of the relativistic formulation of the quantum principles, the intrinsic angular momentum of a particle, or its spin, emerges directly. Spin is an essential ingredient for describing atoms, nuclei and particles; and it is worth noticing that, thanks to the existence of the spin, Pauli formulated his famous Exclusion Principle which explains the stability of atoms and molecules.

• Other important symmetries in Subatomic Physics are the Discrete Symmetries whose experimental tests are still a challenge for Nuclear and Particle Physics. These are : (i) Charge Conjugation, C, which transforms any particle with charge q into its antiparticle with charge -q by keeping its mass and its spin unchanged (it is a consequence of the relavistic Dirac equation). (ii) Parity, P, (or mirror symmetry) which transforms the vectors r , p , and spin s into -r , -p , s. (iii) Time Reversal, T , which transforms r, p, and spin s into r, -p, -s , and exchanges both initial and final states. It is usually argued that a quantum system is invariant by one of these three symmetries, O D = C, P, or T , if its hamiltonian H, which governs its evolution in time, keeps the same form after applying O D on the physical system. Mathematically, it leads to

O D HO D -1 = H =⇒ commutator [O D , H] = 0.
The two symmetries, P and T, are already used in NRQM, especially for establishing selection rules [21]; but their relativistic aspect becomes more crucial in Particle Physics which deals with energetic particles like electrons or neutrinos whose velocities are often very close to the speed of light c.

• In this context, sophisticated experiments have been performed in order to test the validity of these symmetries, as suggested by theorists T.D.Lee and C.N.Yang in 1956 in order to solve the " (θ, τ ) puzzle" [22]. An historical experiment analyzing electron angular distributions from nuclear β decay [23], followed by another one studying weak decay of the muon, µ -→ e -νe ν µ [24], proved in an undoubtful way that both Parity and Charge Conjugation are violated in weak interactions : whatever the initial decaying particle is, the neutrino is a left-handed particle, while the anti-neutrino is a right-handed one ( 4). But the combined operation, CP or PC, is still a good symmetry [START_REF] Landau | Conservation Laws in Weak Interactions[END_REF]. This hypothesis was true until the experimental discovery of Cronin et al (1964) which shown the breakdown of CP symmetry in the neutral meson system, (K 0 K0 ), made out by a pair of particle-antiparticle evolving differently with time [26].

• After the failure of the Discrete Symmetries, a main question arose [START_REF] Wigner | Symmetry and Conservation Laws[END_REF] : Discrete symmetries being only approximate, is there any fundamental law which could relate a particle or a system of particles to their corresponding antiparticles ?

The answer to this crucial question comes again from Relativistic QFT, thanks to the CPT theorem [START_REF]Several authors have postulated the CPT theorem by similar approaches. The pioneering one was the paper of G.Lüders[END_REF] which states the following : "Assuming Lorentz Invariance of the interactions among particles, physical laws are invariant by the product of the three discrete symmetries C, P and T taken in any order : CPT, TPC, TCP, ...". This theorem concerns any local interaction described in the Minkowski 4-space, like Strong, Electromagnetic and Weak interactions, and does not depend on the fact that any discrete symmetry is conserved or violated. It leads to important physical consequences which are still valid on the experimental side : ⋆ The equality of mass of any particle (X) with its antiparticles ( X) [29]. ⋆ The total lifetimes of X and X are identical. ⋆ The intrinsic magnetic moments of X and X are opposite in sign but they have the same absolute value. ⋆ Inverse reaction with antiparticles :

a + b → c + d (1) and c + d → ā + b (2)
Reaction (2) has the same probability of occurrence than reaction (1), even if symmetry CP or T is violated.

Quarks, Leptons and Gauge Fields

Gauge field theories and their applications to Particle Physics represent today the heart of Modern Physics and of all the activities turning around them, both theoretical and experimental. It requires a huge litterature to describe them and, in this paragraph, emphasis will be put on the basic principles and the original ideas which guided their development.

• According to their physical properties (mass, charge, spin) and to their mutual interactions, particles are classified into specific families called multiplets. An evident example is the proton-neutron system which forms a doublet with regard to the strong interaction. Thus, it is said that the relativistic proton-neutron interaction is invariant by the symmetry group SU(2).

After the discovery of many particle families, new quantum numbers arose like strangeness, charm, beauty ..., and their classification become an urgent problem for Theoretical Physics. Despite their differences, particles present several common aspects and notably symmetry properties which lead to the use of Group Theory techniques. Finally, two important families emerged : quarks and leptons, and their interactions are described by gauge fields.

Gauge Field

A gauge field is any field (classical or quantized) which remains invariant by a gauge transformation, like Classical Electrodynamics or QED. Let us consider the electromagnetic 4-potential A = (A µ ) = (V /c, A). If each component A µ is transformed like : A µ → A ′µ = A µ + ∂ µ f , f being any continuous function of (x), the electromagnetic tensor F µν (defined in sect.2) will remain unchanged. This simple transformation (abelian in the language of Group Theory) can be generalized to other fields, like nuclear field or quark field, by taking account of the "internal symmetry" which characterizes particles belonging to the same multiplet [30]. In this last case, the gauge transformation is called a non-abelian one.

