Pascal Barbin

Jean-Luc Cheval

Jean-François Gilis

Pierre Strehaiano

Patricia Taillandier
email: patricia.taillandier@ensiacet.fr

Diversity in Spoilage Yeast Dekkera/Brettanomyces bruxellensis Isolated from French Red Wine. Assessment During Fermentation of Synthetic Wine Medium

Keywords: Brettanomyces bruxellensis, diversity, wine, growth profiles, 4ethylphenol

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The genus Dekkera, and its asporogenous form Brettanomyces, are well recognized as spoilage yeasts of industrial alcoholic fermentations [START_REF] De Miniac | Contamination des fermentations alcooliques industrielles par les levures du genre Brettanomyces[END_REF][START_REF] Délia-Dupuy | Contamination par les levures Brettanomyces dans les fermentations alcooliques[END_REF] and fermented beverages such as beer and wine 5, [START_REF] Fugelsang | Population dynamics and effects of Brettanomyces bruxellensis strains on Pinot Noir wines[END_REF][START_REF] Larue | Incidence du développement de Dekkera/Brettanomyces dans les moûts et les vins[END_REF][START_REF] Loureiro | Spoilage yeast in the wine industry[END_REF] . In red wine this contaminant is responsible for odour and taste defaults qualified as horsy, sweaty and mousy 2 . These unpleasant aromas are due to volatile phenols such as 4-ethylphenol and 4-ethylgaiacol 2 and other volatile compounds, including tetrahydropyridines (the "mouse taint"). B. bruxellensis, the main species naturally present in the wine environment [START_REF] Monje | Determination of ethylphenol compounds in wine by headspace solid-phase microextraction in conjunction with gas chromatography and flame ionization detection[END_REF][START_REF] De Souza Liberal | Identification of Dekkera bruxellensis as a major contaminant yeast in continuous fuel ethanol fermentation[END_REF] , has been shown to synthesize the most important quantities of these phenolic volatile compounds, from the natural hydroxycinnamic acids present in grape musts: namely para-coumaric and ferulic acid [START_REF] Dias | Identification of yeast isolated from wine-related environments and capable of producing 4-ethylphenol[END_REF][START_REF] Heresztyn | Metabolism of phenolic compounds from hydroxycinnamic acids by Brettanomyces yeasts[END_REF] . These faults cause large economical losses [START_REF] Fugelsang | Yeast and molds[END_REF] . In the last fifteen years, studies regarding these microorganisms have multiplied in a significant way 2, 7, [START_REF] Guilloux-Benatier | Influence de l'autolyse des levures après fermentation sur le développement de Brettanomyces/Dekkera dans le vin[END_REF][START_REF] Renouf | Global survey of the microbial ecosystem during alcoholic fermentation in wine making[END_REF][START_REF] De Souza Liberal | Identification of Dekkera bruxellensis as a major contaminant yeast in continuous fuel ethanol fermentation[END_REF] and indicates a desire to understand the behaviour of Brettanomyces and to prevent or at least control its development in the winemaking process.

Nevertheless, if one tries to synthesize and summarize the different works that have been produced regarding Brettanomyces in winemaking, it seems that accurate and definitive conclusions cannot be generalized given that each work has been performed with different strains from private collections. Two questions arise. Do all strains have similar behaviours and also whether in a given cellar? Does a dominant contaminant strain for a given vintage exist? Little data are available in literature regarding these points. Two studies have compared isolates from different wine regions, but not from the same cellars 7,8 . Our study thus consisted of screening B. bruxellensis strains isolated from wines coming from two cellars in order to determine if they had similar or different characteristics, and focusing on their behaviour during fermentation on synthetic wine medium.

MATERIALS AND METHODS

Microorganisms.

Isolation: Strains of Brettanomyces used in this study (designated as strains B to Z) were isolated from red wines. The isolation procedure consisted of micro-morphological and physiological assessments, using selective media (testing resistance to cycloheximidine, acetic acid production, and conversion of p-coumaric into volatile phenols). Strain A was a reference strain isolated from a distillery plant [START_REF] Uscanga | Nutritional requirements of Brettanomyces bruxellensis: growth and physiology in batch and chemostat cultures[END_REF] and identified by the Institute of Hygiene and Epidemiology of Brussels (THEM) as B. bruxellensis strain 6037. A PCR assay, using primers designed to the 26S rRNA gene was used to confirm that the isolated strains belonged to B. bruxellensis species [START_REF] Phister | Real-time PCR assay for detection and enumeration of Dekkera bruxellensis in wine[END_REF] . Specific primers were: DBRUXF: 5'-GGATGGGTGCACCTGGTTTACAC-3' and DBRUXR: 5'-GAAGGGCCACATTCACGAACCCCG -3'.

Molecular typing by PCR Internal Transcribed Spacer (ITS), Enzymatic

Restriction (ER) and Pulsed Field Electrophoresis (PFE):

After DNA isolation and dilution [START_REF] Querol | Molecular monitoring of wine fermentations conducted by dry yeasts strains[END_REF] , the rRNA coding region was amplified with primer pairs ITS1 (5' TCC GTA GGT GAA CCT GCG G 3') and ITS4 (5' TCC TCC GCT TAT TGA TAT GC 3') [START_REF] Guillamón | Rapid identification of wine yeast species based on the ribosomal internal transcribed spacer (ITS) region[END_REF] .

PCR products were digested with restriction enzymes HinfI and TaqI. Restriction segments were pulsed-field-electrophoresed and ethidium bromide was used for staining before photography.

Media and culture conditions.

Strain Conservation: All strains were stored at 4°C on Petri dishes containing a medium consisting of 10 g of glucose, 5 g of yeast extract (Oxoid,UK ), 20 g of agar and 1 L of H 2 0.

Preculture: A two step preculture was performed for activation and acclimatation.

The first one was carried out in 150 mL Erlenmeyer flasks with 75 mL of liquid medium [PreLM1] consisting of : 10 g of glucose, 10 g fructose, 1 g yeast extract, 2 g (NH 4 ) 2 SO 4 , 5 g KH 2 PO 4 , 0.4 g MgSO 4 .7H 2 O, 3 g tartaric acid, 3 g malic acid, 0.3 g citric acid and 1 L of H 2 0. After inoculation by a single colony from an agar plate, each flask was incubated at 30°C and magnetically stirred at 150 rpm for 72 h using a magnetic stirrer. The second preculture was inoculated at 10 6 cells mL -1 , incubated at 20°C and stirred at 150 rpm. For this step, 250 mL Erlenmeyer flasks with 150 mL of [PreLM2] medium were employed, similar to [PreLM1] but with 4 % (v/v) ethanol. This preculture was used for inoculating the main fermentation, when it reached the mid-exponential growth phase.

Fermentations: Fermentations were carried out in 500 mL Erlenmeyer flasks with a 300 mL initial working volume of Synthetic Wine Medium [SWM] containing the following: 2 g glucose, 8 g fructose, 0.5 g yeast extract, 0.4 g (NH 4 ) 2 SO 4 , 5 g of KH 2 PO 4 , 0.4 g MgSO 4 .7H 2 O, 3 g tartaric acid, 3 g malic acid, 0.3 g citric acid, 6 g glycerol, 5 mg p-coumaric acid, 10% (v/v) ethanol and 1 L of H 2 0.. Batch fermentations were inoculated with 10 5 cells mL -1 and were incubated at 20°C with shaking at 150 rpm.

