Andrzej Zywocinski

Katarzyna Pawlak

Robert Holyst

Patrick Oswald

Andrzej Żywociński^

Katarzyna Pawlak^

Robert Hołyst^&

Chirality-Biased Point Defects Dynamics on a Disclination Line in a Nematic Liquid Crystal

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I. Introduction

Disclination lines are very common defects in nematic liquid crystals. They usually form a characteristic "thread-like texture", which is easily observable in the microscope when a droplet of nematic phase is deposited on a glass plate or introduced by capillarity between two glass plates. Unfortunately, this texture is too complicated to study the fine structure of defects as they are strongly curved and strongly interact with each other. In particular, their molecular configuration is difficult to determine in those conditions. For that reason, physicists looked for more controlled configurations. One classical way to force the formation of a single defect is to confine the nematic in a cylindrical capillary with homeotropic anchoring on internal wall [1][2][3][4] . In that way, the molecules orient perpendicularly to the surface of the capillary which leads to the formation of a wedge disclination line 5 of topological strength +1. A similar method with flat capillary tubes was used subsequently to control the formation of +1/2 disclination lines. 6 All these experiments together with general topological considerations 7 showed that defects of integer strengths are topologically unstable, i.e. can disappear by a continuous reorientation of the molecules, whereas defects of half-integer strengths are stable, i.e. impossible to eliminate in a similar way. This difference between defects of integer and half-integer strengths is essential and has many experimental consequences. One of them is that ±1 disclination lines have in usual conditions a non singular core, whereas ±1/2 disclination lines possess necessarily a singular core. This difference is flagrant in thread-like textures in which ±1 line defects form "thick threads" (whose diameter and contrast are not well defined) whereas ±1/2 lines form "thin threads" with a sharp core which strongly scatters the light. This topological difference also arises in cylindrical capillary. In particular, it is very well known that the +1 line which has non-singular core provided that the radius of the capillary is large enough (typically more than a few hundreds of Å). The reason is that the director can freely rotate to align parallel to the capillary axis in its center. This phenomenon has been nicely named "escape in the third dimension" by Meyer 3 . It leads to the suppression of the core singularity in defects of integer strengths, which is very favorable energetically. It has also been shown that defects of halfinteger strengths can partly escape in the third dimension for similar energetic reasons, although their singular core subsists because it cannot disappear topologically (the anisotropy of the Frank elastic constants is responsible for these distortions as shown first by Dzyaloshinskii 8 ). As a result, the director field around a disclination line is rarely planar, even in the case of defects of half-integer strengths. [START_REF] Oswald | For a discussion about topological and energetic stability of defects in nematics[END_REF] This complete or partial rotation of the director along the defect axis has another consequence. Indeed, the director may "escape" in two opposite directions. For that reason, the point defects separating segments of line escaping in two different directions can form along the line. These point defects were known for a long time in cylindrical capillaries and have been studied intensively. [2][3][4][5] In particular, they nucleate in a large number when the sample is "quenched" abruptly in the nematic phase either from the isotropic phase, or from the smectic A phase. In both cases point defects occur randomly along the disclination line, with a density strongly depending on the quench velocity. In a cylindrical capillary, point defects are characterized by a radial or a hyperbolic molecular orientation around their singular centers. For that reason they are called R-defect (radial or "hedgehog" defect) and H-defect (hyperbolic or "antihedgehog" defect), respectively. These defects are directly observable in the microscope if we use a capillary with a large radius [2][3][4][5] (several micrometers). But they were proven to form also in cylindrical micropores of radius 0.05 µm < R < 0.5 µm of porous media as the Vicor. This result was obtained by Crawford et al. [START_REF] Crawford | [END_REF]11 from NMR studies. These authors observed that defects nucleate after a rapid quench from the isotropic liquid and then annihilate by pairs of opposite signs when their mean distance is less than typically the pore diameter. On the other hand, they did not observe their complete disappearance, even after a long annealing of their samples.

The question that immediately arises from these experiments is how the point defects interact. This problem was studied intensively by theorists. Two different answers are given in the literature. According to Peroli and Virga [12][13][14] the defects of opposite signs attract at short distance and do not interact when their distance typically exceeds the capillary diameter (screening effect). On the other hand numerical simulations by Semenov 15 show that defects attract at short distance and repel at long distance. These two types of interaction may explain as well the experimental results of Crawford et al. [START_REF] Crawford | [END_REF]11 in porous media as they lead both to a metastable array of defects after annealing of the sample. Finally, the attraction at short distances between R-and H-defects was studied experimentally by Cladis and Brand 4 in a large-radius capillary. They observed that the motion of the two defects is asymmetric, the Rdefect moving much faster than the H-one. This phenomenon has not yet been explained but is certainly due to some backflow effects neglected in the calculations up to now. 16 Recently, an alternative method was proposed to study the nucleation and the annihilation processes of point defects on a -1/2 disclination line [17][18][19][20] . As shown in Fig. 1, such a line occurs spontaneously near the tip of the meniscus which forms at the edge of a homeotropic nematic sample. In this experiment, the sample, which consists of two parallel glass plates with inner surfaces forcing a homeotropic anchoring, is partly filled with liquid crystal by capillarity. This geometry is interesting because it allows direct observations in the microscope of the point defects and their movements. top view of the sample when observed in the microscope with the polarizer (P) and the analyzer (A) partially crossed. The black region corresponds to the meniscus with the air. A point defect is visible on the disclination line, separating two zones tilted in opposed directions; c) calculated projection of the director field on the yz plane. The director is tangent to the solid lines. The intensity of the gray shading is an encoding for the magnitude of the escaped component of the director field along the line (x-component). The corresponding angle between the director and the yz plane is shown in (d) (from ref. [17]). Note that due to "escape in the third dimension", the director field is twisted, in particular between the line and the bulk sample. In addition, the twist (figured on the photo by the nails which represent the director field in the middle plane of the sample) changes sign on both sides of the point defects.

The main results obtained in this geometry are summarized as follows. First, the mechanisms of formation of the disclination line were analyzed both experimentally and theoretically. 19 It was shown that the line forms only if the anchoring of the molecules at the nematic-air interface is homeotropic (or sufficiently close to this orientation). Another observation was that the director field is not planar at equilibrium, but also escapes in the third dimension. In addition, the line can disappear (i.e. become virtual, which means that it is expelled from the nematic phase) when the anchoring penetration length K/W diverges (K is a Frank elastic constant and W the anchoring energy). The divergence leads to breaking of the homeotropic anchoring at the nematic-air interface. This phenomenon happens when approaching from a smectic-A phase because of the divergence of the elastic constant K (and more precisely of the bend and twist constants K 2 and K 3 ). 21 Similarly, the line can be expulsed from the sample by applying a strong enough electric field which favors the normal orientation of the molecules in the bulk sample.

Second, it has been observed that a great number of point defects can be nucleated along the line by the transient application of a strong enough electric field, or after an excursion into the smectic A phase. In both cases, the defects spontaneously nucleate when the line reappears in the sample. Note that the nucleation mechanisms of the point defects are still unknown in those conditions in spite of recent advances made in this field in directional melting. 20 It was then observed that the point defects collapse by pairs of opposite signs till forming a lattice of uniformly spaced defects. Finally, and this is the most important result for what follows, experimental studies of the time evolution of the defect distribution, coupled with numerical simulations, strongly suggest that defects attract at short distances and repel at large distances. This repulsion at large distance between defects of opposite signs is counterintuitive, even if it seems quite well established in this special geometry. However, for R-and Hdefects observed in the cylindrical geometry there is no definite answer concerning this point.

Faced with this confused situation, we addressed again this problem of the repulsion between defects and searched for a more direct way to prove its existence.

To reach this goal we doped our liquid crystal with a very small amount of a chiral component. Chiral molecules transform a nematic phase into a cholesteric one. To avoid complications, the concentration of chiral molecules was properly adjusted in order to obtain a pitch of the helix larger than the sample thickness. A direct consequence is that the director field remains homeotropic in the bulk sample (the cholesteric phase is unwound). On the other hand, right-hand and left-hand twist distortions are no longer equivalent in the vicinity of the disclination line because of the material chirality. A direct consequence is that two adjacent line segments separated by a point defect will have different energies per unit length as they involve twist deformations of opposite signs. The net result is that a constant force must act, for instance to the right, on all the defects of a given sign, whereas a force of opposite sign (but with equal magnitude) acts to the left on all the other defects of opposite sign. A crucial point is that these forces are independent of the distance between the defects.

As a consequence, they will tend to make all the defects disappeared, whatever their distances, till all the line segments with the good twist merge in a single domain. This scenario should systematically be observed if there is no repulsion between defects, or if the "chiral force" is larger than the maximal repulsion force between defects of opposite signs, by admitting that it exists at large distances. On the other hand, the system should tend towards a metastable configuration consisting of an alternation of small and large domains (having, respectively, the "bad" and the "good" chirality), as long as the chiral force can be equilibrated by the repulsion force.

To test this idea, we performed the experiment and extended our previous numerical simulations by accounting for this constant additional force. Experimental results are presented in section II. The origin of the chirality-induced force on the defects is discussed in section III. The results of the simulations are given in section IV and conclusions are drawn in section V.

II. Experiment

As in previous experiments, [18][19][20]22 liquid crystal 8CB (4-n-octyl-4'-cyanobiphenyl, K24 from Merck Co.) was used. To induce chirality, 8CB was doped with a very small amount of the chiral compound S811 (from Merck Co.). Note that the concentration of S811 (0.15% by weight) was chosen in order to avoid the formation of cholesteric fingers in the bulk sample 21 (the cholesteric pitch is always larger than the sample thickness). The cell consists of two parallel rectangular glass plates treated for homeotropic anchoring with a solution of polyimide (3% mixture of compound #0626 in solvent #26 from Nissan Co.). The glass plates were successively spin-coated with a drop of a polyimide solution, dried under vacuum for 15 min at 80°C and polymerized during 45 min at 180°C. They were then glued together along their longer sides. Two 25-µm-in-diameter nickel wires served as spacers to fix the sample thickness. As explained in the introduction, the sample must be filled partially in order to obtain a meniscus between the nematic and the air. To realize this operation the cell was placed in an apparatus for directional solidification and filled in the presence of a large temperature gradient. The procedure, which is described in detail in ref. [17], allowed us to obtain a straight meniscus parallel to the longer side of the cell and perpendicular to the imposed temperature gradient (see Fig. 2). Experimental setup: cell in the directional solidification apparatus. The sample can be moved in the temperature gradient imposed by the two ovens. This way, the temperature of the meniscus can be easily and rapidly changed.

