Congrès de l'AFS 2023 -RT 11 : Consommation et numérique La place du numérique dans les choix de consommation alimentaire des familles de classes moyennes et supérieures Intro duction (1') Développement massif des outils numériques ces 20 dernières années → multiplication des sources d'information sur l'alimentation : * pages de conseils gouvernementaux ou professionnels * articles scientifiques en ligne * reportages et documentaires * vidéos YouTube * blogs d'information ou de recettes * sites de consommateurices * applications fournissant des informations sur les produits de la grande distribution

→ foisonnement de ce type d'informations prescriptives fondées sur des données « scientifiques », « médicales » ou sur des « journalistique » Séverine Gojard : parents de classes moyenne et supérieure plus susceptibles que les classes populaires de chercher de l'information scientifique ou journalistique sur laquelle fonder leurs choix de consommation PB : Comment les outils numériques sont-ils appropriés au quotidien par les parents de classes moyenne et supérieure dans le cadre de leurs choix de consommation alimentaire ? Comment cela oriente-t-il leurs choix de consommation quotidiens ? → On ne s'intéresse qu'aux outils permettant d'orienter les choix de consommation des familles, pas aux outils destinés à simplifier la logistique d'approvisionnement (ex : Drive).

Objectif : On présentera de trois formes d'appropriation des outils numériques participant au choix des produits alimentaires consommés par les familles. *normes nutritionnelles qui imposent de faire attention à l'équilibre globale des repas, à la diversité des apports, mais aussi à la qualité de chaque produits -ex pesticides, additifs) *normes écologiques et éthiques (bien-être animal, modes de production, origine des produits, etc.) -différenciation sociale des outils utilisés en fonction du capital culturel : plus le capital culturel est élevé, moins les parents utilisent des applications comme Yuka, fortement prescriptives, et plus semblent se tourner vers des documentaires, des reportages ou des vidéos YouTube, plus orientées vers l'information quant aux modes de production et aux conséquences écologiques.

-différenciation genrée des outils : les blogs de cuisine sont mobilisées par les femmes, en charge de la planification des repas |vs| Yuka majoritairement utilisée par les pères Travaux sur détachement et innovations par détachement → favoriser le fait d'apprendre à faire moins, se détacher de mauvaises consommations « Faire sans avec moins » Articulation des supports ? Complémentarité des usages ? Place des réseaux sociaux ? Articulation dans démarches plus globales de transition (moins prendre l'avion/la voiture) ?

