Sebastián Marzetti

Pierre-Alexandre Peyronnet

Florent Barthélemy

Valentin Gies

Valentin Barchasz

Thomas Delaey-Langlois

Daniel Peloux

Philippe Arlotto

Hervé Barthélemy

Low Cost Artificial Ventilator Embedding Unsupervised Learning for Hardware Failure Detection Team

Keywords: Open source, Artificial Ventilator, Breathing Machine, Unsupervised Learning, K-Means, Machine Learning, Embedded Learning, Embedded system

In this paper, a less than $200 artificial ventilator that can be used against COVID-19 pandemic is presented. Using low-cost easy-to-find materials, it has been designed for helping developing countries where supplies for building new medical equipment are limited. It complies with medical requirements, allowing to monitor and adjust ventilation parameters such as tidal volume, maximum intra-lung pressure and breath rate. Even if this ventilator is low cost, focus has been placed on improving its global reliability. Using low-cost recycled materials may lead to mechanical failures, this potential drawback is addressed with an intelligent embedded hardware failure detector implemented inside the microcontroller. Using K-means optimized algorithm, it learns in a short time normal operation corresponding to the couple formed by a given ventilator set-up and a patient. In case of a mechanical breakdown, an alert is generated to inform medical staff. First, mechanical, electrical and software architectures of the system are presented, then hardware failure detection algorithm is detailed. Finally, test results done at IRBA using an artificial lung are discussed. The overall project has been published as an open source one on GitHub: https://github.com/iutgeiitoulon/ArtificialVentilator.

I. INTRODUCTION

During COVID-19 initial lockdown period, a low cost (less than $200) artificial ventilator has been designed at Toulon University, in France. It implements fully autonomous ventilation mode for reanimation, and assisted ventilation mode where patient breath start is detected and assisted by the ventilator. It complies with artificial ventilator medical requirements: limitation of inspired tidal volume, intra-lung pressure and adjustable breath rate [START_REF] Saxena | Respirator design[END_REF]- [START_REF] Stork | Artificial lung ventilator working in the hyperbaric chamber[END_REF].

Using easy-to-find low-cost materials such as fishing wire, aquarium tubing, hose and 3D printer stepper motors, this artificial ventilator can be built anywhere in the world through standard tools such as a jigsaw or a laser cutter. This makes it useful in emergency situations where it is not possible to have an access to a medically certified ventilator, such as during COVID-19 pandemic. However, quality of these materials is unknown. Therefore, in order to improve robustness, a In order to help people, Toulon University students and researchers have decided to open source this project. Electronics, mechanics as well as embedded and monitoring software have been shared on GitHub: https://github.com/iutgeiitoulon/ArtificialVentilator. They can be used or modified freely.

The article is divided in five parts. Sections III and IV present ventilator mechanics, electronics and embedded software architecture. Section V presents monitoring software structure. Then section VI focuses on how to improve system robustness with an embedded failure detector using K-Means optimized algorithm. Finally, in section VII, results are presented and discussed.

II. DISCLAIMER

This artificial ventilator is not intended to become a commercial product used in a modern hospital. It has not been certified as compliant to state of art medical regulations and will not be asked for. However, it has been functionally tested successfully on an artificial lung as shown on Fig. 1 at IRBA (Institut de Recherche Biomédicale des Armées) in HIA (Hopital Inter-Armées) Sainte-Anne, Toulon, France. This test has been detailed in written report. However, Toulon University students and researchers cannot be liable of any damage, injury or death resulting from the use of this device.