Point defects on the line are then prepared as follows. The sample was moved toward the cold side of the directional solidification apparatus till the temperature of the meniscus reached the temperature of the transition to a smectic A phase (T NA ≈ 33.4°C). During this passage the line disappears for the reasons explained in the introduction. The sample was then moved in the opposite direction till the meniscus reaches temperature T NA + ∆T (∆T values are given in Table 1). This temperature is known precisely as the position of the sample in the temperature gradient is measured within ±1 µm with an LVDT sensor (as for the temperature gradient, of the order of 10 K/cm, it is equally known within ±2%). Once the meniscus has reached its final temperature (e.g. T=T NA + 0.32K), the disclination line A quantitative analysis of the images showing the trajectories of the defects was performed using Igor-Pro Wavemetrics program. The images were analyzed and the positions of the defects were detected as a function of time. An example of such an analysis is presented in Fig. 4. The defects were detected using always the same algorithm with the same criteria of recognition. Note that at the beginning of the process many defects were not detected because of the too weak contrast of the images. In addition, defects separated by a distance smaller than 3 pixels were detected as one defect.

Similar experiments were performed at 5 different temperatures ranging between T AN + 0.32K and T AN + 1.93K. The most stable distribution of the defects was obtained at T = T AN + 0.32K and is shown (after 4 hours of equilibration) in Fig. 5. In general, the defects distribution is very stable if there is no temperature gradient along the disclination line. The equilibrium distribution is then reached after 3 ÷ 4 minutes and remains stable during several hours (as shown in Fig. 5). In some of the registered experiments the disclination line was not perfectly perpendicular to the temperature gradient. As a result the defects were very slowly drifting along the line, causing their disappearance at the end of the disclination line i.e. at the edge of the sample. This drift was already observed in previous experiment performed in directional solidification and was attributed to the fact that the point defect energy decreases when the temperature decreases (whence a constant drift of the defects towards the cold part of the sample).

The defects distribution and the drift velocity were found to depend on temperature. We did not performed systematic measurements so far but we observed that at temperature T ≥ T max = T NA + 1.54K all the defects rapidly disappear, even in the absence of a drift. This result shows that at temperature T ≥ T max , the repulsion force between defects can no longer equilibrate the "chirality-induced" force. In that case, all the defects collide and annihilate in a few minutes after their nucleation. The result is the formation of a straight line with the director escaping all along in the same direction (Fig. 6). The best recording of the evolution process was obtained at T = T AN + 0.54K. Its analysis, starting 40 seconds after the nucleation, is shown in Fig. 7. One observes that during a relatively short period of time, four pairs of defects annihilate. It is worth noting that each annihilation gives more free space for the other defects, causing an increase of their separation distance. This observation is important as it is a direct proof for the existence of a repulsive interaction force between the defects. To complete these observations, we measured the distance distribution over sequences of about 25 pairs of defects at different temperatures. The values of the ratio, R = l/s, of the averaged longer to shorter distances (l and s, respectively) between the defects measured at five temperatures are given in Table 1. The increase of the ratio, R, which is a direct measurement of the asymmetry in the defects distribution, shows that the chiral force increases with respect to the repulsion forces as the temperature increases. This property will be qualitatively explained in section III. It must be noted that the disclination line was generally not perfectly straight (this condition is very difficult to fulfill experimentally). For that reason, all the defects were not exactly at the same temperature what explains a variation of their distances (ca. 10%), the larger the higher was temperature. Additionally, the defects were often very slowly drifting.

In spite of these artifacts, we observed that the distribution was always bimodal. An example of such a distribution measured 4 hours after the nucleation process is shown in Fig. 8. It is worth pointing out 18,20 that in absence of the chiral agent the distribution was unimodal with only one characteristic distance between the defects. In the next section the role of the chiral agent is explained and the chirality-induced force acting on each defect is estimated.

III. Interactions induced by chiral additives

In the introduction, we have explained that the disclination line is not planar and tends to twist the director field (see fig. 1). This twist deformation is essentially localized between the line and the bulk sample. It typically extends over a surface area of size d × d. Let ±α be the tilt angle of the director close to the core of the disclination. The twist deformation energy per unit length of the line associated to this distortion is of the order of:

2 2 2 2 K 2 1 d d d K 2 1 α α = ×       ( 1
)
where 2 K is the twist constant of the usual Frank elasticity.

In the absence of chiral additives, this distortion energy is obviously the same on both sides of any point defect in spite of the change of sign of the angle α . For that reason, there is no force acting on the point defect (other than the forces exerted on it by the other defects).

Let us now consider the case of a cholesteric. In that case the material possesses a spontaneous twist proportional to the concentration of chiral dopant. As a result, the segment of a line situated on one side of the point defect where the tilt angle is o q +α must be distinguished from the segment of the line situated on the other side of the defect where the tilt angle is -α . Indeed the distortion energy per unit length of a line becomes in the first segment α α α

d q K K 2 1 d q d K 2 1 o 2 2 2 2 2 o 2 + ≈       + , (2)
whereas it is given by formula

α α α d q K K 2 1 d q d K 2 1 o 2 2 2 2 2 o 2 - ≈       - (3)
in the other segment.

This energy difference is responsible for a net force acting on the defect of constant magnitude .

α d q K 2 f o 2 chiral ≈ (4)
This force acts on all the defects of the same sign (let us say +1). The same reasoning shows that defects of opposite sign (-1) experience a force of opposite sign but of the same magnitude.

It must be noted that close to the transition to the smectic phase diverges (it becomes infinite because twist distortion is forbidden in smectics) whereas q tends to 0 (the cholesteric pitch diverges). On the other hand, the product remains constant as it only depends on the chiral power of the dopant (this is in fact the reason why q tends to 0). As a

2 K o o 2 q K o
consequence the chiral force acting on the defect is independent of temperature as a first approximation. The situation is completely different for the interaction force between two neighboring defects of opposite signs. Indeed, this force scales as the elastic constants (i = 1,2,3) which all strongly increase K i 21 when approaching the transition to smectic-A. One thus expects that close enough to the transition, the chiral force can be equilibrated by the repulsion force, whereas at high temperature, the chiral force is always larger than the repulsion force. This qualitatively explains why stable patterns of defects form only close to the transition (at temperature T < T max ), whereas all the defects disappear when T > T max .

In the following section we present a numerical simulation of the defect dynamics in the presence of a chiral agent.

IV. Computer simulations of the dynamics of point defects

We have studied the evolution of point defects distributed on the line interacting with a force f(x) given by the formula 23 ( ) ( ) ( )

x exp x 1 x f - - - = , ( 5
)
where x>0 is the distance between neighboring defects. This force is attractive at short distances and repulsive at longer distances. 23,24 We have assumed that the forces act between the nearest neighbors, but we have also verified that even if we assume that every defect interacts with every other defect on the line it does not change qualitatively and also quantitatively the results. Because the interactions between the point defects are related to the 3D distribution of the director field it is reasonable to assume that the interactions between the defects are strongly screened by the nearest neighbors and therefore this justifies the assumption of the nearest neighbors interactions. Our experiments also support correctness of such assumption, as can be seen from Fig. 7. Additionally, the chiral additives induce a constant force on the defects, as explained in the previous section. We assume that we have N defects of alternating type (⋅⋅+1⋅-1⋅⋅+1⋅-1⋅⋅+1⋅-1⋅⋅) on the line of length L. The situation is schematically shown in Fig. 9. (top view of the sample); star represents the radial defect (let's say "+") and rhomb represents the hyperbolic defect (consequently "-"). The director (represented by a nail in sketch "c") in the vicinity of the disclination line is, on the side of the bulk sample, alternately tilted to the top and to the bottom between each defect. On the other hand, it is perpendicular to the glass plates (and so represented by a dot) in each plane perpendicular to the line containing a point defect. Note that the director is similarly perpendicular to the glass plates far from the line in the bulk homeotropic nematic phase.

For that reason, regions between the defects are alternately right-handed and left-handed twisted.

In the model we simply assume that the defects move under the influence of the force f with a velocity proportional to it. When the friction coefficient is independent of both the distance between the moving defects and the type of the defect, we can write the following equations for the velocity v i of the i-th defect in the array (Stokes dynamics):

chiral i , 1 i 1 i , i i f ) x ( f ) x ( f v ± + - = - + , (6)
Here, f(x) is the force acting between the defects (given by eq 5), attractive at short distances and repulsive at large distances, and f chiral (eq 4) is the constant force caused by chiral additives with "+" sign for one type of defects (say +1 defects) and "-" for the other (say -1 defects). As for velocity v , it equals

i i i v dt dy = . (7)
Here

j i j , i y y x - =
is the distance between the neighboring defects i-th and j-th and y i is the location of the i-th defect on the line.

In general all constants in the model such as viscosity, strength of the potential or its range rescale the time and the distance and therefore they do not explicitly appear in the model. It follows that all quantities in Eqs. (5÷7) are dimensionless. We took a reflecting boundary conditions with zero force acting on the defect at the boundaries and verified that the results do not change if we take as the boundary force a force produced by the mirror image of the last defect. In general the final results do not depend on the particular form of the interaction between the last (or first) defect on the line with the boundary.

As typical parameters we have used L = 1000 and N = 1500 with a random initial distribution of the defects on the line. Each defect (e.g. "+1") had an opposite defect ("-1") at each of its sides. Two defects were annihilated when the distance between them was less than 0.1. We used a simple Euler scheme to solve the equations with a time step of 0.1 and checked the results for the time step 0.01. We averaged the results typically over 400 runs i.e. 400 initial configurations. We have verified that the final results did not depend strongly on the initial distribution; even for an almost equidistant distribution of defects with small standard deviation (1 percent of the average distance) we have got the same results as with the random distribution. We have also noted that the final distribution of defects did not depend on the initial number of defects, N, on the line, providing that N was large enough.

The dimensionless force, f(x), given by eq (5) has a maximum of repulsion at the distance x = 2 between the defects. At this distance it assumes a value f(x = 2) = 0.135. If the constant force f 0 exceeds the value -0.135, all the defects should annihilate since this force would overcome the repulsive barrier present in f(x). Because f chiral is proportional to the concentration of chiral additives and presumably weakly dependent on temperature (see eq 4)

we expect that at high temperature and/or high concentration of chiral additives all the defects should annihilate. This fact was observed in our experiments.

The histograms of the distances between the defects are shown in Fig. 10. For very small force, f chiral , we find that the defects tend to form an equidistant array similarly as in the case for the nematic without the chiral additives. For larger f chiral the defects tend to group in pairs "+1-1". The distribution of distances is bimodal i.e. there is a shorter distance between the defects in pair and a longer distance between pairs. And this fact was also observed in our experiments.