  Thèse : On montrera que les usages du numérique dans le cadre des choix de consommation alimentaire des familles diffèrent selon deux modalités principales : le capital culturel et le genre. Méthode d'enquête (2') -enquête par entretiens auprès des parents et des enfants de 20 familles de classes moyenne et supérieure ayant au moins un enfant en primaire et habitant en zones urbaines ou périurbaines de la région lyonnaise -les deux parents ont été interrogés (si possible et généralement séparément) -les enfants ont également été interrogés (si consentant.es, ensemble) + observations informelles des espaces de vie et de cuisine → dans cet article, on mobilise principalement les données issues des entretiens avec les parents -les questions portaient sur : * les choix alimentaires des familles * l'éducation des enfants * le travail domestique impliqué par les repas : approvisionnement, recherche d'idées, préparation, rangement, etc. * recherche d'information et discussions autour de l'alimentation au sein de la famille et en-dehors 3 minutes 1 ) Les blogs culinaires et les réseaux sociaux (3'30) -usage dans presque toutes les familles rencontrées -permet d'alléger leur charge mentale tout leur permettant de proposer des menus diversifiés, qui est une des exigences des normes alimentaires dominantes : * équilibre des apports nutritionnels * diversité comme source de plaisir (éviter de s'ennuyer, éviter la routine) → Sanaz : « [Javed], il aime bien de faire quelque chose qu'il connait et tout, donc... C'est moi plutôt qui... qui vais vers l'aventure. » * important de faire aimer des saveurs diverses aux enfants (éduquer leur palais, leur permettre de manger chez des gens sans se plaindre) -mettre en avant les compétences de cuisine (occasions spéciales, notamment pour recevoir) en proposant des recettes originales -quasi-exclusivement les mères : * ce sont elles qui sont le plus souvent en charge de choisir les repas dans les familles : les deux parents cuisinent, mais la charge mentale de l'imagination revient aux femmes. Plusieurs disent que c'est une charge difficile de trouver des idées chaque jour. → notent les idées quand elles les voient passer/quand elles ont le temps pour avoir un porte-feuille d'idées quand elles en ont besoin. → manière d'être des « bonnes mères », nouvelle manière d'être les « mères nourricières » tout en partageant la charge culinaire avec leur conjoint → Nadège : « c'est ça la plaie en fait. C'est de toujours trouver l'idée » […] « je vais m'enregistrer sur Facebook, en fait... je suis dans ce groupe et puis... au fil des... des jours, si je trouve des recettes intéressantes, je vais me les enregistrer et puis je vais essayer de les faire assez rapidement... » → au contraire, son mari (un des rares à regarder des recettes en ligne) « flâne » pour trouver « une sauce » ou une recette particulière, pas de pression à faire un planning de repas pour la semaine, plutôt un repas du week-end occasionnellement. → Virginie : « Ben mon meilleur ami est Pinterest. Sur Pintesrest, voilà, y'a plein de recettes. Donc des bonnes alternatives si y'a des plats... Surtout pour les alternatives à tout ce qui est plats avec des produits laitiers.» * revalorisation de la cuisine domestique, qui reste relativement perçue comme féminine, dans le cadre de la sociabilité avec la famille élargie et les ami.es * blog majoritairement écrits par des femmes et adressés à des femmes 6'30 minutes 2 ) Yuka and co. (3'30) -usage dans plusieurs familles rencontrées -permet de choisir des produits conformes aux normes nutritionnelles dominantes : facilite les choix de consommation dans un contexte d'abondance et de diversité des produits proposés par la grande distribution et d'abondance normative contradictoire (Déchaux) : * manger des produits sains (équilibre nutritionnel, pesticides, additifs) * se faire plaisir et faire plaisir aux enfants → Yasmine : « quand je veux découvrir un produit ou pas, et ben je regarde toujours de toute façon. Et puis... et puiiiiis... et puis pour me rappeler aussi certains : c'est bien de revenir dessus et puis de se dire "au final, c'est vraiment pas très bon... on aime bien, mais faut pas non plus...". Donc ouais, je regarde tout le temps. » -avoir des arguments pour dire non aux enfants → Virginie : « J'ai ma petite application où je regarde un petit peu tout ce qui est édulcorants, additifs. Où là effectivement, je vais... mais je vais lui expliquer pourquoi, je vais lui dire : "Bah tu sais, ça, on peut peut-être essayer de le faire nous.".» * ne pas passer trop de temps à faire les courses ? : double emploi + temps familial -principalement dans des familles de classe moyenne → moindre aisance des classes moyennes avec les injonctions nutritionnelles mises en avant dans l'espace public et une « bonne volonté culturelle » (Bourdieu, 1979) |vs| tandis que ces pratiques sont intériorisées chez les classes supérieures, à la fois grâce à des connaissances scientifiques (Isabelle), un budget plus large permettant des achats de qualité supérieure (Anne et Albin, Pénélope), et un plus grand sentiment de maîtrise (capital culturel intériorisé, maîtrise des consommations distinctives, des normes, plus de facilité à jouer avec les normes distinctives) -plus utilisé par les pères que par les mères : → Nadège « Mon mari, c'est un fou de Yuka.» → moindre sentiment de maîtrise d'un domaine traditionnellement réservé aux femmes ? 10 minutes 3 ) Documentaires et vidéos YouTube (3,30) -usage moins fréquemment mentionné par les familles -utilisé principalement comme source d'information, notamment sur les questions écologiques et les questions touchant aux modes de production des produits alimentaires → entre dans un processus plus globale de transformation des pratiques alimentaires dans une perspective éthique et écologique, notamment diminution de la viande et des produits ultra-transformés -pas assez de données pour savoir si utilisations différentes selon le genre → mais rôle de prescription alimentaire plutôt adopté par les femmes, donc potentiellement plus la cible de ces contenus -outil éducatif : mobilisé par les parents, notamment les mères, pour faire circuler de l'information au sein de la famille, auprès des hommes pour faciliter leur adhésion aux transformations des pratiques alimentaires, et auprès des enfants pour transmettre les normes alimentaires, les valeurs éthiques et le capital culturel et politique. → exemple : Isabelle -plutôt classes supérieures à fort capital culturel → /!\ Faible nombre de familles qui mentionnent l'utiliser, donc besoin de plus d'enquêtes sur le sujet pour vérifier 13'30 minutes Conclusion (1') -outils utilisés dans comme soulagement de la charge mentale impliquée par les normes dominantes :