III. MECHANICAL ARCHITECTURE

Proposed system mechanical architecture is presented in Fig. 2. It can be built using only material plates such as wood or plastic, using a jigsaw or a laser cutter. Mechanical structure is based on three bellows made from inner tire tube replacing the AMBU bag that is often used in artificial ventilators [START_REF] Shahid | Prototyping of artificial respiration machine using ambu bag compression[END_REF]- [START_REF] Capelli | A novel system to assist in manual resuscitation and detect spontaneous breathing[END_REF]. Apart from being easier to supply, using home-made bellows instead of AMBU bags add reproducibility to the system, since AMBU bags are quite different from each other, depending on the local provider. These differences have a highly negative impact on the maximum tidal volume capacity, which undermines the control system. Each bellow uses two limits sensors as shown in Fig. 2 and is operated by a three 3D printer stepper motor using fishing wire. Office rubber bands are used for repositioning bellows in their maximum volume position. Air or gas is directed by garden hoses and watering couplers, allowing important cost reduction, as well as easy implementation.

IV. ELECTRICAL ARCHITECTURE Proposed electrical architecture is presented in Fig. 3. Supplied with a hand-held power tool battery delivering 18V DC, electronic system control is based on a Microchip PIC24FJ64GB004 microcontroller. There are two main sensors in the design: Silicon Micro Structures SM6221 for monitoring intra-lung pressure, and Omron D6F-PH0505AD3 for monitoring inspired tidal volume using a home-made Pitot tube, both connected in I2C. Omron D6F-PH0505AD3 is a precise differential pressure sensor having a measurement range from -50 to 50 Pa, allowing to build a Pitot tube as shown in Fig. 4. Measured differential pressure ∆P is converted into air velocity using Euler equation [START_REF] Lellis | The euler equations as a differential inclusion[END_REF]. Using fluid velocity and knowing hose pipe section, inspired volume D can be calculated as shown in Eq. 1, where t 0 is the start of compression, t 1 its end, V Air the air velocity and S P ipe the hose pipe section.

D[m 3 ] = t1 t0 V Air * S P ipe dt = t1 t0 2∆P ρ * S P ipe dt (1)
Actuators are Sanyo Denki 103H5208-5240 3D printer stepper motors driven by a Allegro A4988 motor driver. A serial link has been implemented in order to monitor all parameters on a standard PC, without adding a difficult to find or expensive LCD directly to the ventilator.

V. USER INTERFACE

User graphic interfaces for supervision and set-up are developed in C#, thus it can be run on a standard PC connected to the ventilator through a RS232 link. Sensors data is monitored in real time (intra-lung pressure and tidal volume). This interface allows the user to configure the following ventilator medical parameters:

• Inspired volume: can be set-up between 0.2 and 1.5 L.

• Maximum intra-lung pressure (in millibars).

• Breath cycles per minute. Interface also allows to set ventilator active mode among the following two ones:

• Reanimation: breath is initiated by the ventilator (usually used when patient is unconscious) • Assistance: breath is initiated by the patient (patient is conscious and needs only an assistance) It is important to note that computer interface is only used for monitoring and set-up. Artificial ventilator is fully controlled by its microcontroller, allowing an autonomous and reliable behaviour, even if computer link is lost for any reason. The full interface program is open source and published in [START_REF] Gies | Smiot: Scientific microsystems for the internet of things, technological platform[END_REF].

VI. HARDWARE FAILURE AUTOMATIC DETECTION

As low-cost recycled materials quality is unknown, a mechanical breakdown may occur. Therefore, in order to improve robustness, a solution based on cooperation between embedded artificial intelligence (AI) and medical staff is proposed.

Embedded AI algorithm is based on characterizing interactions between the patient and the ventilator. This is done by extracting features from sensors signals as described in Section VI-A during breathing cycles, in order to later detect hardware failures or anomalies. It uses in situ unsupervised learning: every patient is different, so that it is not possible to learn system behaviour once and for all. Consequently, training process will be done every time the ventilator is used, under supervision of medical staff for a short time (about one minute), to check its proper functioning during that period.

Life of the patient cannot depend on the serial link between embedded microcontroller and computer interface. Thus, failure detection is fully implemented on the embedded microcontroller, so that a problem with RS232 communication will not affect ventilator function, nor anomalies detection. Computer user interface software is used only for supervision and control. Hardware failure automatic detection system is presented in Fig. 5. Each block is described in the following subsections: Fig. 5: Hardware failure automatic detection system.