V. Discussion and conclusions

Our experimental and theoretical studies indicate that chiral additives allows the control of the distribution of defects on the line. Moreover, further experimental studies with chiral additives could help to determine in details the interaction potential between the defects. At present, we are certain that "+1" and "-1" defects attract each other at short distances and repel at large distances. Moreover, we have shown that the chiral additives can overcome the repulsive barrier and lead to the annihilation of all defects. More information will be obtained if we could predict the time and length scales from the microscopic calculations. So far microscopic calculations gave only conflicting results as to the nature of the interaction potential between the defects. 12,15,24 The dynamics of point defects is also more complex than described in the simulations. For example the mobility of defects (defined as the ratio between the force acting on the defect and its velocity) is certainly not a constant but must depend on the nature of the defects (because of probable backflow effects which should depend on the nature of the defect). We also expect that the mobility increases when the defects are very closely together as the distortions of the field in the bulk decrease in this limit. In the first approximation let us assume that the mobility is inversely proportional to the distance between the defects although a logarithmic dependence should be more appropriate. Even in this extreme case, we have verified that it does not change qualitatively the salient feature of the evolution, the main influence being to increase the time scale of the evolution. Another issue, not discussed in this paper in detail, concerns the core region of point defects. Its size might influence the interaction potential and dynamics of the defects.

Finally we have not discussed the possible asymmetry of the dynamics of +1 and -1 defects on a -1/2 disclination line, although we observed that in our experiments. Such asymmetry exists in our geometry, as in cylindrical capillaries 4 between R-and H-defects, but is difficult to study in detail because of the presence of the temperature gradient which often causes a slow drift of the whole pattern. Further experiments at constant and homogeneous temperature are thus necessary to address seriously this point.

To conclude this article, let us emphasize that our experiment has a direct relation to one of the most important problems in condensed matter science. Namely, how to prepare a sample to achieve ordering at large distance while eliminating the defects. We have shown that, in our case, adding a very small amount of chiral impurities is sufficient to reach this objective.

Figure 1 .

 1 Figure 1. a) Formation of a -1/2 disclination line near the edge of a homeotropic nematic sample; b)

Figure 2 .

 2 Figure 2.Experimental setup: cell in the directional solidification apparatus. The sample can be

Figure 3 .

 3 Figure 3.Time evolution of the point defects nucleated after a rapid passage from the smectic-A

Figure 4 .

 4 Figure 4. Results of the computerized analysis of the movie showing the early stage of evolution

Figure 5 .

 5 Figure 5. Equilibrium distribution of the defects at T=T AN + 0.32K; the image was taken 4 hours after

Figure 6 .

 6 Figure 6.Disclination line after all the defects have disappeared. This final state is systematically

Figure 7 .

 7 Figure 7. Position of the defects as a function of time starting 40 seconds after their nucleation at

Figure 8 .

 8 Figure 8. Histogram of the experimental distribution of the defects measured 4 hours after their

Figure 9 .

 9 Figure 9. Schematic presentation of the defects self-assembling in pairs because of the chirality force

Figure 10 .

 10 Figure 10. Histograms of simulated distribution of the defects for a constant chiral force: a) unimodal

TABLE 1 : Ratio, R = l/s, of the longer to shorter distances (l and s, respectively) between the defects as measured for different temperatures; temperatures are given as an increment over smectic-A -to -nematic phase transition, T NA = 33.43 K

 1

	∆T (K)	R = l/s
	0.31	2.0 ± 0.4
	0.32	2.4 ± 0.4
	0.54	3.3 ± 0.6
	0.99	4.7 ± 0.8
	1.54	5.2 ± 1.6

Acknowledgments

This work was supported by the Committee of Scientific Research (Poland, KBN 2P03B00923 (2003÷2005), as well as the Polonium Program for bilateral cooperation between France and Poland. One of us (AŻ) would like to thank the people from École Normale Supérieure de Lyon for their hospitality.


domain: spi sdv

text:
Reina Aydé
email: reina.ayde@creatis.insa-lyon.fr

Gwenaël Gaborit
email: gwenael.gaborit@univ-savoie.fr

Jean Dahdah
email: jean.dahdah@kapteos.com

Lionel Duvillaret
email: lionel.duvillaret@kapteos.com

Nadège Courjal

Clement Guyot
email: clement.guyot@femto-st.fr

Raphaël Sablong
email: raphael.sablong@univ-lyon1.fr

Anne-Laure Perrier
email: anne-laure.perrier@creatis.insa-lyon.fr

Olivier Beuf

A.-L Perrier

Unbiased Electro-Optic Waveguide as a Sensitive Nuclear Magnetic Resonance Sensor

Keywords: Electro-optical devices, optical sensors, magnetic field measurement, nuclear magnetic resonance, magnetic resonance imaging

A pigtailed Ti:LiNbO 3 waveguide is here associated to a specific nuclear magnetic resonant coil to perform a low invasive magnetic field measurement. The developed device exploits a passive electro-optic transduction between the measured magnetic field and polarization state modulation of a laser probe beam. Because of the use of integrated optics, the coil electromotive force induces a dramatically enhanced electric field, thus leading to sensitivity improvement. A minimum detectable magnetic field lower than 60 fT • Hz -1/2 is achieved at the resonant frequency of 128 MHz. A dynamic range exceeding 100 dB is experimentally demonstrated.

I. INTRODUCTION

M AGNETIC Resonance Imaging (MRI) constitutes a non invasive technique which provides information related to the anatomy of a living being and allows to diagnosis certain diseases thanks to the analysis of soft tissues. Based on nuclear magnetic resonance (NMR), MRI consists in the study of magnetic modification properties of a nuclei which reflects its interaction with the environment (other nuclei and lattice) [START_REF] Kastler | Understanding MRI[END_REF]. In order to extract this information, it is necessary to modify the magnetization from its equilibrium state, and then to detect its return to this equilibrium state driven by magnetic resonance frequency and different relaxations processes.

Conventionally, an inductive loop is used to characterize the magnetization variations via the induced electromotive force (EMF). The measurement is usually performed with an external coil to image the biological media [START_REF] Vaughan | RF Coils for MRI[END_REF]. However, despite technical MRI improvements, spatial resolution and achievable image quality with external coil are still limited for the examination of deep regions. It has been demonstrated that an endoluminal (internal) coil provides an important increase in local signal to noise ratio (SNR), and enables very high resolution images [START_REF] Beuf | In vivo colon wall imaging using endoluminal coils: Feasibility study on rabbits[END_REF]. Nevertheless, the use of metallic coaxial cable, connecting the coil to the MR system could induce a considerable local specific absorption rate (SAR) especially around this cable [START_REF] Yeung | RF safety of wires in interventional MRI: Using a safety index[END_REF], [START_REF] Armenean | RF-induced temperature elevation along metallic wires in clinical magnetic resonance imaging: Influence of diameter and length[END_REF]. In fact, during a MRI experiment, the radiofrequency (RF) magnetic field is accompanied by an electric (E) field, thus inducing current flow in the metallic cable at the same frequency called common mode. Number of solutions were proposed to reduce this current induced heating [START_REF] Ladd | Reduction of resonant RF heating in intravascular catheters using coaxial chokes[END_REF]- [START_REF] Peterson | Common mode signal rejection methods for MRI: Reduction of cable shield currents for high static magnetic field systems[END_REF]. However, these solutions were not efficient all along the coaxial cable. Optical fibre link can be used to overcome heating problems and to ensure patients safety. In general, to transmit NMR signal from coil to MR system, the direct modulation of laser diode intensity fixed on the probe is applied [START_REF] Du | Comparison of FP, VCEL and DFB laser diode in optical transmission for MR RF coil array[END_REF]- [START_REF] Memis | Miniaturized fiberoptic transmission system for MRI signals[END_REF]. Since direct current supply is required, these methods remain invasive.

Magnetic field characterization involving passive optical system has been already performed [START_REF] Suzuki | Integrated optics magnetic sensor from 2 kHz to 9 GHz[END_REF], [START_REF] Mhapatra | Electro-optic electromagnetic field sensor system with optical bias adjustment[END_REF]. This type of magnetic probe is based on an electro-optic (EO) material associated to a magnetic loop inducing an electric (E)-field proportional to a magnetic field component. This experimental configuration allows to minimize the invasiveness: the probe can be pigtailed, does not need power supply and includes metallic element much shorter than the wavelength of the field to be measured. This kind of sensor has been studied by Suzuki et al. [START_REF] Suzuki | Optical magnetic field probe with a loop antenna element doubly loaded with electrooptic crystals[END_REF] and exploits a magnetic loop doubly loaded with LiNbO 3 crystal, acting as the capacitance of the resonator. The realized set-up is dedicated to measure the magnetic field propagating along a stripline, over a wideband of frequencies in the microwave range. The goal is here to develop a resonant EO probe dedicated to RF magnetic field measurement. As our field of interest concerns MRI, the resonant frequency is precisely defined (e.g. 128 MHz corresponding to the proton frequency at 3 T). Moreover, this resonant frequency is lower in MRI system compared to the microwave frequencies, thus inducing a lower electromotive force applied to the crystal ( = -dφ dt with φ the magnetic flux). A setup involving a bulk millimetre-sized LiTaO 3 crystal has been already analysed [START_REF] Aydé | Potentialities of an electro-optic crystal fed by nuclear magnetic resonant coil for remote and low-invasive magnetic field characterization[END_REF]. The results were in a very good agreement with the theoretical expectations. But using a thinner crystal will lead to a higher internal electric field, thus to an enhanced sensitivity. Moreover, a smaller sensor induces lower disturbances on the external field to be measured. Hence, the purpose of the study is to develop an equivalent set up associated to a matched resonant loop dedicated to magnetic field measurements, where a Ti:LiNbO 3 waveguide replaces a bulk crystal. In the first section, the waveguide transducer performs the E-field induced polarization state modulation (PSM) of a laser beam, and its intrinsic E-field sensitivity is analysed. Subsequently, the EO chip is coupled to the resonant coil to perform magnetic field measurement. The device is optimized in terms of impedance matching. Its linearity response with respect to the magnetic field magnitude is then analysed. The sensitivity is compared to the one obtained with a bulk crystal instead of the optical EO waveguide. The result demonstrates the sensitivity improvement.

II. EO TRANSDUCTION USING AN INTEGRATED LiNbO 3 WAVEGUIDE

The Pockels effect acts in non-centrosymetric crystals. This EO effect traduces the linear link between the field -→ E to be measured and the induced modification of index ellipsoid of the crystal. A detailed study of EO crystal can be found in Ref. [START_REF] Yariv | Optical Electronics[END_REF] and [START_REF] Duvillaret | Electro-optic sensors for electric field measurements. II. Choice of the crystals and complete optimization of their orientation[END_REF]. The developed transducer is based on a LiNbO 3 crystal. The corresponding EO tensor which gives the vectorial dependence of index ellipsoid with the applied electric field

-→ E is given below: ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 -r 22 r 13 0 r 22 r 13 0 0 r 33 0 r 51 0 r 51 0 0 -r 22 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (1)
This tensors involves the EO coefficients r i j providing the vectorial behaviour of the EO measurement, i.e. the dependence of the refractive indices n i to the components E j of the field vector:

δ( 1 n 2 ) i = 3 j =1 r i j E j (2)
The optical wave crossing the crystal sees actually two eigen indices n + and n -. As we here exploit the E-field induced modification of the polarization state of a laser beam, the relevant information is δn(

-→ E ) = δ(n + ( -→ E ) -n -( -→ E )
. The E-field induced modifications of the eigen indices are much lower than the intrinsic indices and this latter equation can be rewritten as follow [START_REF] Duvillaret | Electro-optic sensors for electric field measurements. II. Choice of the crystals and complete optimization of their orientation[END_REF]:

δn( -→ E ) = --→ K . -→ E (3)
The differential sensitivity vector --→ K gives the direction of electric field component probed by the EO crystal and its modulus leads to the sensitivity of the EO conversion.