Zhengrui Tu
email: zhengrui.tu@universite-paris-saclay

Jianhao Zhang

Carlos Alonso-Ramos

Xavier Le Roux

Laurent Vivien

Eric Cassan

On-chip dual-band waveguide Bragg filter with identical subnanometer-bandwidth stopbands near 1310nm and 1950nm wavelengths

We propose the realization of an on-chip dual identical narrowband Bragg filter at ~1310nm and ~1950nm wavelengths simultaneously based on the silicon-on-insulator (SOI) platform. By taking advantage of a subwavelength corrugation behavior at large wavelengths and a huge difference in the mode areas of the involved modes at the two widely separated wavelengths, undesired diffraction losses are circumvented while achieving Bragg resonances at the two wavelengths simultaneously. A double corrugation Bragg grating rib waveguide filter is proposed, with two sets of gratings, the inner one close to the rib operating near 1310nm wavelength while the outer grating being designed to achieve a transmission dip around 1950nm. Introducing a proper lateral misalignment to the set of inner grating indents, a dual identical narrowband Bragg filter owning ~0.47nm 3dB bandwidth at ~1310nm and ~1950nm is achieved. The proposed design strategy based on SOI platform relies on a single-etching fabrication process and presents potential applications in all situations where equal-bandwidth filtering is needed in on-chip communications and sensing.

Introduction

Bragg gratings waveguide structures on different platforms, such as silicon-on-insulator (SOI), silicon nitride, Ⅲ-Ⅴ, etc., play a great role in various application fields, including lasers [START_REF] Pollnau | Optically pumped rare-earth-doped Al2O3 distributed-feedback lasers on silicon [Invited[END_REF], modulators [START_REF] Bedard | Dual phase-shift Bragg grating silicon photonic modulator operating up to 60 Gb/s[END_REF][START_REF] Jafari | Mach-Zehnder Silicon Photonic Modulator Assisted by Phase-Shifted Bragg Gratings[END_REF], filters [START_REF] Oser | Coherency-broken Bragg filters: overcoming on-chip rejection limitations[END_REF][START_REF] Zamek | Compact chip-scale filter based on curved waveguide Bragg gratings[END_REF], signal processing circuits [START_REF] Zhang | A fully reconfigurable waveguide Bragg grating for programmable photonic signal processing[END_REF], optical delay lines [START_REF] Khan | Demonstration of complementary apodized cascaded grating waveguides for tunable optical delay lines[END_REF][START_REF] Sun | Integrated Discretely Tunable Optical Delay Line Based on Step-Chirped Subwavelength Grating Waveguide Bragg Gratings[END_REF], on-chip sensors [START_REF] Prabhathan | Compact SOI nanowire refractive index sensor using phase shifted Bragg grating[END_REF][START_REF] Wang | A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide[END_REF], wavelength division multiplexers [START_REF] Wang | Low crosstalk Bragg grating/Mach-Zehnder interferometer optical add-drop multiplexer in silicon photonics[END_REF]. Several types of Bragg waveguide gratings, such as phased-shift [START_REF] Prabhathan | Compact SOI nanowire refractive index sensor using phase shifted Bragg grating[END_REF][START_REF] Wang | A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide[END_REF], apodized [START_REF] Khan | Demonstration of complementary apodized cascaded grating waveguides for tunable optical delay lines[END_REF], Moiré [START_REF] Cheng | Fully tailorable integrated-optic resonators based on chirped waveguide Moiré gratings[END_REF], sampled [START_REF] Wang | Uniform and sampled Bragg gratings in SOI strip waveguides with sidewall corrugations[END_REF], or claddingmodulated gratings [START_REF] Tan | Cladding-modulated Bragg gratings in silicon waveguides[END_REF][START_REF] Hung | Narrowband reflection from weakly coupled cladding-modulated Bragg gratings[END_REF][START_REF] Yen | Narrowband dual-wavelength silicon waveguide Bragg reflectors[END_REF], have been demonstrated and each of them can provide its own special advantages. The studies based on Bragg gratings are thriving in the past decades. Multiband operation is a valid option to significantly extend optical communication capacity [START_REF] Napoli | Towards multiband optical systems[END_REF]. However, there are few works only addressing how to successfully circumvent diffraction limitation resulting in achieving an on-chip dual identical narrowband Bragg filtering simultaneously at two separated far away wavelengths. A straightforward approach is to implement both gratings on the same waveguide simultaneously, but this is not without difficulty. Due to the huge difference between these involved wavelengths, the Bragg grating period for the longer wavelength can then be responsible for diffraction at the short aiming wavelengths, thus hindering the target dual function.