The initial pH of all liquid cultures was adjusted to pH 3.4 with 10 M NaOH. All liquid media were sterilized at 120°C for 30 min (when containing ethanol, this was added after autoclaving and the ethanol was previously sterilized by filtration through a 0.2 µm porosity filters).

The reproducibility of the fermentations was tested by conducting the B strain experiments in triplicate under standard conditions. The standard deviation was less than 10% for fermentation duration, biomass dry weight, sugars and ethanol concentration measurements.

Analytical techniques.

Microorganisms. Growth was followed by numeration (cells mL -1 ) using a Thoma hemacytometer and an Olympus BH-2 microscope -x 400 enlargement. Standard deviation was less than 13 % for all cultivated strains. Final biomass concentration was also determined as dry weight, using a gravimetric method. To determine dry weight, 10 mL of culture sample was centrifuged for 10 min at 2500 rpm. The cells were washed twice with distilled water and dried using a humidity analyser (model HA60, Precisa, Zurich, Switzerland).

Sugars. Total sugar concentration was determined using the dinitrosalicylic colorimetric method [START_REF] Millet | The viable but non-culturable state of wine micro-organisms during storage[END_REF] . Glucose was assayed with an enzymatic glucose analyser 2700 Select (YSI, Yellow Springs, Ohio) using glucose-oxydase.

4-ethylphenol. The concentration of 4-ethylphenol was measured by headspace

Solid-Phase Micro-Extraction coupled with Gas Chromatography and Flame Ionization Detection [START_REF] Monje | Determination of ethylphenol compounds in wine by headspace solid-phase microextraction in conjunction with gas chromatography and flame ionization detection[END_REF] . Calibrations were performed in the synthetic wine medium. The standard deviation was less than 6%.

RESULTS

Genetic identification and typing.

Brettanomyces strains used in this study (designated as strains B to Z) were isolated from post alcoholic fermentation red wines, sampled from two cellars in the south-western region of France (Madiran and Buzet) between 2001 and 2004 (Table I).

The first PCR step confirmed that all 24 isolates belonged to the Brettanomyces bruxellensis species, validating the isolation procedures (results not shown).

Further genetic typing using the PCR ITS and an enzymatic restriction (HinfI and TaqI) coupled with PFE was performed in order to screen the isolates at an intraspecies level. Electrophoresis profiles are presented in Fig. 1 and there were no strict identical isolates with the exception of two strains. Strains B and M appeared to have exactly the same profile. Since they were isolated from the same sample, we can assume that the 2 isolates are the same strain. Nine strains were selected from the 23 identified. To compare their behaviour during fermentation, two strains (D and H) among the 10 strains isolated from Madiran Winery and seven strains (B, O, Q, S, V, W and X) among the fourteen from the Buzet winery, were chosen arbitrarily from both vintages.

Behaviour during fermentation.

In order to control the parameters, the experiment was performed with a synthetic wine medium [SWM]. To test the strains in an enology context, the synthetic medium was close to a wine at the final phase of alcoholic fermentation (but without any polyphenols). Total cell numeration was chosen to avoid under evaluating the population in regard to cells in a viable but non-cultivable state (viable and active cells that are not able to grow on classical solid medium), since this is often the case in long term cultures and under stressful conditions such as in alcohol-containing medium 7,25 , Moreover high metabolic activity was noted until the end of the experiments.

Growth profiles. Fig. 2 shows the growth profile comparison for the strains and indicates that there was no unique profile. Even if all of the strains had presented a long lag phase (the shortest being strain X with 200 h) and despite an inoculation from a preculture in mid-exponential growth phase, the profiles differed for both the kinetics and the population levels. The four main growth patterns could be differentiated.

Type 1: a single growth period with a long stationary phase after an exponential growth phase (strains B and W).

Type 2: two growth periods. A first phase with a classical exponential growth period followed by a constant and almost linear second growth phase (strains D and Q).

Type 3: three periods. A first phase with an exponential growth period, followed by a relatively long stationary phase, then a second growth with another exponential phase (strains H and S).

Type 4: close to Type 3, but with a second growth phase that was not exponential, it was slower (strains O, V and X).

Sugars consumption.

As shown in Fig. 3, several behaviours were detected for the sugar consumption profiles.

Type 1: the rapid exhaustion of both glucose and fructose in one phase ; with strains B, O and X there was a very fast utilization of the sugars, strain Q was slower.

Type 2: a very slow and progressive consumption of sugars; strain D to 0 g L -1 , while strain W was unable to consume all the initial sugar and left 3.4 g L -1 of residual sugar.

Type 3: a two step consumption of sugars, separated by a long stationary phase. At the end of fermentation the concentration was 0 g L -1 ; strain S entered stationary phase at 6 g L -1 at 500 h and strain V at 500 h was at stationary phase with 0.5 g L -1 . Strain H was classified in the last group with a slow first step sugar consumption, followed by a short 250 h stationary phase, and then a second sugar consumption to a residual concentration of 0.2 g L -1 .

All nine strains consumed both glucose and fructose. While glucose was always depleted before 1400 h, fructose was not depleted with strains H and W. Moreover, all strains started consuming glucose before fructose, but fructose consumption always started before glucose totally disappeared from the medium (results not shown). For strains B, O, Q and X, the total depletion of sugars occurred simultaneously to the first growth phase (Fig. 2 and Fig. 3).

4-ethylphenol profiles.

Table II shows significant differences between the nine strains in term of 4-ethylphenol production. Both the maximal concentration (from 0.35 mg L -1 to 2.773 mg L -1 ) and the time necessary to reach this level differed from one strain to the next (from 600 h to 1400 h with the majority at 1300 h). Moreover, when growth profile is superimposed on the 4-ethylphenol production pattern, several observations can be made. Indeed, some strains such as B, W and X appeared to have a direct "coupling" between growth and 4-ethylphenol production as shown in Figure 4.A.

for strain B. But for the other strains (H, O, S, V, X) the production of 4-ethylphenol was detected without observing any growth between 500 and 900 h (as illustrated for strain V on Fig. 4.B.). Nevertheless the production of 4-ethylphenol during the stationary growth phase was very weak for strains O and X (data not shown). On the opposite, strain Q presented significant growth without significant production of 4ethylphenol between 300 h and 500 h of culture (Figure 4.C.).

DISCUSSION

Except for strain B also referenced as M, the genetic screening of the 24 isolates showed [START_REF] Shinohara | Distribution of phenolic yeasts and production of phenolic off-flavors in wine fermentations[END_REF] has also shown intraspecific variability among industrial isolates contaminating one bio-ethanol plant. On the contrary, Mitrakul et al. [START_REF] Mitrakul | Discrimination of Dekkera/Brettanomyces yeast isolates from wine by using various DNA fingerprinting methods[END_REF] suggested the predominance of a single strain on spoiled wines from several vintages produced in a California winery. Our observations suggest that different kinds of contamination (single or multiple strain/s) may occur depending on the winery. Moreover one can hypothesize that the origin of the contamination could be different according to the tanks, and coming not only from the cellar, but for example, also from the vineyard. However, the genotype of the isolates was not further studied. In a recent publication 8 it was reported that 244 isolates from 31 winemaking regions in Australia, resulted in 8 genotypes being found, suggesting common points of origin of some strains.