--→ K is given by: --→ K depends only on EO crystal properties and on the probe beam direction -→ k relatively to the crystal axis. Considering an x-cut wafer of LiNbO 3 with the optical wave propagating along the y-axis of the crystal (see Fig. 1), the sensitivity to the field component E z takes the value | --→ K y-prop,E z |. As r i j n -2 , the modulus of this sensitivity vector becomes:

--→ K = -→ ∇ δn( -→ E )|-→ E = 0 (4)
| --→ K y-prop,E z | = | 1 2 (n z 3 r 33 -n x 3 r 13 )| (5)
with n x = n + = 2.20 and n z = n -= 2.13, the eigen indices seen by the optical wave, r 13 = 8.6 pm/V and r 33 = 30.8 pm/V, the EO coefficients involved in the E z component probing. The field |E z | to be probed is induced by the EMF = V z , corresponding to the potential difference applied to the gold electrodes. Hence, the separating distance d, between the electrodes is here of major importance in order to reach very high sensitivity: while electrodes placed on sides of a bulk crystal are usually separated by a distance lying in the millimetre range, the optical waveguide transverse dimensions are approximately 10 μm (for the optical wavelength λ opt = 1550 nm), leading potentially to a significant enhancement of the induced field (E z ∝ V z d ), so that to the response of the sensor.

The transverse transducer consists in a Ti-diffused LiNbO 3 waveguide (length L y = 6 mm, in agreement with the NMR coil dimensions) confined in between two coplanar electrodes lying in the yz plane and separated by a distance d = 18 μm. This distance ensures a sensitivity enhancement together with an homogeneous E-field (in modulus and direction) seen by the optical beam. Moreover this configuration induces a capacitance which can be compensated (see section III). A schematic of the EO chip is proposed in Fig. 1. A FDTD simulation (Quickfield ) has been performed in order to assess the induced electric field -→ E z between electrodes, in the optical waveguide location. The result is given is in Fig. 2.

This E-field E z will lead to a dephasing between the two allowed polarization state inside the EO crystal and is finally written:

ϕ = 2π λ opt | --→ K y-prop,E z |.E z L y ( 6
)
The PSM is finally optically treated with a quarter wave plate to compensate the static birefringence of the crystal, and with a polarizer to convert the PSM into a modulation of optical power. An amplified high speed photodiode ensures the last conversion to an electrical signal.

The response of the realized waveguide EO transducer has been characterized. For that purpose, a synthesizer feeds the electrodes at the frequency of interest ( f = 128 MHz) and the EO signal P E O,[d Bm] is recorded with a spectrum analyser as a function of the delivered power P in, [d Bm] . Fig. 3 illustrates the obtained result. The experimental characterization can be fitted with the equation involving the delivered power itself as well as the noise contribution:

P E O[d Bm] = 10 log 10 (10 Noise [d Bm] 10 + 10 α P in,[d Bm] +β 10
) (7)
The coefficients α and β are the linearity factor and the conversion efficiency, respectively. This latter one is determined by the EO conversion itself ( ϕ), but also to the optical power carrying the modulation (1 mW) and to the photodiode responsivity (0.85 A/W). The measurements dynamic range exceeds 100 dB. The minimum detectable field E min is obtained equalizing the linear behaviour of the chip and the Johnson-Nyquist noise floor of the spectrum analyser and of the photodiode electronics (Noise [d Bm] = -105 dBm). E min is here lower than 10 mV.m -1 .Hz -1/2 . This latter value corresponds to a root mean square (RMS) voltage V z < 0.3 μV. Moreover the linear slope of the measurement is α = 0.98 and the standard deviation of the error distribution is about 0.5 dB (see inset of Fig. 3). The conversion efficiency between P in and P E O is β = -3 dB.

III. THE ELECTRO-OPTIC RESONANT COIL

Based on previous work [START_REF] Armenean | Optimization of endoluminal loop radiofrequency coils for gastrointestinal wall MR imaging[END_REF], a reference rectangular coil was built with an external width set to 5.1 mm and a length set to 47 mm. The coil is constituted with 35 μm thickness copper tapes on a substrate FR4 (relative permittivity r = 4 and thickness T = 0.8 mm). These size and geometry provide good image with adequate penetration depth and exploration length. Moreover, the dimensions are in agreement with further developments dedicated to endoluminal sensor. The coil is matched to 50 at a frequency of 128 MHz thanks to matching and tuning nonmagnetic capacitors on distal and proximal part of the coil. The initial values of these capacitors are C t = 36.2 pF and C m = 186.8 pF for tuning and matching respectively. The resonance of the coil itself has been firstly analysed via the measurement of its S11 parameter. The design actually leads to a minimum return loss at the expected frequency (see dashed curve on Fig. 5). In order to realize a magnetic sensor, the coil is connected to the EO chip via bonding wires. The experimental setup dedicated to characterize the magnetic sensor is prensented in Fig. 4. A RF synthesizer feeds an emitting coil, providing a 128 MHz RF magnetic field B y . The endoluminal coil converts the B y field into a RF E-field. The EO signal P E O is recorded with a spectrum analyser. The measurement obtained with the sensor is presented in Fig. 5 Fig. 6.

EO NMR coil response as a function of the applied magnetic field. ( f = 128 MHz, R BW = 30 Hz). Black dots and gray dashed line corresponds to measurement data and fitting curve respectively. Inset illustrates the deviation between measurement data and theoretical fitting curve. Additional gray curve is linked to a previous measurement obtained with a bulk LiTaO 3 crystal 4 mm-thick. Dot-dashed line indicates the intrinsic resonant coil (galvanic linked) response.

(gray line). The curve shows that the addition of the EO chip shifts the sensor resonance toward lower frequencies, due to its intrinsic and non-negligible capacitance. In order to retrieve the nominal resonance, C t has been adjusted to 35.2 pF to compensate capacitance value of the EO chip and inductance of bonding connection. Measurement of P E O in this case is presented with the black line of Fig. 5.

The linearity response of the realized EO coil has been investigated as a function of the applied magnetic field. The result is presented in Fig. 6. Once again, the linearity measurement exhibits a dynamic range greater than 100 dB. The fitting curve gives the magnetic field B response and is written:

P E O[d Bm] = 10 log 10 (10 Noise [d Bm] 10 + 10 3 α E Ocoil B 2 ) (8)
The coefficient α E Ocoil is the link between the EO resonant coil signal and the magnetic field to be probed. α E Ocoil takes here a value of 1.27 × 10 11 W.T -2 , which corresponds to a linear coefficient of

√ Rα E Ocoil = 2.5 × 10 6 V.T -1 , R = 50
being the resistive load. This measurement allows to determine the minimum detectable magnetic field B min = 0.3 pT in the 30 Hz resolution bandwidth (RBW) leading to B min = 56 fT.Hz -1/2 . Furthermore, this measurement is compared to a similar setup involving a bulk LiTaO 3 crystal (4 mm thick) [START_REF] Aydé | Potentialities of an electro-optic crystal fed by nuclear magnetic resonant coil for remote and low-invasive magnetic field characterization[END_REF]. An increase in sensitivity is demonstrated and reaches more than 57 dB, without any additional amplification of the signal. Finally, the equation 8 can be applied for the magnetic field response of the intrinsic NMR coil in its nominal use (galvanic linked and 50 loaded, see dotdashed line on Fig. 6). It allows to determine α Coil = 1.23 × 10 10 W.T -2 , ten times lower than the achieved value of α E Ocoil (with the fibre link and the EO conversion).

IV. CONCLUSION

An EO waveguide has been coupled to a resonant coil to design a pigtailed deported and sensitive magnetic field sensor. The use of integrated optics allows to apply the coil EMF on a weak separating distance leading to the enhancement of the field strength seen by the laser probe beam. The passive EO chip exhibits a linear response over a dynamic range exceeding 100 dB. A magnetic field as low as 300 fT remains detectable at the resonant frequency f = 128 MHz. The sensor response is increased by more than 50 dB compared to the use of a bulk crystal and by 10 dB compared to the galvanic linked NMR coil. Finally, the millimetre size and the composition of the device are fully suitable for further developments concerning low-invasive and endoluminal MR coils.

Fig. 1 .

 1 Fig. 1. Schematic of the LiNbO 3 chip ensuring the EO transduction.

Fig. 2 .

 2 Fig. 2. Transverse FDTD simulations of the field induced by the electrode potentials and surrounding the LiNbO 3 chip. A potential difference of 1 V is applied on the electrodes. Equipotential lines are indicated in black and field strength is linked to the color, from purple to red. The inset corresponds to a zoom on the location of the optical probe beam, in between the electrodes. The respective field scales are given on the left side of each image.

Fig. 3 .

 3 Fig. 3. EO response as a function of the frequency synthesizer delivered power ( f = 128.2 MHz, R BW = 30 Hz). Measurement data and theoretical fit are linked to black dots and gray dashed line respectively. The inset corresponds to the deviation between measurement data and theoretical fitting curve.

Fig. 4 .

 4 Fig. 4. Implementation scheme of the EO LiNbO 3 waveguide and experimental setup dedicated to the magnetic field response analysis. DFB Laser: distributed feeback Laser; PM fibre: polarization maintaining fibre.

Fig. 5 .

 5 Fig. 5. Frequency response of the EO coil (P in = 14 dBm, R BW = 30 Hz). The black dashed line corresponds to the return loss S11 of the initially designed coil itself. Gray curve and black curves correspond to the unmatched sensor and to the LiNbO 3 chip matched coil, respectively. An increase of 19.5 dB is obtained at the nominal frequency of 128 MHz.

This work was supported in part by the Region Rhne-Alpes through the ADR and CIBLE Project, in part by the French Military Programs Management and Procurement Agency-Project RAPID SNIFER, in part by the Directorate General of Commercial Intelligence and Statistics, and in part by the LABEX PRIMES under Grant ANR-11-LABX-0063 through the Université de Lyon, within the Program Investissements d'Avenir under Grant ANR-11-IDEX-0007, operated by the French National Research Agency.