In this work, based on SOI platform, we fix as a typical example objective case where filtering at ~1310nm and ~1950nm with identical narrow stopbands are simultaneously needed. A rib waveguide geometry containing two sets of gratings is chosen (see Fig. 1 (a)) as in this type of waveguide, where a mode field can spread widely laterally in the slab, the introduction of corrugation that disturbs the profile of the guided mode is simpler than in a fully etched strip waveguide. One of the set of gratings (inner gratings), which is considered just close to the rib, is designed for ~1310nm Bragg wavelength operation. Another one is located far away enough to the rib (outer gratings) to generate the ~1950nm wavelength stopband. Section 2 describes the proposed methodology applied to obtain two stopbands with identical 3dB bandwidth (~0.47nm) at ~1310nm and ~1950nm over a ~1.3mm long rib slightly single-etched SOI waveguide. In section 3, simulation results are reported, demonstrating the soundness and the efficiency of proposed design strategy. Then, discussion and conclusion are given in section 4.

Principles and design

The proposed methodology is illustrated relying on a 220nm thick (Hcore) SOI wafer case. Preliminarily, in order to obtain a waveguide that can well support a TE-like polarization mode within a broadband wavelength range (covering the ~1310nm-~1950nm range in our case), a rib waveguide with rib width (Wrib) of 500nm and rib height (Hrib) of 100nm are chosen. The 3D schematic and 2D cross-section pictures of the rib waveguide are shown in Fig. 1(a) and Fig1. (b), respectively. Based on this rib waveguide structure, the effective refractive index for wavelengths ranging from 1250nm to 2000nm in TE-like foundational mode was calculated and the result is shown as the red curve in Fig. 2. According to the Bragg's law [START_REF] Bragg | The reflection of X-rays by crystals[END_REF], the corresponding grating period for the 1 st order Bragg wavelength can be derived from equation (1):

2 B eff n   (1)
where Λ is the grating period, λB is the 1 st order Bragg wavelength, and neff is the effective refractive index at λB. In Fig. 2, the blue curve gives out the relationship between the Bragg grating period and light wavelength. Two dashed vertical lines are labeled: the green one corresponds to a stopband center wavelength of 1310nm (B-) and the yellow one corresponds to a similar stopband wavelength of 1950nm (B+). From Fig. 2, one can find that the Bragg period for achieving a 1310nm stopband wavelength is around 240nm. This period is short enough to act as a sub-wavelength corrugation resulting in a homogenized effective material with respect to wavelengths around 1950 nm (/neff  870 nm>>240nm=B-) [START_REF] Tu | Doubly Resonant Distributed Feedback Cavity with Controllable Wide Wavelength Separation[END_REF][START_REF] Bock | Subwavelength grating periodic structures in silicon-on-insulator: a new type of microphotonic waveguide[END_REF][START_REF] Halir | Waveguide sub-wavelength structures: a review of principles and applications[END_REF][START_REF] Tu | High-sensitivity complex refractive index sensing based on Fano resonance in the subwavelength grating waveguide micro-ring resonator[END_REF]. However, the large period grating needed for achieving a stopband around B+=1950nm is simultaneously around 425 nm and can obviously give rise to diffractive effects [START_REF] Taillaert | An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers[END_REF] for short wavelengths around 1310nm. Fig. 3 shows the classical diffraction process diagram and the diffractive condition that can be expressed by equation ( 2