In the second part of this study, nine strains were tested during the fermentation of synthetic wine medium, and nine different profiles were obtained, reflecting a significant diversity in metabolic and physiological behaviour. All strains presented a very long lag phase, and some strains presented a stationary period between two growth steps. The existence of such long periods, without any noticeable population increase, before significant growth must be taken into account during monitoring in winemaking.

It suggests that a low cell concentration of Brettanomyces observed on one day, which would suggest a no-risk situation in short term, might result in high contamination on the following days, leading to spoilage if preventive procedures and care are not taken.

Among the nine growth profiles four types were distinguished. One could hypothesize what might have occurred during the different growth steps. For growth with one single step, such as with strain B, a main substrate could be converted into biomass.

Comparisons between the growth profile and sugar consumption profile clearly identified the sugars [glucose + fructose] as the limiting substrate. Once these substrates were depleted, the strain was unable to use the other substrates to increase its population.

Strain W also had single growth period, but it behaved differently in its sugar consumption. Sugars were progressively depleted both during growth and the stationary phase. A limiting substrate, different from sugars, could cause the growth stop. For two step growth profiles, with a linear second phase (growth pattern of type 2) one can propose that it was a limitation of several necessary compounds (or at least one). These compounds may be provided progressively in limited proportions, for example dissolved oxygen. Growth profiles of type 3 appear like diauxic profiles. Indeed previous works have reported that yeasts of the genera Dekkera/Brettanomyces were able to shift from glucose, to a secondary substrate, to start a second growth phase. The two growth phases were separated by a new latent period, necessary for enzymatic adaptation. Such diauxic growth has been described when specific sugars, [glucose + sucrose] or [glucose + cellobiose], were used as the carbon source 1, [START_REF] Gilis | Etude de contamination de fermentations alcooliques industrielles par les levures Brettanomyces[END_REF][START_REF] Gondé | Selection of yeast strains for cellobiose alcoholic fermentation[END_REF] , with glucose always being consumed first. Glucose and fructose were the only sugars used in the medium for the experiments reported here and the diauxic profile did not match with a sequential consumption of these two sugars. It appeared that for strains H, S and V, the two phases of sugars consumption (both glucose and fructose), were separated by a constant sugar concentration phase, and were in parallel with the two phases of growth (Fig. 2 and3). These three strains appeared to use the sugars for the two steps of growth.

Until now, no clear explanation can be proposed for the two separated-phase phenomenom. One can assume that a compound necessary for growth was in default and that the yeast readapted its metabolism before pursuing growth (eg. nitrogen sources have to been known to have a limiting compound effect). Growth profiles of type 4, could be explained as a combination of diauxy and a limitation during the second growth phase. Sugars were depleted during the first growth step of strains O and X. and their stationary phase lasted over 500 h, before there was a new slow and progressive increase in cellular concentration.

For all strains examined, glucose was always consumed before fructose, but fructose was consumed while glucose was still available at low concentrations. Such observations were also made by Gilis 15 who suggested a competition between these two sugars for the same transporter. Other authors have demonstrated the co-existence of two transport systems for glucose [START_REF] Silva | Studies on the wine spoilage capacity of Brettanomyces/Dekkera spp[END_REF] .

The different hypotheses proposed suggest the existence of other substrates that have not yet been identified. Ethanol is well known for being a carbon [START_REF] Rodrigues | Development and use of a new medium to detect yeasts of the genera Dekkera/Brettanomyces[END_REF] and energy source for Brettanomyces spp. under both aerobic and semi -aerobic conditions [START_REF] Freer | Acetic acid production by Dekkera/Brettanomyces yeasts[END_REF][START_REF] Geros | Biochemical studies on the production of acetic acid by the yeast Dekkera anomala[END_REF] . Cottonplugged and shake flask fermentations would be considered as micro-aerobic, resulting in favorable conditions for ethanol reduction and acetic acid production 6 . But in this case, ethanol analysis (data not shown) showed that ethanol was consumed by all the strains in the second part of fermentation (13 to 25 g/L) without any correlation to the second growth phase when it existed.

Moreover results from Table III lay the emphasis on significant diversity with 4ethylphenol production and show important differences between the strains, both in terms of the maximal amount of volatile phenol and the time necessary to reach this concentration. The biotransformation leading to volatile phenol synthesis implies two enzymatic reactions: the first step is a decarboxylation of the phenolic acids into vinyl phenols, the latter being transformed during the second step into ethyl phenols by a vinyl reductase. The second reaction is very active in B. bruxellensis, compared to other yeasts or bacteria present in a wine related environment [START_REF] Dias | Identification of yeast isolated from wine-related environments and capable of producing 4-ethylphenol[END_REF] . Investigating p-coumaric acid metabolism by this yeast, the results suggest that the conversion is not directly linked to the metabolism pathways for growth (e.g, strains produce 4-ethylphenol during the nongrowth phase). It is also important to note that the highest 4-ethylphenol production was not linked to the highest cell concentrations (Table II). For instance, strain W had a maximum population of 0.5 g.L -1 and produced 2.749 mg.L -1 of 4-ethylphenol, while strain H population reached 2.6 g.L -1 and produced 0.35 mg L -1 of 4-ethylphenol.

Brettanomyces strains appear to have a different specific efficiency for converting pcoumaric acid. This result was consistent with a study by Conterno et al 8 who showed that one of the most discriminating characteristics for physiological diversity among 47 strains was the production of 4-ethylphenol. A study comparing 6 strains 14 pointed out that significant variation was noted for growth rate, final population level and 4ethylphenol production. Similar observations have also been reported from wineries. Some spoiled wines present high concentration of 4-ethylphenol, but a relatively low cell concentration of Brettanomyces (suggests there are high producing strains), while others have a high Brettanomyces population but relatively low 4-ethylphenol quantities and no "Brett taint" (suggests low producing strains). Consequently, a low level of contaminant in the wine may not prevent from the wine spoiling if the implicated strain is a strong 4-ethylphenol producer.

A major expectation for winemakers is to have effective tools in hand to prevent and control any risk of microbial spoilage, especially regarding Brettanomyces. This study shows that it would be hasty to draw conclusions from experiments with only one or two strains as being accurate and representative since strains appear to behave differently one from another, thus reflecting the huge diversity amongst the species. This diversity also recently been reported by Conterno et al. 8 for other characteristics, such as potential substrates, and it makes it more difficult to propose a curative method to protect against the risk of Brettanomyces contamination.

ACKNOWLEDGEMENTS.

This study was partly supported by La Cave Coopérative des Vignerons de Buzet (47160 Buzet sur Baïse, France) who provided technical help and samples.

REFERENCES.