A-L Perrier
email: anne-laure.perrier@etu.univ-savoie.fr

J-M Duchamp

P Ferrari

A Small-size Semi-lumped Three-port Tunable Power Divider

Keywords: Power divider, tunable device, tuner, varactors

In this paper, a small-size semi-lumped three-port power divider with both power and frequency tune is designed and measured. The proposed topology uses complex impedance transformers in each output branch. A 3D CPW prototype is realized with varactor diodes. It exhibits a wide tunable power, from -1.3 dB to -21 dB, and a wide working frequency tune, from 0.8 GHz to 1.4 GHz. Measurements and simulations are in good agreement.

I. INTRODUCTION

Tunable miniaturized devices constitute a great challenge in the field of modern RF/microwave telecommunications. In the near future, more and more applications will use multiple operating frequencies, with hard miniaturization constraints for MMIC applications. In this context, some researches are carried out to show the feasibility of various tunable devices that can be embedded in RF/microwave telecommunication systems like phase shifters, impedance transformers, filters, couplers, and power dividers. Several technologies can be used to achieve such tunable devices: ferroelectric and ferromagnetic materials, MEMS, piezoelectric devices, optical control and semiconductors.

The maturity of the power dividers goes back to the years 1960 when Wilkinson realized a N-way power divider [START_REF] Wilkinson | An N-way hybrid power divider[END_REF]. Since, many researches were made in order to reduce the size of the devices [2], [START_REF] Cohn | A class of broadband thr -ports TEM-m brids[END_REF], [START_REF] Yi | Modified Wilkinson power divider for nth harmonic suppression[END_REF], to increase their bandwidth [5], to eliminate the frequency harmonics [6] and for an unbalance distribution of the power in each output ports [START_REF] Ahn | General design equations, small-sized impedance transformers, and their application to small-sized three-port 3-dB power dividers[END_REF].

To our knowledge, few works proposed a tunable power distribution. In [8] a four-port divider with a tunable distribution of the power was presented. However no work was made on tunable three-port dividers.

In this paper, a three-port tunable power divider is presented. It can be tuned both in power distribution and in term of working frequency. As any power divider, our device is based on impedance transformation but it does not use quarter-wave transmission lines. Two semi-lumped complex impedance transformer, with a λ/10 length, constitute each output branch of our divider, leading to a miniaturized device. The paper is organized as follows: in section II, the principle of the power divider is described. In section III, the design of a prototype based on a 3D CPW (three dimensions coplanar waveguide) technology is described. Measurements are shown in section IVand some perspectives are discussed in section .

II. PRINCIPLE

A three-port tunable power divider is used to split a power P 1 at the input port throughout the two output ports, with powers P 2 and P 3 , respectively. In our design, this power distribution is tunable from balanced (P 2 =P 3 ) to maximum unbalanced distribution (P 2 >>P 3 or P 3 >>P 2 ). The principle of the three-port power divider is described in Fig. 1. A CLC tunable impedance transformer constitutes each output branch of the divider. Let call Z 2 and Z 3 the input impedances of the two output branches.

The available power at the input port 1 can be written as:

2 1 1 V P Z =
where V is the voltage at the junction point J.

In the same manner:

2 i i V P Z =
with i=2 or 3 for the output branches.

So

1 1 i i P Z P Z = (1)
.

If losses are neglected, then: P 1 =P 2 +P 3 , leading to:

1 2 1 1 1 3 Z Z Z = + (2) or Y Y 1 2 3 Y = + (3), where 1 i i Y Z = is the input admittance of branch i.
The input port is matched when Z 1 is equal to the conjugate of the reference impedance, ie 50 Ω in the general case that will be considered in this paper. Next, the power distribution is realized by tuning Z 2 and Z 3 input impedances. When

1 3 2 2Z Z Z = =
, the input power is equally distributed throughout ports 2 and 3. We speak of a balanced distribution

C 0 C 1 L C 0 ' C 1 ' L ' Z 2 Z 3 Port 2 Port 3 Z 1 Port 1
J of the power. The divider works in a balanced mode.

When Z 2 (resp. Z 3 ) = Z 1 and Z 3 (resp. Z 2 ) >> Z 1 , there is an un pical case, the evolution of Z 2 and Z 3 from balanced to balanced distribution of the input power. Port 3 (resp. port 2) is considered as open. In this case, all the power is sent into port 2 (resp. port 3). The divider works in an unbalanced mode. For a ty unbalanced mode is simulated for all ports loaded by 50 Ω, and for an input matching dB 20 11 -< S

. Simulations results are shown in Fig. 2 Smith chart at a fixed frequency. In the balanced mode, Z 2 = Z 3 ≈ 100Ω and P 2 = P 3 . For the maximum unbalanced mode, Z 2 ≈ 50 Ω and Z 3 ≈ ∞, leading to P 2 ≈ P 1 and P 3 ≈ 0. Between these two extreme cases, Z 2 and Z 3 are complex, they are linked by relation (2).

III. CONCEPTION

The power divid by two impedance tra ig. [START_REF] Yi | Modified Wilkinson power divider for nth harmonic suppression[END_REF] The tunable power divider.

For the realized prototype, Macom™ MA4ST-1240 va In order to minimize losses, the inductor L is realized by a se ider was designed on a Rogers™ RO4003 su Z balanced mode (white) to a maximum unbalanced mode (black), with intermediate points (Grey).

nsformers. These Π transformers with three tunable elements allow a large complex impedance coverage on the Smith chart. Fig. 3 gives the realization principle of the impedance transformers. The variable capacitors are realized by varactor diodes and the tunable inductance is realized by a varactor in series with an inductor [9]. er is constituted nsformers as shown in Fig. 4. A pair of capacitors C is necessary to separate the diode's bias voltages of the two impedance transformers while maintaining the divider symmetry.

F ractor diodes are used. The equivalent electrical circuit of these diodes (Fig. 5) is constituted by a series inductor (L S =1.8nH), a series resistance (R s =1.6Ω), a case capacitance (C c =0.11pF) and a tunable capacitance (C(V)) varying from 1.5 to 8.6 pF with a bias voltage V varying from 12 V to 0 V. ction of transmission line of characteristic impedance Z c and electrical length θ, leading to a semi-lumped device. We chose a CPW technology to achieve an easier soldering of the varactor diodes. The used of a 3D junction leads to an easy connection between the three branches of the divider. Note that a e gaps in the ground planes permit to separate diodes biases. SMC (Surface Mounted Component) capacitors maintain the ground continuity in high frequency.

V 0 V 1 V 2 L V 0 V 0 ' V 1 ' V 2 ' V 1 V 2 Port 1 C Port 2 Port 3 Z c , θ Z c , θ Port 2 Port 3 Port 1 SMC capacitors Varactors R s L s C c C(V) Z 2 =Z 3 ≈100Ω P 2 =P 3 Z 2 ≈50 Ω P 2 ≈P 1 Z 3 ≈∞ P 3 ≈0 Gaps m
First the tunable dist h s shown at a fixed frequency. Sec ng of the working fr tunable di of the power from a balanced mode when strib different icrostrip technology could also be used, with via holes for the connection of the shunt varactor diodes.

IV. RESULTS ribution of t e power i ondly the tuni equency for a balanced and for a maximum unbalanced distribution of the power is presented.

A. Tunable power distribution for a fixed working frequency. At a fixed working frequency, the divider permits a stribution The differences between simulations and measurements can be attributed to the varactor diodes parasitic elements inaccuracies and the 3D junction that has been considered as ideal in the simulations.

The same study of the working frequency tunability was realized considering the maximum unbalanced mode. Measurement results of 21 S and 21 S are given in Fig. 9.

From 0.8 GHz to 1.4 GHz, it can be considered that the unbalanced mode is possible, with ity of the tunable power divider topology. To go further, it could be interesting t power divider in a m

|S21|

The realized prototype has shown the valid proposed o realize the icrostrip technology, in order to obtain a 2D planar device and a much simpler J junction.

It is also interesting to point out the origin of insertion loss.

Frequency (GHz) ). The difference between simulation and measurement res obably due to the 3D J junction that was not at all optimized.

Ω = 6 . 1 s R Ω = 4 . 0 s R 0.8 -4.8 dB -3.4 dB 1.1 -3.4 dB -3.1 dB 1.4 -3.7 dB -3.2 dB
These simulation results show that the use of a MEMS technology to realize tunable power dividers for working frequencies around 20 GHz could lead to very interesting results in term of power tuning and insertion loss. ults are pr otype has been realized in a 3D put power can be t 8, pp. 116-118, Jan. 1960.

[2] L. I. Parad and R. L. Moynihan, "Split-tee power divider," IEEE Trans.

Microwave Theory and T , pp. 91-95, Jan. 1965. ode hy etters,

[5]

[6] June 1989.

hniques,

[8] EE Trans.

[9] VI. CONCLUSION

A new topology of a power divider with both power and working frequency tune has been proposed. A prot CPW technology. The out .3 dB t uned between -1 o -17 dB over a wide working frequency range, from 0.8 GHz to 1.4 GHz. The semi-lumped approach used for the design leads to a small-sized device with less than λ/10 output branches.

Insertion loss is lower than 1.8 dB. It is due to the CPW 3D junction and the varactor diode's parasitic series resistance. The insertion loss could be reduced to less than 0.4 dB if the junction was correctly optimized and diodes with a better quality factor were used. Finally, a MEMS power divider based upon the proposed technology could be designed with interesting expected characteristics in term of power and working frequency tuning range.

Fig. 1

 1 Fig.1Principle of the proposed tunable three-port power divider.

Fig. 2

 2 Fig. 2 Input impedance Z 2 (▲) and 3 (•) of branches 2 and 3 for a

Fig. 3

 3 Fig.3The impedance transformer realization principle.

Fig. 5 .

 5 Fig. 5. Macom MA4ST-1240 Varactor equivalent electrical circuit.

  Fig. 6 Photography of the power divider.

SFig. 7 |Fig

 7 Fig. 2. This ution of the power is obtained by varying the diode's tunable di biases. Fig. 7 (a) shows measurement results of S 21 and S 31 magnitude for the balanced and the maximum unbalanced modes, and for intermediate mode. As expected in the balanced mode, the input power is equally distributed to the two output ports. The same bias voltages are applied to the diodes of the two transformers: V 0 =V 0 ', V 1 =V 1 ' and V 2 =V 2 '. We measured dB 8 . 3 31 21 -= = S S at 1.1 GHz. So insertion

  Fig. 9 |S 21 | and |S 31 | for the maximum unbalanced mode.