): , , , m z out z K K qK    ( 0, 1, 2,...) q    ( 2
)
where Km is the light waveguide mode wavevector, Km,z is the projection component of Km in the z direction along propagation; Kout is the wavevector of the outgoing light from the waveguide into the cladding, Kout,z is its projection component in z direction; KΛ is the grating vector and q is the diffraction order. These related parameters can be expressed as follows:

, , 2
2 sin sin 2 m z m eff out z out c K K n K K n K π λ π θ θ λ π                 (3)
where nc is the index of cladding and θ is the diffraction angle. Then, a further relationship can be derived from equation ( 2) and ( 3) to ( 4):

( sin ) In the typical studied example case made of an air cladded rib SOI waveguide, Λ  425nm can be adopted for the outer Bragg grating period for 1950nm, which leads to the conclusion that -1 st order diffraction effect can in principle happen at 1310nm wavelength with a diffraction angle θ  -16.25°. Obviously, this diffraction-based leakage channel can severely limit the target application of the realization of a low-loss dual wavelength rejection filter. It is thus necessary to circumvent this possible drawback.

c eff n n q      (4)
In fact, it turns out that it is sufficient to move the outer gratings far enough away from the waveguide axis to effectively limit this penalty. This simple approach leads to inner and outer gratings of quite distinct coupling strengths due to the imposed distance of the outer gratings. This discrepancy can then be compensated for by an asymmetrical realization of the internal gratings, which deterministically reduces its coupling strength, and ultimately equalizes the properties of the two gratings and thus equalizes their bandwidths. The practical implementation of this approach is illustrated in Fig. 4. With the SOI substrate parameters fixed, the main lines of the design are controlled by two parameters only, the lateral dimension of any corrugation (WG) and the lateral distance from the rib to the outer gratings (Gap). The grating corrugation width is set as WG = 50nm, thus remaining acceptable with respect to the fabrication process . Then the mode profiles at both target wavelengths are calculated as well as their corresponding effective mode areas through equation ( 5): To further realize dual identical narrowband Bragg filtering, the other key issue is to match the grating coupling coefficients of the two gratings. As well known, the bandwidth of a Bragg reflector can be derived from equation (6) [START_REF] Flanders | Grating filters for thin-film optical waveguides[END_REF]:

2 B g n L λ π λ κ π       ( 6
)
where Δλ is the bandwidth between the first nulls among the reflection spectrum, ng is the group index of the waveguide modes, κ is the coupling coefficient provided by the grating structure, and L represents the total grating length. From equation ( 6), one can see that both the coupling coefficient and the Bragg reflector total length determine the Bragg bandwidth.

We estimated the coupling coefficients at λBand λB+ based on the proposed rib waveguide including two sets of gratings for Gap = 400nm of each a 50% filling factor. With all the determined parameters, the coupling coefficients could be estimated and derived through coupled mode theory (CMT) [START_REF] Murphy | Design, Fabrication and Measurement of Integrated Bragg Grating Optical Filters[END_REF][START_REF] Wang | Silicon photonic waveguide Bragg gratings[END_REF][START_REF] Tu | Potential for sub-mm long erbium-doped composite silicon waveguide DFB lasers[END_REF]. κB-= 6.73×10 4 m -1 and κB+ = 1.13×10 3 m -1 were obtained, respectively. These different values are not surprising, as the two gratings are indeed positioned at very different distances from the center of the waveguide, but this situation does not allow, except for additional adjustment, an equalization of the bandwidths provided by each of the two gratings. To balance the two coupling coefficients, a misalignment between the left and right sides of the inner grating was further introduced [16,28] (see Figs. 6 (a) and (b)). The 500 nm wide SOI rib waveguide being single-mode at both wavelengths including 1310nm, this asymmetry does not create mode conversion between guided modes. In case a L misalignment is introduced between the two waveguide corrugated sidewalls, the grating coupling coefficient can be rewritten as follows:

0 0 0 exp( 2 / ) cos 22 L i L κ κ π κ π κ               ( 7
)
where κ0 is the coupling coefficient for a normal grating structure (i.e. misalignment), Λ the grating period, and 2πΔL/Λ is the phase offset between the two misaligned grating sidewall profiles. Accordingly, the inner grating coupling strength can be minimized to match the one of the outer grating. This approach is further developed in section 3 following.