Quarks

Quarks [START_REF] Gellmann | Schematic Model of Baryons and Mesons[END_REF] are the fundamental constituents of the Hadrons, heavy particles like Mesons and Baryons which are produced by strong interactions. When interacting among themselves, quarks exchange gluons (massless partcles like the photon) and this kind of interaction requires a new quantum number called color. The color is described by a relativistic gauge field theory which is similar to QED (describing electron and photon interactions). It is called Quantum Chromodynamics (QCD) [32].

All the quark properties are deduced from symmetry principles underlying the physical properties of the hadrons [START_REF] Lichtenberg | Unitary Symmetry and Elementary particles[END_REF]. Their charges are fractions of the elementary electric charge e, and they can be classified into two groups, light quarks and heavy ones. They are respectively : u(2/3), d(-1/3), s(-1/3), c(2/3), b(-1/3), t(2/3) . Number inside parenthesis indicates the charge of the quark in unit of e; the three first quarks, u, d and s , are the light ones while the three last ones are the heavy quarks.

Quarks have another phyical property :they undergo electromagnetic and weak interactions like the hadrons in which they are confined. Thus, in the Electroweak Model of Glashow-Weinberg-Salam [START_REF] Weinberg | Summary of the Standard Electro-Weak Model[END_REF], quarks are classified into doublets, each doublet having an Up quark with charge +2/3 and a Down quark with charge -1/3.

Leptons

The word Lepton means initially a light particle, like the electron whose mass is approximately 1856 smaller than the proton mass. The neutrino, which is simultaneously produced with the electron in β decay, is a lepton too. Then, after the discovery of the muon, µ ± , which has the same properties than the electron except its mass (the muon is 210 heavier than the electron), "lepton" usually indicates particles which are not sensitive to the strong interaction. Leptons are sensitive only to electromagnetic and weak interactions; and each lepton has its own quantum number called the "leptonic number". At this level, particle physicists noticed a stringent similarity between Leptons and Quarks in the electroweak domain; it was the birth of the Standard Model which classifies all the fundamental particles into doublets according to their electroweak interactions [START_REF]For a clear and concise introduction to the Standard Model, one can consult the excellent book of C.Quigg[END_REF].

The discovery of the third charged (and heavy !)lepton, the τ ± , with its associated neutrino the ν τ [36], followed by those of the quarks b (beauty) and t (top), confirmed in a brilliant manner the exactness and the accuracy of the Standard Model (SM).

Needless to say that the SM is now a true theory. All its predictions have been verified experimentally : aside heavy quarks cited above, there were the discovery of the massive intermediate vector bosons, W ± , Z 0 (CERN 1983) and the one, more spectacular, of the Higgs boson (CERN, 2012-2013) whose field generates the mass for the fundamental constituents of matter.

Conclusion

Attempt has been made to give a large survey on the importance and intrication of Relativity Theory with Symmetry in the physical processes. From Classical to Modern Physics especially Particle Physics, numerous aspects of symmetry would not be understood even discovered without the principles of (Special) Relativity. Thanks to the wedding of Relativity and Quantum Mechanics and the inclusion of new symmetry principles, the resulting quantum fields get a predictive power like the Standard Model one. However, several fundamental problems remain unsolved : Do magnetic monopoles exist in Nature ? Their absence still indicates a dissymmetry between Electricity and Magnetism. Could modern Theoretical Physics realize a complete synthesis between General Relativity and Quantum Mechanics, as Dirac achieved eighty years ago ? At a deep level of Quantum Mechanics, is there a real symmetry, or Supersymmetry, between Bosons and Fermions ??

All these open problems require a large use of symmetry principles including sophisticated methods from Group Theory; which is nowadays visible with the impressive development of Mathematical Physics. On another side, experimental physics could answer and eventually solve all these passionating problems.

(A µ ) is the four-vector potential given by (V /c, A) and ∂ µ is the partial derivation according to the component x µ

v is the velocity of the massive particle which is identified with the group velocity of the wave, while V is the phase velocity which could be greater than c , h is the Planck constant and γ = 1/ 1 -v 2 /c 2

Recalling that helicity is given by λ = s• p p = ±1/2 , only neutrinos with λ = -1/2 called left neutrino ν L exists, while only anti-neutrinos with λ = +1/2 called right anti-neutrino νR exists in Nature. The transformation allowing the transition from ν L to νR is the product of C and P, or CP Symmetry.

Acknowledgments

The author would like to thank many colleagues and friends for stimulating discussions about symmetries in Physics. He is particularly very grateful to V.Morenas whose comments and remarks helped him in clarifying difficult problems.