Table 1

 1 Simulation of |S 21 | for two different series resistances of the diodes in the balanced mode.

Table 1

 1

	shows simulation results of the insertion loss
	obtained in the balanced mode. Simulations have been carried
	ut for two different values e o i 1.6 Ω (actual case) and 0.4 Ω. For of the diode's series resistance R s , Ω = 4 . 0 s R , insertion
	loss are smaller than 0.4 dB for the whole tuning frequency
	band, from 0.8 GHz to 1.4 GHz. For	s R	= 6 . 1	Ω	, insertion
	loss are equal to 0.4 dB, compared to the 1.2 dB measured
	insertion loss (see Fig. 8(b):	S	21	≈	-	. 4	2	dB


Christos Pelekis
email: pelekis.chr@gmail.com

AND Jan Ramon
email: jan.ramon@inria.fr

Yuyi Wang
email: yuwang@ethz.ch

ON THE BERNSTEIN-HOEFFDING METHOD

Keywords: 2010 Mathematics Subject Classification. 60E15 Hoeffding's inequality, convex order, Bernstein polynomials

We consider extensions of Hoeffding's "exponential method" approach for obtaining upper estimates on the probability that a sum of independent and bounded random variables is significantly larger than its mean. We show that the exponential function in Hoeffding's approach can be replaced with any function which is non-negative, increasing and convex. As a result we generalize and improve upon Hoeffding's inequality. Our approach allows to obtain "missing factors" in Hoeffding's inequality. The later result is a rather weaker version of a theorem that is due to Michel Talagrand. Moreover, we characterize the class of functions with respect to which our method yields optimal concentration bounds. Finally, using ideas from the theory of Bernstein polynomials, we show that similar ideas apply under information on higher moments of the random variables.

Prologue, related work and main results

Given a real number p ∈ (0, 1) let B(p) denote the set consisting of all [0, 1]-valued random variables whose mean is equal to p. This work is motivated by the problem of obtaining sharp upper bounds on the "tail probability"

P n i=1 X i ≥ t ,
where t is a fixed real number t such that n i=1 p i < t < n and X 1 , . . . , X n are independent random variables such that X i ∈ B(p i ), for i ∈ {1, . . . , n}.

If t ≤ i p i , then the problem is trivial; just choose X i to be equal to p i with probability 1. Throughout the text, whenever a random variable X ∈ B(p) is under consideration, it will be tacitly assumed that p ∈ (0, 1), thus excluding the uninteresting cases where X is either identically equal to zero, or identically equal to one.

There is a vast amount of literature that is related to the aforementioned problem and the interested reader is invited to take a look at the works of Bentkus [START_REF] Bentkus | A remark on Bernstein, Prokhorov, Bennett, Hoeffding and Talagrand inequalities[END_REF], [START_REF] Bentkus | On Hoeffding's inequalities[END_REF], [START_REF] Bentkus | Optimal Hoeffding-like inequalities under a symmetry assumption[END_REF], Fan et al. [START_REF] Fan | Sharp large deviation probabilities for sums of independent bounded random variables[END_REF], From [START_REF] From | An Improved Hoeffding's Inequality of Closed Form Using Refinemets of the Arithmetic Mean-Geometric Mean Inequality[END_REF], From et al. [START_REF] From | A refinement of Hoeffding's inequality[END_REF], Györfi et al. [START_REF] Györfi | Some refinements of large deviation tail probabilities[END_REF], Hoeffding [START_REF] Hoeffding | Probability inequalities for sums of bounded random variables[END_REF], Kha et al. [START_REF] Kha | Probability inequalities for sums of independent random variables[END_REF], Krafft et al. [START_REF] Krafft | A note on Hoeffding's inequality[END_REF],

McDiarmid [START_REF] Mcdiarmid | On the method of bounded differences[END_REF], Pinelis [START_REF] Pinelis | On inequalities for sums of bounded random variables[END_REF], [START_REF] Pinelis | On the Bennett-Hoeffding inequality[END_REF], Schmidt et al. [START_REF] Schmidt | Chernoff-Hoeffding bounds for applications with limited independence[END_REF], Siegel [START_REF] Siegel | Towards a usable theory of Chernoff-Hoeffding bounds for heterogeneous and partially dependent random variables[END_REF], Talagrand [START_REF] Talagrand | The missing factor in Hoeffding's inequalities[END_REF], Xia [START_REF] Xia | Two refinements of the Chernoff bound for the sum of nonidentical Bernoulli random variables[END_REF], Zheng [START_REF] Zheng | A refined Hoeffding's upper tail probability bound for sums of independent random variables[END_REF], among many others.

Probably the first systematic approach that allows one to obtain upper bounds on the probability that a sum of independent [0, 1]-valued random variables is larger than its mean, was devised by Hoeffding in [START_REF] Hoeffding | Probability inequalities for sums of bounded random variables[END_REF]. Hoeffding's approach is based on a method of Bernstein (see [10, page 14]) and from now on will be referred to as the Bernstein-Hoeffding method. The Bernstein-Hoeffding method is, briefly, the following.

Markov's inequality and the assumption that the random variables are independent imply that

P n i=1 X i ≥ t ≤ e -ht n i=1 E[e hX i ] ≤ e -ht 1 n n i=1 E[e hX i ]
n , for all h > 0,

where the last inequality comes from the arithmetic-geometric means inequality. Using the fact that the function f (t) = e ht is convex one can show (see Lemma 2.2 below) that

E[e hX i ] ≤ E[e hB i ]
, where B i is a Bernoulli random variable of mean p i . Therefore, it follows that P n i=1 X i ≥ t ≤ e -ht {(1 -p) + pe h } n , for all h > 0, where p = 1 n n i=1 p i . Minimizing the expression in the right hand side of the last inequality with respect to h, we find e h = t (1-p) p(n-t) and hence we obtain the following celebrated result of Hoeffding (see [START_REF] Hoeffding | Probability inequalities for sums of bounded random variables[END_REF]Theorem 1]).

Theorem 1.1 [START_REF] Hoeffding | Probability inequalities for sums of bounded random variables[END_REF]. Let the random variables X 1 , . . . , X n be independent and such that 0

≤ X i ≤ 1, for each i = 1, . . . , n. Set p = 1 n n i=1 E[X i ].
Then for any t such that np < t < n we have

P n i=1 X i ≥ t ≤ inf h>0 e -ht 1 -p + pe h n .
Furthermore,

inf h>0 e -ht 1 -p + pe h n = p(n -t) t(1 -p) t (1 -p)n n -t n := H(n, p, t).
The function H(n, p, t) in the last expression is the so-called Hoeffding bound (or Hoeffding's function) on tail probabilities for sums of independent, bounded random variables. Here and later, we denote by Ber(q) a Bernoulli random variable with mean q and by Bin(n, q) a binomial random variable of parameters n and q. If two random variables W, Z have the same distribution we will write W ∼ Z. Let us remark that the Hoeffding bound is sharp, in the sense that the Bernoulli random variables Ber(p i ) attain the bound, i.e., inf

h>0 e -ht 1 n n i=1 E[e hB i ] n = H(n, p, t),
where B i is a Ber(p i ) random variable. The main ideas behind this work are hidden in the fact that

i E e hB i ≤ E e hB ,
where

B ∼ Bin(n, p) with p = 1 n n i=1 E[B i ],
and the fact that the function f (x) = e hx , h > 0, is non-negative, increasing and convex. In a subsequent section we will show that, while applying the Bernstein-Hoeffding method, one can replace the exponential function f (x) = e hx , h > 0, with any function f (•) having the aforementioned properties. Let us also remark that a slightly looser but more widely used version of Hoeffding's bound is the bound exp -2n(t/n -p) 2 , which follows from the fact that H(n, p, t) ≤ exp -2n t n -p 2 (see [10, formula (2.3)]).

In this article we shall be interested in improvements upon Hoeffding's theorem. This is a topic that has attracted the attention of several authors (see, for example [START_REF] Bentkus | On Hoeffding's inequalities[END_REF][START_REF] Pinelis | On the Bennett-Hoeffding inequality[END_REF][START_REF] Siegel | Towards a usable theory of Chernoff-Hoeffding bounds for heterogeneous and partially dependent random variables[END_REF][START_REF] Talagrand | The missing factor in Hoeffding's inequalities[END_REF]). Let us bring to the reader's attention the following result which is due to Talagrand (see [22, Theorem 1.2]). Talagrand's work focuses on obtaining a "missing factor" in Hoeffding's inequality. The missing factor is obtained by combining the Bernstein-Hoeffding method together with a technique (i.e., suitable change of probability measure) that is used in the proof of Cramér's theorem on large deviations, yielding the following.

Theorem 1.2 [START_REF] Talagrand | The missing factor in Hoeffding's inequalities[END_REF]. Let the random variables X 1 , . . . , X n be independent and such that 0

≤ X i ≤ 1, for each i = 1, . . . , n. Set p = 1 n n i=1 E [X i ].
Then, for some absolute constant K, and every real number t such that np + K ≤ t ≤ np + np(1 -p)/K, we have

P n i=1 X i ≥ t ≤ θ t -np np(1 -p) + K np(1 -p) • H(n, p, t),
where H(n, p, t) is the Hoeffding bound and θ(•) is a non-negative function such that

1 √ 2π(1 + x) ≤ 2 √ 2π(x + √ x 2 + 4) ≤ θ(x) ≤ 4 √ 2π(3x + √ x 2 + 8) , for x ≥ 0.
See [START_REF] Talagrand | The missing factor in Hoeffding's inequalities[END_REF] for the proof of this theorem and the precise definition of the function θ(•). In other words, Talagrand's result improves upon Hoeffding's by inserting a "missing" factor of order

≈ √ np(1-p) √ np(1-p)+t-np
< 1 in the Hoeffding bound. Notice that Talagrand's result holds true for moderate values of t, i.e., for t ∈ [np + K, np + np(1 -p)/K], for some absolute constant K whose value does not seem to be known. Talagrand (see [22, page 692]) mentions that one can obtain a rather small numerical value for K, but numerical computations are left to others with the talent for it. One of the purposes of this article is to improve upon Hoeffding's inequality by obtaining "missing" factors with explicit numerical value for the constant.

Another improvement upon Hoeffding's theorem is due to Bentkus. In the proof of [2, Theorem 1.2] (see [2, page 1659]) Bentkus, implicitly, obtains the following result.

Theorem 1.3 [START_REF] Bentkus | On Hoeffding's inequalities[END_REF]. Let the random variables X 1 , . . . , X n be independent and such that 0

≤ X i ≤ 1, for each i = 1, . . . , n. Set p = 1 n n i=1 E [X i ].
Then, for any positive real t such that np < t < n, we have

P n i=1 X i ≥ t ≤ inf a<t 1 t -a E [max{0, B -a}] ,
where B ∼ Bin(n, p).

The quantity inf a<t

1 t-a E [max{0, B -a}] is estimated in [2, Lemma 4.2]
. We will see in the forthcoming sections that Bentkus' result is optimal in a slightly broader sense, i.e., it is the best bound that can be obtained from the inequality

P n i=1 X i ≥ t ≤ 1 f (t) E[f (B)],
where f is a non-negative, convex and increasing function.