Simulation results and discussion

To study and verify the proposed design strategy, device simulations based on 2.5D-FDTD [29] were implemented on 1.3mm long structures. 2.5-FDTD was applied due to the very heavy cost of full 3D electromagnetic simulation of Bragg mirrors of several hundreds of µm lengths. It is based on collapsing the 3D geometry into a 2D set of effective indices calculated by relying on a variational approach that can be solved with 2D FDTD [START_REF] Hammer | Effective index approximation of photonic crystal slabs: a 2-to-1-D assessment[END_REF]. The main hypothesis of this approach is that there is a weak coupling between different supported vertical slab modes. The method allowed us to study the different Bragg mirrors with sufficient accuracy to validate their principles of operation and to compare the initial designs.

For clarity, all the related simulation parameters are listed in Table 1. Meanwhile, the grating offset value of the inner grating is noted as follows: In a first step, no offset was introduced while the inter-grating distance value (Gap) was scanned from 100nm to 600nm. A series of transmission spectra were then acquired through 2.5D-FDTD simulation covering the 1200nm-2000nm range. The related results are plotted in Fig. 7(a) and Fig. 7(b) around 1310nm and 1950nm wavelengths, respectively. No attempt to apodize the spectral response of the studied filters [START_REF] Hastings | Optical waveguides with apodized sidewall gratings via spatial-phase-locked electron-beam lithography[END_REF] to minimize spectral sidelobes was tried as it was not the main purpose of the proposed study. From Fig. 7(a), it is observed that when Gap = 100nm, huge diffraction loss occurs around 1310nm, responsible for a low transmission level down to 15%, while losses are effectively blocked by moving away from the outer gratings as early as Gap = 300nm. On the other hand, in Fig. 7(b), the stopband around 1950nm becomes narrower and narrower with increasing Gap values, thereby progressively hiding the presence of a transmission dip around 1950nm wavelength. In view of previous achievements, the same fabrication constraints apply to these Bragg mirrors in terms of possible fabrication errors (under or over etching, stitching errors) [START_REF] Coppola | Effect of fabrication errors in channel waveguide Bragg gratings[END_REF]. Careful realization using DUV, or E-beam lithography and proper etching is necessary for their realization without affecting the principle of their implementation. As visible, the mirror reflectivity values at the Bragg wavelength do not reach 100% in several of the studied geometries. Previous works showed that narrow bandwidth Bragg filters with large reflectivity need long light pathways (e.g. [START_REF] Taillaert | An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers[END_REF] and equation ( 6)). To solutions could thus apply to improve the Bragg filter reflectivity levels: either increase the coupling strength of both gratings and match them in a second step while maintaining a constant grating overall length or, indeed, increase the Bragg filter lengths following previously proposed approaches to deepen the optical integration density [START_REF] Zamek | Compact chip-scale filter based on curved waveguide Bragg gratings[END_REF].

1 2 90 °L Offset    (8)
In a second step, Gap was thus set to 400nm while the inner grating offset was scanned from 0° to 90°. As visible in Fig. 8, the stopband near 1310nm then becomes narrower and narrower with increasing offset values, thus bringing a confirmation of a grating asymmetry. Meanwhile, we verified that in all conditions the stop band near 1950nm keeps constant whatever the inner grating offset value is. Based on the acquired data, 3dB bandwidth values of all configurations were gathered (shown in Fig. 9(a)). As visible and understandable, the 3dB bandwidth of the short wavelength stopband decreases with the inner grating offset increase while the second stopband near 1950nm keeps a constant 3dB bandwidth. As observable in Fig. 9(b), which is a zoom-in view from the green dot line window of Fig. 9(a), an inner grating offset value around 84° was identified to provide two stopbands near 1310nm and 1950nm with equivalent 3dB bandwidth (~0.47nm).