1-Blondin, B., Ratomahenina, R., Arnoud, A. and Galzy P., Purification and properties of the β-glucosidase of a yeast capable of fermenting cellobiose to ethanol: Dekkera intermedia. Van Der Walt. Appl. Microbiol. Biotechnol., 1993, 7, 1-6.

  23 different ER-PFE profiles, implying that 23 different strains of Brettanomyces bruxellensis were present. No prevalent strain was identified from either a given site or for a given vintage. From a given cellar and during one vintage, isolates coming from different tanks (five from Madiran 2001, five from Madiran 2002, six from Buzet 2003 and eight from Buzet 2004) were identified as different strains, proving the richness of the contamination pool. Research by de Souza et al.

FIGURESFig. 1 .

 1 FIGURES

Fig. 2 .

 2 Fig. 2. Growth profiles for nine strains of Brettanomyces bruxellensis during fermentation of synthetic wine medium at 20°C.

Figure 2 .

 2 Figure 2.A: Strains O (Buzet 2003), strains W and X and S (Buzet 2004).

Figure 2 .

 2 Figure 2.B: Strains B, Q (Buzet 2003), H (Madiran 2002), V (Buzet 2004) and D (Madiran 2001).

Fig. 3 .

 3 Fig. 3. Total sugar utilization by nine strains of Brettanomyces bruxellensis during fermentation of synthetic wine medium at 20°C.

Figure 3

 3 Figure 3.a: Sugar for strains O (Buzet 2003), strains W and X and S (Buzet 2004).

Figure 3

 3 Figure 3.b: Sugar for strains B, Q (Buzet 2003), H (Madiran 2002), V (Buzet 2004) and D (Madiran 2001).

Fig. 4 .

 4 Fig. 4. Growth profiles and 4-ethylphenol production by three strains of Brettanomyces bruxellensis during fermentation of synthetic wine medium at 20°C

Figure 1 .Figure 2

 12 FIGURES

Figure 2

 2 Figure 2.B: Strains B, Q (Buzet 2003), H (Madiran 2002), V (Buzet 2004) and D (Madiran 2001).

Figure 2 .Figure 3

 23 Figure 2. Growth profiles for nine strains of Brettanomyces bruxellensis during fermentation of synthetic wine medium at 20°C.

Figure 3

 3 Figure 3.b: Sugar consumption by strains B, Q (Buzet 2003), H (Madiran 2002), V (Buzet 2004) and D (Madiran 2001).

Figure 3 :

 3 Figure 3: Total sugars consumptions by nine strains of Brettanomyces bruxellensis during fermentation of synthetic wine medium at 20°C.

Figure 4

 4 Figure 4.a: Strain B.

Figure 4 .

 4 Figure 4. Growth profiles and 4-ethylphenol productions by three strains of Brettanomyces bruxellensis during fermentation of synthetic wine medium at 20°C.

Table II .

 II Maximal 4-ethylphenol concentration and associated time and biomass concentration.

Table II .

 II Maximal 4-ethylphenol concentration and associated time and biomass concentration.

	Strains	Cmax (mg/L)	Biomass (g/L)	Time (h)
	B	2.59	1.88	600
	D	2.773	1.35	1300
	H	0.35	2.6	1400
	O	2.762	1.55	1300
	Q	1.895	1.54	1300
	S	2.022	0.68	1300
	V	0.453	2.08	1300
	W	2.749	0.5	800
	X	1.228	0.94	1400


L Amour

L Jager

J Nourrigat

Ground state photon number at large distance

Keywords: Ground state, photon localization, photon number, photon density, photon asymptotic, nuclear magnetic resonance, quantum electrodynamics

Ground state photon number at large distance

1 Introduction.

This article is concerned with the ground states of a Hamiltonian describing Nuclear Magnetic Resonance (NMR) in the framework of Quantum Electrodynamics (QED), in particular with the localization in position space in R 3 of the photons in these states.

NMR is the spin interaction of a finite number of static particles of R 3 with a constant magnetic field and with the quantized magnetic field. This phenomenon is represented by a Hamiltonian H(g) depending on a positive parameter g (the coupling constant) and acting on a Hilbert space H tot . This Hamiltonian, introduced by Cohen-Tannoudji, Dupont-Roc and Grynberg [START_REF] Cohen-Tannoudji | Photons et atomes. Introduction à l'électrodynamique quantique, InterEditions[END_REF] (see also Reuse [START_REF] Reuse | Electrodynamique et Optique Quantiques[END_REF]), may also be seen as a simplification of the Pauli-Fierz Hamiltonian [6,12]. It is recalled in Section 2.

The Hilbert space H tot is a completed tensor product H tot = H ph ⊗ H sp , where H ph is the space of the photons and H sp the space of the particles with spin (see Section 2). The space H ph is a Fock space, in which the usual operators, as the number operator N or the annihilation operators a(k), k ∈ R 3 , are defined.

If f belongs to the domain D N 1/2 ⊗ I then one can localize the photons in a given state f of the Hilbert space H tot , in momentum space, as well as in position space. We recall in Appendix A that, for all f in D N 1/2 , one can define a function (annihilation operator) denoted k → a(k)f defined almost everywhere on R 3 and taking values in H 3 ph , belonging to L 2 (R 3 , H 3 ph ) and that, the expectation of photon number in this state f is given by:

N 1/2 f 2 = R 3 a(k)f 2 dk.
For our model, if f ∈ D N 1/2 ⊗ I then the expectation of the photon number is:

(N 1/2 ⊗ I)f 2 = R 3 (a(k) ⊗ I)f 2 dk. 1
One can localize photons in momentum space in the following way. If E is a Borel set of R 3 , one usually considers that, in the state f ∈ D(N 1/2 ⊗ I), the expectation of the number of photons with a momentum belonging to E is given by:

E (a(k) ⊗ I)f 2 dk.
The localization in position space is less studied but it can be nevertheless analyzed using the Fourier transform.

For all f in D(N 1/2 ⊗ I), since the function k → (a(k) ⊗ I)f belongs to L 2 (R 3 , H 3 tot ), one can define its Fourier transform, which is also in L 2 (R 3 , H 3 tot ). We denote this transform by x → ( a(x) ⊗ I)f . By abuse of notation, one usually writes:

( a(x) ⊗ I)f = R 3 e -ix•k (a(k) ⊗ I)f dk. (1.1)
One has:

(2π) -3 R 3 ( a(x) ⊗ I)f 2 dx = (N ⊗ I)f, f .
Since (N ⊗I)f, f is the average number of photons, one can consider that the average number of photons with a position lying in a Borel set E of R 3 is given by:

N E f, f = (2π) -3 E a(x) ⊗ I)f 2 dx. (1.2)
In this way, the function x → ( a(x) ⊗ I)f 2 can be seen as the density, in position space, of the number of photons in the state f ∈ H tot . We note that, the problem of the localization of photons is mentioned in Reuse [START_REF] Reuse | Electrodynamique et Optique Quantiques[END_REF] (page 296-297). See also [26] and [4]. Let us mention that the operator N E defined in (1.2) is identical to the operator dΓ(1 E (i∇ k )) introduced in [11] for other issues. Our objective is to study this function when f is a state of minimal energy of the Hamiltonian given in [START_REF] Cohen-Tannoudji | Photons et atomes. Introduction à l'électrodynamique quantique, InterEditions[END_REF]. It was shown in [6,19,[START_REF] Spohn | [END_REF]13] that the infimum of the spectrum is an eigenvalue. This result is recalled in Theorem 2.1.