We shall be interested in employing the Bernstein-Hoeffding method to a larger class of generalized moments. Such approaches have been already considered by Bentkus [START_REF] Bentkus | On Hoeffding's inequalities[END_REF], Eaton [START_REF] Eaton | A probability inequality for linear combinations of bounded random variables[END_REF], Pinelis [START_REF] Pinelis | Exact inequalities for sums of asymmetric random variables, with applications[END_REF], [START_REF] Pinelis | On the Bennett-Hoeffding inequality[END_REF]. Nevertheless, we were not able to find a systematic study of the classes of functions that are considered in our paper. We now proceed by defining a class of functions that pertains to the Berstein-Hoeffding method. For fixed t > 0, we denote by F ic (t) the class of functions

F ic (t) := {f : [0, ∞) → [0, ∞) : f is convex, increasing on [t, +∞) and f (t) = 1}.
Examples of functions belonging to the class F ic (t) are: -t) for h > 0 and so on. Our first result shows that the Bernstein-Hoeffding method can be adapted to the class F ic (t).

f (x) = |x-| t-for fixed < t, f (x) = 1 t-max(0, x -) for fixed < t, f (x) = e h(x
Theorem 1.4. Let the random variables X 1 , . . . , X n be independent and such that 0

≤ X i ≤ 1, for each i = 1, . . . , n. Set p = 1 n n i=1 E[X i ].
Then the following hold true. a) For any fixed real number t such that np < t < n, we have

P n i=1 X i ≥ t ≤ inf f ∈F ic (t) E[f (B)],
where B ∼ Bin(n, p) is a binomial random variable and F ic (t) is the class of functions defined above. b) Moreover, we have

inf f ∈F ic (t) E[f (B)] = inf a<t 1 t -a E [max{0, B -a}] .
By employing Theorem 1.4 to a particular function from the class F ic (t), we deduce the following improvement upon Hoeffding's inequality.

Theorem 1.5. Let the random variables X 1 , . . . , X n be independent and such that 0 ≤ X i ≤ 1, for each i = 1, . . . , n. Set p = 1 n n i=1 E[X i ] and let t be a fixed positive integer such that enp ep-p+1 ≤ t < n. Then

P n i=1 X i ≥ t ≤ 1 + h e h • (H(n, p, t) -T (n, p, t; h)) + 1 - 1 + h e h P [B n,p = t] ,
where H(n, p, t) is the Hoeffding function, B n,p is a binomial random variable of parameters n and p,

T (n, p, t; h) = t-1 i=0 e h(i-t) P [B n,p = i] ,
and h is such that e h = t(1-p) p(n-t) , i.e., it is the optimal real such that

1 e ht E[e hB ] = inf s>0 1 e st E[e sB ], with B ∼ Bin(n, p).
Let us illustrate that the bound of the previous result is indeed an improvement upon Hoeffding's inequality. To see this, notice that the bound provided by Theorem 1.5 is

≤ 1 + h e h • H(n, p, t) + 1 - 1 + h e h P [B n,p = t]
and the later quantity is a convex combination of

X i ≥ t ≤ P n i=1 X i ≥ t .
In other words, Theorem 1.5 improves upon Hoeffding's by adjusting "missing factors" in the Hoeffding bound. Notice that the bound provided by Theorem 1.5 holds true for large t, i.e., for t in the interval enp ep-p+1 , n , in contrast to Talagrand's result which holds true for moderate values of t, and so it may be seen as complementary to Theorem 1.2. It is unclear how to see whether the intervals [np + K, np + np(1 -p)/K] and enp ep-p+1 , n overlap without knowing the constant K. In case the intervals do overlap, it may be informative to include a few words on comparison between the two factors. Since e h = t(1-p) p(n-t) it follows that the "missing" factor of Theorem 1.5 can be written as

1 + h e h = p 1 -p n t -1 1 + ln 1 -p p(n/t -1)
.

On the other hand, Talagrand's result provides a factor that is approximately

np(1 -p) √ 2π( np(1 -p) + t -np) + K np(1 -p) .
If we assume that K = 0 then elementary, though quite tedious, calculations show that Talagrand's bound is sharper than the bound of Theorem 1.5. Our bound has the advantage that it does not involve unknown constants and that it is obtained using a rather simple argument (see also Fan et al. [START_REF] Fan | Sharp large deviation probabilities for sums of independent bounded random variables[END_REF] for refinements of Talagrand's inequality having explicit values for the constants).

Our last result may be seen as an extension of Theorem 1.1 for sums of bounded, independent random variables whose first m moments are known. Before being more precise, let us first fix some notation. Given real numbers µ 1 , . . . , µ m ∈ (0, 1), we denote by B(µ 1 , . . . , µ m ) the set of all [0, 1]-valued random variables whose i-th moment equals µ i , i = 1, . . . , m. Formally,

B(µ 1 , . . . , µ m ) := {X : 0 ≤ X ≤ 1, E[X] = µ 1 , E[X 2 ] = µ 2 . . . , E[X m ] = µ m }.
Notice that the set may be empty. Note also that if B(µ 1 , . . . , µ m ) is non-empty then we have

µ 1 ≥ µ 2 , ≥ • • • ≥ µ m .
Recall the definition of the class F ic (t), defined above.

Theorem 1.6. Fix positive integers, n, m ≥ 2 and for i = 1, . . . , n let {µ ij } m j=1 be a finite sequence of real numbers such that the class B(µ i1 . . . , µ im ) is non-empty. Let X 1 , . . . , X n be independent random variables such that X i ∈ B(µ i1 , . . . , µ im ), for i = 1, . . . , n, and fix t ∈ (µ, n), where µ = i µ i1 . Then

P n i=1 X i ≥ t ≤ inf f ∈F ic (t) E [f (Z nm )] ,
where Z nm = n i=1 Z i is an independent sum of random variables Z i such that

P [Z i = j/m] = m j • E X j i (1 -X i ) m-j , for j = 0, 1, . . . , m.
Moreover,

inf f ∈F ic (t) E [f (Z nm )] = inf a<t 1 t -a E [max{0, Z nm -a}] .
To the best of our knowledge, this is the first result that considers the performance of the Bernstein-Hoeffding method under additional information on higher moments. Notice that the probability distribution of the random variable Z nm depends solely on the given sequence of moments {µ ij } i,j . Indeed, using the binomial formula, it is easy to see that

E X j i (1 -X i ) m-j = m-j k=0 m -j k (-1) m-j-k µ i,m-k
and so each Z i is uniquely determined by the given sequence of moments {µ ij } i,j . Let us also mention that the random variables Z i arise in the study of the so-called Hausdorff moment problem (see Feller [START_REF] Feller | An introduction to probability theory and its applications[END_REF]).

The remaining part of our article is organized as follows. In Section 2 we prove Theorem 1.4, by employing ideas from the theory of convex orders. Moreover, we show that the functions from )] are those that occur in the aforementioned result of Bentkus, i.e., Theorem 1.3. In Section 3 we prove Theorem 1.5 by employing Theorem 1.4 to a suitable class of functions. In Section 4, we prove Theorem 1.6 using ideas from the theory of Bernstein polynomials. Finally, in Section 5, we provide some pictorial comparisons between the bound given in Theorem 1.5 and a refinement of Hoeffding's that is due to Zheng [START_REF] Zheng | A refined Hoeffding's upper tail probability bound for sums of independent random variables[END_REF].

F ic (t) that minimize 1 f (t) E[f (B

Proof of Theorem 1.4

In this section we prove Theorem 1.4, which allows to improve upon Hoeffding' bound by suitably choosing function from the class F ic (t). Notice that Theorem 1.4 implies that there may be some space for improvement upon Hoeffding's bound. We will employ this result and en route find a function φ ∈ F ic (t) such that

E[φ(B)] < inf h>0 e -ht E[e hB ],
where B ∼ Bin(n, p). Hence there is indeed space for improvement upon Hoeffding's bound.

The proof of Theorem 1.4 will require some well-known results and the following notion of ordering between random variables (see [START_REF] Shaked | Stochastic Orders[END_REF]). Definition 2.1. Let X and Y be two random variables such that

E[f (X)] ≤ E[f (Y )]
, for all convex functions f : R → R, provided the expectations exist. Then X is said to be smaller than Y in the convex order, denoted X ≤ cx Y .

We begin with the following, well-known, result whose proof is included for the sake of completeness.

E[f (X)] ≤ E[f (B)].
Proof. Given X, we couple the random variables by setting B X to be either equal to a with probability b-X b-a , or equal to b with probability X-a b-a . It is easy to see that E[B X |X] = X and so

E[B X ] = E[E[B X |X]] = E[X] = p. Jensen's inequality now implies that E[f (X)] = E[f (E[B X |X])] ≤ E [E[f (B X |X)]] = E[f (B X )],
as required.

The following two results are well-known (see Theorems 3.A.12 and 3.A.37 in [START_REF] Shaked | Stochastic Orders[END_REF] and Theorem 4 in [START_REF] Hoeffding | Probability inequalities for sums of bounded random variables[END_REF]). The first one shows that convex order is closed under convolutions.

Lemma 2.3. Let X 1 , . . . , X n be a set of independent random variables and let Y 1 , . . . , Y n be another set of independent random variables.

If X i ≤ cx Y i , for i = 1, . . . , n, then n i=1 X i ≤ cx n i=1 Y i .
The next lemma shows that a sum of independent Bernoulli random variables is dominated, in the sense of convex order, by a certain binomial random variable.

Lemma 2.4. Fix n real numbers p 1 , . . . , p n from (0, 1). Let B 1 , . . . , B n be independent Bernoulli random variables with B i ∼ Ber(p i ). Then

n i=1 B i ≤ cx B,
where B ∼ Bin(n, p) is a binomial random variable of parameters n and p := 1 n i p i . We are now ready to prove the first statement of Theorem 1.4.

Proof of Theorem 1.4, a). Fix f ∈ F ic (t). Since f (•) is non-negative, increasing in [t, ∞) and f (t) = 1, Markov's inequality implies that

P n i=1 X i ≥ t ≤ E f n i=1 X i . Since f (•) is convex, Lemmata 2.2 and 2.3 imply that E f n i=1 X i ≤ E f n i=1 B i ,
where

B i ∼ Ber(E[X i ]), i = 1, . . . , n. Now Lemma 2.4 yields E f n i=1 B i ≤ E [f (B)]
and the result follows.

Similar ideas as above have been employed to sums of independent Bernoulli random variables by León and Perron in [START_REF] León | Extremal properties of sums of Bernoulli random variables[END_REF].

We now proceed with the second statement of Theorem 1.4. Let the random variables X 1 , . . . , X n be independent and such that 0 ≤ X i ≤ 1, for each i = 1, . . . , n. Set p = 1 n n i=1 E[X i ] and fix a real number t such that np < t < n. We have already seen that, for every f ∈ F ic (t), it holds

P n i=1 X i ≥ t ≤ E [f (B)]
, where B ∼ Bin(n, p).