Conclusion

To conclude, a design strategy to successfully simultaneously realize a dual identical narrowband Bragg reflection filter at two widely separated wavelengths (~1310nm and ~1950nm in the investigated example study case) only requiring a single-etching fabrication process is proposed. A rib waveguide containing two sets of Bragg gratings was designed: the inner grating is close to the rib and owns a smaller period, thus providing a Bragg mirror effect for short wavelengths (~1310nm) while operating as a sub-wavelength effective index medium for longer wavelengths (~1950nm). The outer grating is located enough far away from the rib to avoid diffracting effects from the short wavelength compressed mode while providing a second Bragg stopband at and around ~1950nm. Diffraction effects being efficiently circumvented through a spatial lateral shift of the outer grating, the two sets Bragg gratings operate nearly independently. Additionally, a misalignment was introduced on the inner grating to balance the coupling strength coefficients between these two Bragg reflectors. Based on the determined parameters, simulation were carried to verify the feasibility of our proposed design and evaluate the needed offset value for the inner grating. As desired, the followed approach provided two narrow stopbands near 1310nm and 1950nm with identical 3dB bandwidth (~0.47nm). The proposed design strategy based on the silicon photonics platform relies on a single-etching fabrication process and presents potential applications in all situations where equal-bandwidth filtering is needed in on-chip communications and sensing.

Declaration of competing interest:

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 1 (

 1 Fig. 1 (a) 3D schematic (b) 2D cross-section view of the SOI rib waveguide.

Fig. 2

 2 Fig. 2 Red curve: relationship between wavelength and effective refractive index; blue curve: relationship between wavelength and Bragg grating period.

Fig. 3

 3 Fig. 3 Diagram for diffraction process.

Fig. 4

 4 Fig. 4 (a) 3D schematic (b) 2D cross-section view of the rib waveguide with two sets gratings.

  x,y) is the electrical field intensity and x, y are the dimensions along the waveguide cross-section. As expected, strongly different effective areas can be observed at B-= 1310nm (0.1554 μm 2 ) and B+=1950 nm (0.3614 μm 2 ) consistently with the mode profiles visible in Figs.5 (a) and (b). The Gap parameter was investigated to figure out the spacing required to optically isolate the second gratings. It can be seen in Fig.5 (c) that the effective index of the short-wavelength mode stabilizes beyond a gap of 400 nm. This is clearly the signature that the influence of the external corrugation on the short wavelength propagative mode then vanishes, thus favoring a reduction of the diffractive propagation loss level. An evaluation of this result is given hereafter in support of FDTD simulation.

Fig. 5 (

 5 Fig. 5 (a) Mode profile at λB-= 1310nm. (b) Mode profile at B+=1950nm.(c) Relationship between effective refractive index at 1310nm and different Gap values.

Fig. 6

 6 Fig. 6 Top view grating schematic (a) without misalignment; (b) with misalignment for inner grating.

Table 1 Parameters

 1 Rib width: Wrib = 500nm Rib height: Hrib = 100nm Both gratings' single pillar width: Wg = 50nm Inner grating's period: Λ1 = 230nm Single inner grating's length: Lg1 = 0.5•Λ1 = 115nm Outer grating's period: Λ2 = 426nm Single outer grating's length: Lg2 = 0.5•Λ2 = 213nm Number of inner grating periods: N1 = 5556 Number of outer grating periods: N2 = 3000 Dual wavelengths Bragg filter's total length: L = 1278μm

Fig. 7 2

 7 Fig. 7 2.5 FDTD-simulation results without offset on the inner grating while Gap varies from 100nm to 600nm with a 100nm step. (a) Normalized transmission near 1310nm wavelength; (b) Normalized transmission near 1950nm wavelength.

Fig. 8 2

 8 Fig. 8 2.5-FDTD simulation results with Gap = 400nm while the offset value varies from 0° to 90°: normalized transmission near 1310nm.

Fig. 9

 9 Fig.9 3dB bandwidth analysis derived from simulation results: (a) whole offset value window; (b) window near the cross-point between those two 3dB bandwidth stopbands

Acknowledgments:

We thank the OpticAll ANR French national project as well as the CSC for supporting this work.

Reference