The eigenvectors are called ground states. Their properties are studied in, e.g., [23,5,24,10]. It is proved in [13] that, if the constant external field is not vanishing and if the coupling constant is sufficiently small then the subspace of ground states is one dimensional. It is well known (see [10,21], see also [19] and [6,13]) that every ground state U g of the Hamiltonian H(g) belongs to the domain D(N m ⊗ I) for an arbitrary m ≥ 0. It is also well known that the ground state is not in the photon vacuum. See [13] for estimates of the average number of photons in the state U g .

Consequently, if the constant external field is not vanishing and if the coupling constant g is small enough, one can study the function x → ( a(x) ⊗ I)U g 2 when U g is a normalized ground state, which is the purpose of our main result, namely Theorem 1.1. This function is supposed to describe the localization (in position space) of the photons in a ground state.

The main result is the following. Theorem 1.1. Suppose that B ext = 0. Let U g be a normalized ground state of Theorem 2.1, where the coupling constant g is small enough to ensure that the space of ground states has dimension 1. Then:

1. The function x → |x| 5/2 ( a(x) ⊗ I)U g is bounded and continuous on R 3 , with values in H 3 tot .

2. For all unit vectors v of R 3 , one has:

lim |x|→∞ |x| 5/2 ( a(|x|v) ⊗ I)U g = - 3 √ 2 χ(0)(v × S [tot] )U g
where χ is the "ultraviolet cut-off function", used in (2.1) for the definition of the magnetic field operator, and where S [tot] is the expectation of total spin in the ground state U g , i.e., the vector of R 3 with coordinates:

S [tot] j = P λ=1 (I ⊗ σ [λ] j )U g , U g
where σ

[λ] j is defined in (2.3).
In physics, one often supposes that χ(0) = 1. In particular, for the local density of the photons of the ground state, there exists C > 0 such that:

( a(x) ⊗ I)U g 2 ≤ C(1 + |x|) -5 .
Since U g is normalized, one has, for all unit vectors v ∈ R 3 :

lim |x|→∞ |x| 5 ( a(|x|v) ⊗ I)U g 2 = (9/2)|χ(0)| 2 |(v × S [tot] )| 2 .
One sees that, for large distances, the photons are fewer in the direction of the expectation of the total spin.

Section 2 is devoted to the precise description of the model, Section 3 gives the proof of the main result (Theorem 1.1) and the appendices recall some facts concerning annihilation operators and the Pull Through Formula.

2 The Hamiltonian and its ground state.

The Hilbert space of the states of our system is a completed tensor product H ph ⊗ H sp , where H ph is the Hilbert space of the free photons and H sp , the space of the particles with spin.

Photons. The one photon configuration Hilbert space H is the set of mappings

f ∈ L 2 (R 3 , R 3 ) satisfying k • f (k) = 0 almost everywhere in k ∈ R 3 (see [29]) where |f | 2 = R 3 |f (k)| 2 dk.
Then, H C stands for its standard complexification and one denotes by f, g the scalar product of two elements f and g of H C , where the mapping g → f, g is antilinear. The Hilbert space H ph of photon quantum states is the symmetrized Fock space F s (H C ) over H C the complexified space of H. We follow [START_REF] Reed | Methods of modern mathematical physics[END_REF] for Fock spaces considerations and notations, in particular, for the usual operators in these spaces: the Segal field Φ S (V ) associated with an element V in H 2 , the Γ(T ) and dΓ(T ) operators associated with some operator T acting in H 2 . Note that, throughout this paper, the space H 2 is sometimes identified to the complexified space H C but this identification is not everywhere systematically effectuated in order to avoid possible confusions.

Let M ω be the operator with domain

D(M ω ) ⊂ H such that M ω q(k) = |k|q(k) almost everywhere in k ∈ R 3 .
We denote in the same way the analogous operators defined on H 2 or on the complexified space H C . In the Fock space framework, the photon free energy Hamiltonian operator H ph is usually defined as H ph = dΓ(M ω ).

The photon number operator denoted by N is N = dΓ(I).

The three components of the magnetic field at each point x in R 3 are defined using the elements B jx belonging to H 2 and written as follows, when one identifies H 2 with the complexified space H C :

B jx (k) = iχ(|k|)|k| 1 2 (2π) 3 2 e -i(k•x) k × e j |k| , k ∈ R 3 \{0} (2.1)
where the function χ (ultraviolet cutoff) belongs to S(R).

One then defines the magnetic fields components operators at each point x of R 3 by:

B m (x) = Φ S (B mx ), for m = 1, 2, 3.
Spins. The configuration space of the system of P spins is then the space H sp = (C 2 ) ⊗P . The fermion property for the spin-1 2 fixed particles is omitted here. In the space H sp , we use the operators related to the spins of the different particles. Let σ j (1 ≤ j ≤ 3) be the Pauli matrices:

σ 1 = 0 1 1 0 , σ 2 = 0 -i i 0 , σ 3 = 1 0 0 -1 . (2.2)
For all λ ≤ P and all m ≤ 3, we denote by σ

[λ]
m the operator in H sp defined by:

σ [λ] m = I ⊗ • • • ⊗ I ⊗ σ m ⊗ I ⊗ • • • ⊗ I, (2.3)
where σ m is located at the λ th position.

The Hamiltonian. This Hamiltonian is often used for modeling NMR in quantum field theory (see [30,27] and Section 4.11 of [START_REF] Reuse | Electrodynamique et Optique Quantiques[END_REF]). It is a selfadjoint extension of the following operator, initially defined in a dense subspace of H ph ⊗ H sp :

H(g) = H 0 + gH int , (2.4)
where g is a positive constant and:

H 0 = H ph ⊗ I + P λ=1 3 m=1 B ext m ⊗ σ [λ] m (2.5)
where H ph = dΓ(M ω ) is the photon free energy operator, acting in a domain D(H ph ) ⊂ H ph and

B ext = (B ext 1 , B ext 2 , B ext
3 ) = 0 is the constant magnetic field. Moreover:

H int = P λ=1 3 m=1 B m (x λ ) ⊗ σ [λ] m (2.6)
and the x λ (1 ≤ λ ≤ N ) are the points of R 3 where the static particles are located.

If an element U of H 2 lies in the domain D(M

-1/2 ω
) then the Segal field Φ S (U ) is bounded from D(H ph ) into H ph , see point ii) of Proposition 3.4 in [1] or see [START_REF] Dereziński | [END_REF]. This is therefore the case for the operators B j (x) according to the assumptions on the ultraviolet cutoff function χ in (2.1). Thus, according to the Kato-Rellich Theorem, H(g) has a selfadjoint extension with the same domain as the free operator H ph ⊗ I domain.