Notice that

E[f (B)] = n i=0 f (i) • P[B = i]. Notice also that a function f ∈ F ic (t) that minimizes E[f (B)] has to be such that E[f (B)] ≤ np t ; indeed, since np t is the bound on P [ i X i ≥ t]
given by Markov's inequality (or by Theorem 1.4,(a) applied to the function f (x) = x/t) it follows that an optimal function has to provide a bound that is at least as good. Hence, for the purpose of characterising the function that minimize E[f (B)], we may assume that f belongs to the class F * ic (t), where

F * ic (t) consists of all functions in f ∈ F ic (t) that satisfy E[f (B)] ≤ np t . The following result characterizes the functions f ∈ F * ic (t) that minimize E[f (B)]. Theorem 2.5. Let f ∈ F * ic (t). Then there exists ε ∈ [0, t) such that E [φ ε (B)] ≤ E [f (B)], where φ ε (x) = max{0, 1 t-ε • (x -ε)}. Proof.
Let m t := min{n ∈ N : t < n} be the smallest positive integer that is strictly larger than t. Note that, by definition, 0 < m t -t ≤ 1. Let ε = tf (mt)-mt f (mt)-1 and for x ≥ 0 define the function

φ ε (x) := max 0, 1 t -ε (x -ε) .
In other words, φ ε (•) equals zero for x < ε and for x ≥ ε it is a straight line starting from point (ε, 0) ∈ R 2 and passing through the points (t, f (t)) and (m t , f (m t )). Since the function f (•) is convex it follows that for every integer i in the interval [0, n] we have φ ε (i) ≤ f (i) and this, in turn, implies

E [φ ε (B)] ≤ E [f (B)]
. Clearly, we have ε < t and it remains to show that ε ≥ 0. Indeed, if ε < 0, then φ ε (0) > 0 and the function f 1 (x) = 1-φε(0)

t x + φ ε (0) is such that np t + φ ε (0) 1 - np t = E[f 1 (B)] ≤ E[f (B)].
This implies that E[f (B)] is even worse than the bound obtained by Markov's inequality, and contradicts the assumption that f ∈ F * ic (t). The result follows. In other words, Theorem 2.5 implies that, in order to minimize

E[f (B)] for f ∈ F * ic (t), it is enough to consider functions of the form max{0, 1 t-ε • (x -ε)}, for ε ∈ [0, t).
The following result is an immediate consequence of Theorem 2.5 and finishes the proof of the second statement of Theorem 1.4.

Corollary 2.6. Let the parameters n, p, t be as in Theorem 1.4. Then for any t ∈ (np, n) we have

inf f E[f (B)] = inf a<t 1 t -a E [max{0, B -a}] ,
where B ∼ Bin(n, p) and the infimum on the left hand side is taken over all functions f ∈ F ic (t).

Notice that we can write the function ρ ε (x) := max{0, 1 t-ε • (x -ε)}, for ε ∈ [0, t), in the form g h (x) := max{0, h • (x -t) + 1}, where h = 1 t-ε , and that this correspondence is injective. Notice also that, since ε ≥ 0, we have h ≥ 1/t. The following question arises naturally from Corollary 2.6. Question 2.7. What is the optimal ε such that

inf a<t 1 t -a E [max{0, B -a}] = E [ρ ε (B)] ?
We remark that such an ε will satisfy ε ≤ t -1, where t := min{k ∈ N : t ≤ k}. To see this notice that if ε > t -1, then ρ ε ( t -1) = 0 and we may decrease ε, until it reaches the point t -1, without increasing the value E [ρ ε (B)]. Since ε ≤ t -1 it follows that h ≤ 1 t-t +1 . Now, finding the optimal ε is equivalent to finding the optimal h. We are not able to find this h. Nevertheless, due to the following result, one can easily find h using, say, a binary search algorithm. Proposition 2.8. Let the parameters n, p, t be as in Theorem 1.4. Let h > 0 be such that

E [max{0, h • (B -t) + 1}] = inf s>0 E [max{0, s • (B -t) + 1}] ,
where B ∼ Bin(n, p). Then we may assume that h = 1 t-j , for some positive integer j ∈ {0, 1, . . . , t -1}.

Proof. Recall that h ∈ 1 t , 1 t+1-t .
We have

E[g h (B)] = n i=0 n i p i (1 -p) n-i • g h (i).
It is clear that the function E(h) := E[g h (B)] is linear on the interval 1 t-j , 1 t-j-1 , for every j ∈ {0, 1, . . . , t -1}. Hence the function E(h) is continuous and piecewise linear on the interval 1 t , 1 t-t +1 and this implies that it attains its minimum at the endpoints of 1 t-j , 1 t-j-1 , for some j ∈ {0, 1, . . . , t -1}. The result follows.

In the next section we obtain an improvement upon Hoeffding's bound.

Proof of Theorem 1.5

This section contains the proof of Theorem 1.5.

Proof of Theorem 1.5. Given h > 0 define the function f (x) = max{0, h(x -t) + 1}, for x ≥ 0. Clearly, we have f ∈ F ic (t). Let m t be the largest positive integer for which f (m t ) = 0. Using Theorem 1.4 and the inequality e x > 1 + x, for x ∈ R, we estimate

P n i=1 X i ≥ t ≤ E[f (B)] = n i=mt+1 (h(i -t) + 1)P[B = i] < n i=mt+1 e h(i-t) P[B = i]
≤ H(n, p, t), which shows that E[f (B)] is strictly smaller than Hoeffding's bound. Since we assume that t ≥ epn ep-p+1 it follows that h ≥ 1 which in turn implies, since t is an integer, that f (i) = 0, for all i ∈ {0, 1, . . . , t -1}. Hence we can write

H(n, p, t) -E[f (B)] = n i=0 e h(i-t) P[B = i] - n i=t+1 (h(i -t) + 1)P[B = i] = t-1 i=0 e h(i-t) P[B = i] + n i=t+1 e h(i-t) -(h(i -t) + 1) P[B = i]. For i ≥ t + 1, we have e h(i-t) -(h(i -t) + 1) = 1 - 1 + h(i -t) e h(i-t) e h(i-t) ≥ 1 - 1 + h e h e h(i-t)
which implies that

H(n, p, t) -E[f (B)] ≥ 1 - 1 + h e h H(n, p, t) + 1 + h e h • t-1 i=0 e h(i-t) P[B = i] - 1 - 1 + h e h P [B n,p = t] .
The result follows.

4. Proof of Theorem 1.6

In this section we prove Theorem 1.6. The proof borrows ideas from the theory of Bernstein polynomials (see Phillips [START_REF] Phillips | Interpolation and approximation by polynomials[END_REF]Chapter 7]). Recall that, for a function f : [0, 1] → R, the Bernstein polynomial corresponding to f is defined as

B m (f, x) = m j=0 m j x j (1 -x) n-j f (j/m) ,
for each positive integer m. The following is a folklore result regarding Bernstein polynomials.

Lemma 4.1. If f : [0, 1] → [0, ∞) is convex, then f (x) ≤ B m (f, x), for all x ∈ [0, 1]. If f : [0, 1] → [0, ∞) is continuous, then sup x∈[0,1] |f (x) -B m (f, x)| → 0, as m → ∞.
Proof. See [START_REF] Phillips | Interpolation and approximation by polynomials[END_REF] Theorems 7.1.5 and 7.1.8. We remark that the first statement is easy to prove and the second arose from Bernstein's search for a proof of Weierstrass' theorem.

We can now provide a proof of Theorem 1.6.

Proof of Theorem 1.6. Let f ∈ F ic (t). Since f is non-negative and increasing and f (t) = 1, Markov's inequality yields

P n i=1 X i ≥ t ≤ E f n i=1 X i . Since f is convex and X i ∈ [0, 1], Lemma 4.1 implies that E [f (X i )] ≤ E [B m (f, X i )] . Now note that E [B m (f, X i )] = m j=0 m j • E X j i (1 -X i ) m-j • f (j/m).
For j = 0, 1 . . . , m let

π j := m j • E X j i (1 -X i ) m-j .
Notice also that

E X j i (1 -X i ) m-j = m-j k=0 m -j k (-1) m-j-k µ i,m-k which implies that E X j i (1 -X i ) m-j
is the same for all random variables from the class B(µ i,1 , . . . , µ i,m ). It is easy to verify that m j=0 π j = 1. Now, if we define the random variable Z i that takes on the value j m with probability π j , j = 0, 1, . . . , m, we have

E[f (X i )] ≤ E[B m (f, X i )] = E[f (Z i )]
, for convex f . Since the convex order is closed under convolutions, by Lemma 2.3, the first statement follows. The proof of the second statement is almost identical to the proof of Theorem 2.5 and is left to the reader.

Comparisons

In this section we perform pictorial comparisons between the bound given by Theorem 1.5 and a refinement of Hoeffding's bound which is due to Zheng [START_REF] Zheng | A refined Hoeffding's upper tail probability bound for sums of independent random variables[END_REF]. In [START_REF] Zheng | A refined Hoeffding's upper tail probability bound for sums of independent random variables[END_REF] it is shown, under the same assumptions as in Theorem 1.5, that certain refinements of the arithmetic-geometric means inequality yield the following estimate: Let us remark that comparisons between the bound provided in Theorem 1.5 and the bound given in (1) require quite tedious calculations. However, it is rather straightforward to put the computer to work and see that the two bounds are quite close to each other.

More precisely, let Z(n, p, t) be the right hand side of (1) and let Y (n, p, t) be the bound given in Theorem 1.5. In Figure 1 we fix the value of n, draw n numbers p 1 , . . . , p n from [0, 1] (which serve as the expected values of the random variables) uniformly at random and then plot the function f (t) = Y (n, p, t ) -Z(n, p, t) for t ∈ enp ep-p+1 , n . For moderate vales of t the bound given by (1) performs slightly better than our bound; while for larger values of t the two bounds are almost equal. In all cases, the two bounds are very close to each other. Notice that as n gets larger the difference between the bounds appears to get closer to zero.

Lemma 2 . 2 .

 22 Fix real numbers a, b such that a < b. Let X be a random variable that takes values on the interval [a, b] and is such that E[X] = p. Let B be the random variable that takes on the values a and b with probabilities b-E[X] b-a and E[X]-a b-a , respectively. Then for any convex function, f : [a, b] → R, we have

X

  i ≥ t ≤ w -t 1 -p + pw -σ 2 (w -1) 2 2w n , for np < t < n, where w = -B+ √ B 2 -4AC 2A and A = (1 -t/n)(p -σ 2 /2), B = -t n (1 -p + σ 2 ), C = σ 2 2 (1 + t/n) and σ 2 = 1 n n i=1 (E[X i ] -p) 2 .

Figure 1 .

 1 Figure 1. Pictorial comparisons between the bound of Theorem 1.5 and the bound given in Zheng [24].

Acknowledgements. The authors were supported by European Research Council Starting Grant 240186 "MiGraNT, Mining Graphs and Networks: a Theory-based approach". We are grateful to Dr. Xiequan Fan and to an anonymous referee for valuable suggestions and comments that improved the presentation of the paper.