The ground state. Let us now recall the results of [19], [START_REF] Spohn | [END_REF] and [13] about the ground state of this Hamiltonian, or more precisely of Hamiltonians very close to this one.

Theorem 2.1. The operator H(g) defined by (2.4), (2.5) and (2.6), admits a selfadjoint extension with the same domain as the free operator H 0 . There exists a unit element U g of D(H 0 ), such that H(g)U g = E(g)U g , where E(g) is the infimum of the spectrum of H(g). This element U g is in the domain of N m ⊗ I, for all m ≥ 0. If g is small enough and if B ext = 0, this element U g is unique up to a phase factor. This theorem follows from [13] (Theorem 1 page 447), and, for the uniqueness, from [START_REF] Spohn | [END_REF] and [22].

3 Asymptotic behavior at infinity of the photon number.

Theorem 1.1 is proved in this Section. We first note that, according to equality (B.1), the mapping k → (a(k) ⊗ I)U g taking values in H 3 tot , belongs to L 1 (R 3 ), and consequently, its Fourier transform denoted by x → a(x)U g , also taking values in H 3 tot , is continuous on R 3 .

The following result is useful in order to prove the other points in Theorem 1.1.

Proposition 3.1. Let (H, D(H)) be a self-adjoint operator in a Hilbert space H and suppose that the spectrum of the operator H is the half line [E, ∞) (with E ∈ R). Then,

F (z, H, E) = z(H -E + z) -1
is a holomorphic mapping on the half plane {z ∈ C, Rez > 0}, continuous on the closure of this half plane without the origin, and with a norm smaller than or equal to one. Let (z n ) be a sequence of complex numbers satisfying Rez n ≥ 0 and converging to zero. Then, for any f ∈ H:

lim n→∞ F (z n , H, E)f = P f
where P is the orthogonal projection on ker(H -E).

Proof. Set ϕ the function defined on the spectrum of H by ϕ(E) = 1 and ϕ(x) = 0 if x > E. For any x belonging to the spectrum of H, set:

ϕ n (x) = z n z n + x -E .
If Rez n ≥ 0 then |ϕ n (x)| ≤ 1 for all x in the spectrum of H. In addition, ϕ n (x) tends to ϕ(x) for each x in the spectrum of H. According to a standard result (for example, see [START_REF] Sh | Spectral theory of selfadjoint operators in Hilbert space[END_REF] Lemma 3, Chapter 5, Section 4), ϕ n (H)f tends to ϕ(H)f for each f . Observe that ϕ n (H)f = F (z n , H, E)f . It is also well known (see [START_REF] Sh | Spectral theory of selfadjoint operators in Hilbert space[END_REF], Theorem 3, Chapter 6, Section 1) that ϕ(H) is the orthogonal projection on ker(H -E). The proof of the Proposition then follows.

The mapping ( a(x) ⊗ I)U g is given by (1.1) and (a(k) ⊗ I)U g by (B.1) and (2.1) where U g is the ground state. That is:

( a(x) ⊗ I)U g = - ig 4π 3 2 P λ=1 3 m=1 R 3 e -ix•k χ(|k|)|k| 1 2 k × e m |k| (H -E + |k|) -1 f [λ] m dk with f [λ] m = (I ⊗ σ [λ]
m )U g . In the sequel, χ(|k|) is approximated by χ(0)e -|k| . Therefore, the following function is under consideration:

b(x)U g = - ig 4π 3 2 P λ=1 3 m=1 R 3 e -ix•k χ(0)e -|k| |k| 1 2 k × e m |k| (H -E + |k|) -1 f [λ] m dk. (3.1)
Lemma 3.2. There exists C > 0 such that:

( a(x) ⊗ I)U g -b(x)U g ≤ Cg |x| 3 .
Proof of the Lemma. The standard measure on the unit sphere S 2 is denoted by µ. For any v ∈ S 2 , m ≤ 3 and λ > 0, one has:

S 2 e -iλv•ω (ω × e m )dµ(ω) = 4iπ(v × e m ) cos λ λ - sin λ λ 2 . (3.2)
One then deduces the existence of C > 0 satisfying:

( a(x) ⊗ I)U -b(x)U ≤ C 3 m=1 |I m (x)|
with, if x = |x|v, |v| = 1:

I m (x) = ∞ 0 cos |x|ρ |x|ρ - sin |x|ρ (|x|ρ) 2 Φ m (ρ)dρ where: Φ m (ρ) = ρ 5/2 χ(ρ) -χ(0)e -ρ (H -E + ρ) -1 f m .
One checks that:

∞ 0 sin(|x|ρ) |x| 2 ρ 2 (Φ m (ρ))dρ = 1 |x| 3 ∞ 0 cos(|x|ρ) d dρ Φ m (ρ) ρ 2 dρ
and:

∞ 0 cos(|x|ρ) |x|ρ (Φ m (ρ))dρ = 1 |x| 3 ∞ 0 cos(|x|ρ) d 2 dρ 2 Φ m (ρ) ρ dρ.
These integrations by parts holds true since as ρ → 0:

d dρ Φ m (ρ) ρ 2 + d 2 dρ 2 Φ m (ρ) ρ ≤ C ρ 1/2 .
Moreover, the left hand side is rapidly decreasing as ρ → ∞. The proof is thus completed.

Proof of Theorem 1.1. It is sufficient to consider b(x)U g . For each v ∈ S 2 , one sees using (3.1) and (3.2):

b(|x|v)U g = χ(0)π -1/2 P λ=1 3 m=1 v × e m I [λ] m,aux (|x|)
with:

I [λ] m,aux (|x|) = ∞ 0 cos |x|ρ |x|ρ - sin |x|ρ (|x|ρ) 2 ρ 5/2 e -ρ (H -E + ρ) -1 f [λ] m dρ.
One has:

I [λ] m,aux (|x|) = 2 |x| 5/2 ∞ 0 (t 2 cos(t 2 ) -sin(t 2 ))e -t 2 /|x| F ( t 2 |x| , H, E)f [λ] m dt (3.3)
where

F (s, H, E) = s(H -E + s) -1 .
We can write

I [λ]
m,aux = I λ+ m,aux + I λ- m,aux setting:

I λε m,aux = 1 |x| 5/2 ∞ 0 e εit 2 (t 2 + εi)e -t 2 /|x| F t 2 |x| , H, E f [λ] m dt (3.4)
with ε = ±. If |x| ≥ 1, a change of contour of integration shows that:

I λε m,aux = 1 |x| 5/2 (εie εiπ/4 ) ∞ 0 (r 2 + 1)e -r 2 e -εir 2 /|x| F εir 2 |x| , H, E f [λ] m dr. (3.5)
It is therefore deduced that:

|x| 5/2 I [λ] m,aux (|x|) ≤ 2|f [λ] m | ∞ 0 (r 2 + 1)e -r 2 dr. (3.6)
Consequently, the function x → |x| 5/2 |b(x)| is bounded on {|x| ≥ 1} proving point i) of the Theorem when using Lemma 3.2.

Using Lebesgue's dominated convergence Theorem, point ii) comes from:

lim |x|→∞ |x| 5/2 I λε m,aux (|x|) = (εie εiπ/4 ) lim z→0,Rez≥0 F (z, H, E)f [λ] m ∞ 0 (r 2 + 1)e -r 2 dr.
The existence of the above limit comes from Proposition 3.1 which also shows:

lim z→0,Rez≥0 F (z, H, E)f [λ] m = P f m = f [λ] m , U g U g
where P is the projection on the eigenspace of the infimum of the spectrum of H(g). Therefore:

lim |x|→∞ |x| 5/2 b(|x|v)U g = -2/πχ(0) P λ=1 3 m=1 v × e m (I ⊗ σ [λ] m )U g , U g U g ∞ 0 (r 2 + 1)e -r 2 dr = -2/πχ(0)(v × S [tot] )U g ∞ 0 (r 2 + 1)e -r 2 dr = - 3 √ 2 χ(0)(v × S) [tot] U g
proving point ii) with the help of Lemma 3.2.

Appendices

A Standard facts on annihilation operators.

The following results are classical but adapted to the Hilbert space H in Section 2. We denote by F reg s (H C ) the subspace of F s (H C ) constituted by the finite linear combinations of the symmetrized products of

g 1 ⊗ • • • ⊗ g m where the g j ∈ S(R 3 , R 3 ) satisfy k • g j (k) = 0 for all k ∈ R 3 .
For all k ∈ R 3 and f ∈ F reg s (H C ), a(k)f is classically defined as follows. For any (non symmetrized) product g = g 1 ⊗ • • • ⊗ g m , with the g j satisfying the above conditions, one set:

a(k)g = √ mg 1 (k)g 2 ⊗ • • • ⊗ g m .
This definition is next adapted to the symmetrized space. Since the function g 1 takes values in R 3 , one sees that k → a(k)f takes values in (F s (H C )) 3 and belongs to S(R 3 ). One also notes:

N f, f = R 3 a(k)f 2 dk.
The next two propositions are useful for the purpose of an extension by density in D(N 1/2 ).

Proposition A.1. S(R 3 , R 3 ) ∩ H is dense in H.

Proof. Set f ∈ H and ρ a C ∞ smooth cut-off function defined on R + , vanishing on [0, 1] and equal to one on [2, +∞). One has:

lim n→∞ R 3 |f (k) -f (k)ρ(n|k| 2 )| 2 dk = 0
using Lebesgue Theorem. The above norm | | is the one in R 3 . Thus, the set H 0 of functions in H vanishing on a ball centered at the origin is dense in H. Take f ∈ H 0 vanishing on a ball centered at 0 with radius η > 0. There exist sequences f

(n) i of S(R 3 , R) converging to f i , for i = 1, 2, 3
(n is the coordinate index for these sequences). It is naturally not clear that:

3 1 k i f (n) i (k) = 0
and therefore that f (n) (whose components are the f (n) i ) belongs to H. We then set, again with a cut-off function:

g (n) = (f (n) - k |k| 2 f (n) • k) ρ( 4 η 2 |k| 2 ). (A.1)
One checks that g (n) ∈ H ∩ S(R 3 , R 3 ) and tends to f in H, i.e., with the L 2 (R 3 , R 3 ) norm.

Proposition A.2. F reg s (H C ) is dense in D(N 1/2 ).
Proof. One chooses a basis (u i ) of H in H ∩ S(R 3 , R 3 ) and sets:

ζ α = |α|! α! S n ⊗ i u ⊗αi i , n = |α|. (A.2)
The set of ζ α (with |α| = n) is an orthonormal basis of ⊗ n s H (see Janson [START_REF] Janson | Gaussian Hilbert spaces[END_REF]). Thus, the set of finite linear combinations of ζ α (with arbitrary |α|) is dense in the set of finite number particles F fin (H) which is dense in D(N 1/2 ). The proof of the Proposition is then complete since the set of finite linear combinations of ζ α is included in F reg s (H C ).

We can now extend the definition of a(k) to D(N 1/2 ).

Proposition A.3. Set f ∈ D(N 1/2 ) and a sequence (f n ) in F reg s (H C ) converging to f in D(N 1/2
). Then, the k → a(k)f n has a limit in L 2 (R 3 , (F s (H C )) 3 ). This limit is denoted (definition) by a(k)f . Indeed, according to the previous points, one has if m < n:

R 3 a(k)(f m -f n ) 2 dk ≤ ( N 1/2 f m + N 1/2 f n ) N 1/2 (f m -f n )
and one also has (ϕ ∈ D(N 1/2 )):

N 1/2 ϕ 2 = R 3
a(k)ϕ 2 dk.

(A.3)

B Pull Through Formula

Let U g be a normalized ground state given by Theorem 2.1. It is recalled in Theorem 2.1 that U g belongs to the domain N m ⊗ I for all integers m. According to Proposition A.3, the function k → (a(k) ⊗ I)U g is well defined as an element of L 2 (R 3 , H 3 tot ). In particular, this function is defined almost everywhere. We give in the next result an explicit expression of this function, recalling the Pull through formula and its proof (see [13] and see also [START_REF] Schweber | An introduction to relativistic quantum field theory[END_REF][START_REF] Glimm | Statistical Mechanics and Quantum Field Theory[END_REF]12,19,[START_REF] Glimm | Quantum field theory and statistical mechanics[END_REF]6]). The uniqueness of the ground state is not useful. For all k = 0, the operator (H(g) -E(g) + |k|) is invertible. One then deduces equality (B.1).

Theorem B. 1 . 3 m=1B 3 m=1 3 m=1[ 3 m=1B

 13333 Let U g be a (normalized) ground state of Theorem 2.1. Then, for almost every k ∈ R 3 \ {0}:(a(k) ⊗ I)U g = -(gm,x λ (k)(H(g) -E(g) + |k|) -1 (I ⊗ σ [λ] m )U g (B.1)where B m,x λ (k) is defined in (2.1). Consequently:(a(k) ⊗ I)U g ≤ (g|B m,x λ (k)| |k| (I ⊗ σ [λ] m )U g . (B.2)Proof. In view of Lemma 2.5 in[START_REF] Dereziński | [END_REF]:[dΓ(M ω ), a(k)] = -|k|a(k). (B.3)Thus, from (2.4) and (2.5), for almost every k in R 3 :(H(g) -E(g) + |k|)(a(k) ⊗ I)U g = (a(k) ⊗ I)(H(g) -E(g))U g + g[H int , (a(k) ⊗ I)]U g .One has:[H int , (a(k) ⊗ I)] = P λ=1 B m (x λ ), a(k)] ⊗ σ [λ] m . Besides: √ 2[B m (x λ ), a(k)] = [a ⋆ (B m,x λ ), a(k)] = -B m,x λ (k). (B.4)Consequently:[H int , (a(k) ⊗ I)] = -(1m,x λ (k)(I ⊗ σ [λ] m ).


Carlos David

Morales Peña

José Bruno De Araújo

Christophe Morlaas

Alexandre Chabory

Romain Pascaud

Marjorie Grzeskowiak

Gautier Mazingue

Polarized Circularly

Huygens

Circularly Polarized Huygens Source Antenna Based on Two Stacked Dielectric Resonators

Keywords: Dielectric resonator antenna, Huygens source, circular polarization, half-space radiation

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I. INTRODUCTION

The interest in small-size autonomous platforms is growing rapidly. UAVs are, for instance, more and more used in the fields of media, agriculture, and industry. Another example concerns the space industry where nanosatellites could yield a significant reduction in the launching and operation costs. These two applications share the common constraint of embedding complex systems without a large metallic plate for antenna support.

This integration constraint is important for antenna design in terms of size, polarization and radiation pattern. Such pattern requires improved radiation properties in the forward direction and low backward radiation despite the lack of a ground plane. These requirements can be satisfied by designing a Huygens source, i.e., an antenna with a half-space radiation pattern.

Based on works related to magneto-electrical sources to obtain circular polarization (CP) [START_REF] Wheeler | A helical antenna for circular polarization[END_REF], and a Huygens source radiation pattern [START_REF] Chlavin | A new antenna feed having equal e-and h-plane patterns[END_REF], some antennas with cardioid radiation pattern and CP properties have been proposed [START_REF] Alitalo | Design and realisation of an electrically small huygens source for circular polarisation[END_REF][4] [START_REF] Morlaas | Helical-ring antenna for hemispherical radiation in circular polarization[END_REF].

In a previous paper [START_REF] Morales | Huygens source antenna using stacked dielectric resonators[END_REF], the authors have presented a single-fed linearly polarized Huygens source antenna based on stacked dielectric resonators (DRs). As a new contribution, we present an original solution to design a Huygens source in CP. It consists of a cross-shaped antenna made of two stacked rectangular DRs with different electrical and physical properties.

II. ANTENNA PRINCIPLE AND DESIGN

A. Antenna principle

The antenna principle to design a Huygens source in CP relies on the orthogonal combination of two linearly polarized Huygens sources. According to the coordinate system shown in Fig. 1, the first Huygens source which is linearly polarized along the xz-plane, is due to the TE x 111 and quasi-TM y 111 modes in the DR, which radiate a magnetic and electric radiation pattern-like, respectively [START_REF] Petosa | Dielectric resonator antenna handbook[END_REF]. The quasi-TM y 111 is denoted here in reference to the corresponding cylindrical resonator mode. On the other hand, the second Huygens source which is linearly polarized along the yz-plane, is generated by the TE y 111 and quasi-TM x 111 modes in the DR.

B. Dielectric resonator antenna design

The dielectric resonator antenna (DRA) at f 0 = 2.45 GHz ISM band is designed to realize a Huygens source pattern, both theoretically using the dielectric waveguide model (DWM) [START_REF] Petosa | Dielectric resonator antenna handbook[END_REF] and numerically using eigen mode analysis in Ansys HFSS. The cross-shaped antenna is made up of two stacked DRs, as shown in Fig. 1. The stacked cross-shaped DR is fed using two coaxial probes 90 • out of phase to excite the TE x 111 and TE y 111 modes with the required phase difference in order to achieve CP. In addition to that, the quasi-TM x 111 and quasi-TM y 111 modes are excited using the electric field distribution of the x-directed and y-directed magnetic dipoles, respectively. Here, a proper frequency shift between TE and quasi-TM modes allows to satisfy the requirements to obtain CP. This frequency shift is accomplished by setting the electrical and physical properties of the stacked DR. A parametric analysis of the antenna dimensions is performed in order to adjust the frequency shift between TE and quasi-TM modes. The electrical and physical properties of the DRs are a = 13 mm, b = d = 31 mm, e = h = 20 mm, g = 13.5 mm, r1 = 11.5, and r2 = 18. Finally, the total electrical antenna dimensions are d + h = 0.41λ 0 , and b = 0.25λ 0 . The probes have a diameter of d f = 1.23 mm, and lengths of l p1 = 15.5 mm and l p2 = 15 mm. The probes are supported by small metallic plates with square dimensions of l m = 0.08λ 0 to ensure the impedance matching. Table I shows the eigen modes for the stacked cross-shaped DRA. As a result, we observe that the desired modes radiate around f 0 .

Fig. 2: Simulated reflection coefficient.

The total simulated front-to-back (F/B) and axial ratio (AR) of the antenna are shown in Fig. 3. The Huygens source antenna achieves the largest F/B ratio of 31.8 dB at 2.43 GHz. Besides, the DRA has a F/B ratio bandwidth (F/B ≥ 15 dB) of 5.3% from 2.38 GHz to 2.51 GHz, and an AR bandwidth (AR ≤ 3 dB) equal to 22.9% from 2.22 GHz to 2.78 GHz.

Fig. 4 illustrates the simulated radiation patterns of the proposed antenna. It corresponds to a Huygens source. Furthermore, the DRA operates in left-hand CP at 2.45 GHz with a maximum simulated gain at boresight of 5.93 dBi.

IV. CONCLUSION

A stacked cross-shaped DRA has been designed in circular polarization in the 2.45 GHz ISM band to achieve a Huygens source radiation pattern. The methodology relies on the combination of two orthogonal pairs of TE and quasi-TM modes.

The proposed DRA has an impedance bandwidth of 21.2%, and a 3-dB axial ratio bandwidth of 22.9%. Furthermore, it operates in left-hand circular polarization with a maximum simulated gain of 5.93 dBi at 2.45 GHz. Due to the need of a heterogeneous dielectric, additive manufacturing can be used to build the proposed antenna. Moreover, in order to manage the permittivity difference between the upper and lower DRs, a periodized structure of several unit cells using 3-D printing of ceramics can be as previously done in [START_REF] Thevenot | A dielectric resonator antenna designed with a structured dielectric material[END_REF].

Circularly

  Polarized Huygens Source Antenna Based on Two Stacked Dielectric Resonators C. D. Morales * † , J. B. de Araújo * † , C. Morlaas * , A. Chabory * , R. Pascaud † , M. Grzeskowiak † , G. Mazingue ‡

Fig. 1 :

 1 Fig. 1: Stacked cross-shaped DRA.

Fig. 3 :

 3 Fig.3: Simulated front-to-back and axial ratios.

Fig. 4 :

 4 Fig. 4: Simulated total antenna gain (dBi) at 2.45 GHz, considering φ = 0 • and φ = 90 • .

TABLE I :

 I Resonance frequency (GHz) of the eigen modes in the optimized cross-shaped DR. 11 and S 21 . The antenna has an impedance bandwidth of 21.2% from 2.26 GHz to 2.78 GHz, for a reflection coefficient of | S 11 |≤ 10 dB. Furthermore, a low coupling between the two coaxial probes ensures a good orthogonality between the desired TE and quasi-TM modes.

	Modes		TE x 111 TE y 111	TM x 111 TM y 111
	Resonance frequency		2.35	2.72
	III. ANTENNA PERFORMANCES
	Fig 2 shows the simulated S 2 2.1 2.2 2.3 2.4 -	2.5	2.6	2.7	2.8	2.9	3
	Frequency (GHz)

V. ACKNOWLEDGEMENT

The authors acknowledge the Région Occitanie and ENAC for their financial support.