A. Features extraction

First step of the proposed algorithm is to extract relevant features from sensors signals on each breath cycle as shown Fig. 6. Following characteristics features are extracted and used as inputs of the unsupervised learning algorithm:

• Maximum intra-lung pressure (P p ) in millibar.

• Expired volume (P v ) in litres.

• Area under the pressure curve (A p ).

• Area under the volume curve (A v ).

• Pressure pulse duration (D p ) in seconds.

• Volume pulse duration (D v ) in seconds.

B. K-Means unsupervised learning and detection algorithm

Considering that the ventilator will be used with different patients and settings, learning use case reference situation is necessary for later detecting issues. This is done in unsupervised learning using a K-Means optimized algorithm, classifying features in real time without storing any dataset [START_REF] Khanmohammadi | An improved overlapping k-means clustering method for medical applications[END_REF]- [START_REF] Kanungo | An efficient k-means clustering algorithm: analysis and implementation[END_REF]. This allows an embedded implementation in a microcontroller, improving system reliability. Training and detection algorithms are presented in Figs. 7a and7b.

Learning phase is preceded by an initialisation phase. First k samples are used for evaluating data dispersion by calculating the standard deviation for each feature (σ = [σ Pp , σ Pv , σ Ap , σ Av , σ Dp , σ Dv ]).

Then in training phase, sets of features (composed by 6 elements having been described in Section VI-A) are assigned to existing clusters, if they are close to an existing one, or to a new one if not. A cluster C j (where j is the cluster ID), represents a behaviour of the couple {ventilator-patient}. A cluster is described by its centroid C j = [P pj , P vj , A pj , A vj , D pj , D vj ] and its dispersion

σ C j = [σ Pp j , σ Pv j , σ Ap j , σ Av j , σ Dp j , σ Dv j ]. Notations C j = [C j 1 , ..., C j 6 ] and σ C j = [σ C j 1 , ..., σ C j 6
] are used for simplifying equations. n j is the number of samples classified in cluster j. Each time a feature has been extracted, distance D between its signature S = [P p , P v , A p , A v , D p , D v ] and each cluster is evaluated using Eq. 2:

D = 6 i=1 S i -C j i σ C j i 2 (2)
Distance D is compared to a threshold D M ax optimized offline using deep learning. Its value is equal to 1.25. This optimization is not the purpose of this paper.

If D ≤ D M ax , sample is assigned to cluster j. Consequently as the sample is assigned to cluster j, each component of its centroid and their corresponding dispersions are updated as shown in Eq. 3a and Eq. 3b :

C j i = (1 -α)C j i + αS i (3a) σ C j i = (1 -α)σ C j i + α S i -C j i ( 3b
)
n i = n i + 1 (3c)
Eq. 3a and 3b shows that C j i and σ C j i are moving averages of S i and S i -C j i , implemented as low pass filters. This implementation is much lighter than using an arithmetical mean and adapts to slow changes in cluster parameters. Number of elements of the cluster n i , incremented in Eq. 3c, gives indications on cluster usefulness. Only most important ones will be kept at the end of learning phase.

If D > D M ax , there are two options:

• If algorithm is in learning phase, a new cluster C j is created as shown Fig. 7a, its centroid parameters are initialized with the new sample signature and its disper-sion is initialized considering the overall data dispersion calculated in the initialisation phase, as shown in Eq. 4.

C j i = S i σ C j i = σ i 2 (4)
• If algorithm is in detection phase, an alert is generated as shown in Fig. 7b. Fig. 8 shows a screen-shot of the interface during monitoring, two clusters can be observed represented in 3D with the most relevant features. VII. RESULTS Proposed artificial ventilator has been successfully tested at IRBA (Institute de Recherche Biomédical des Armées) using a reference artificial lung. For evaluating the performance of the fault detector, a learning phase without failures has been done. Then, several tests were performed simulating mechanical issues after learning the baseline condition: disconnection or closure of pipes, structure breaking, fishing wires cut as well as cut of fellows elastic strap. Every failure was detected except in one situation: when a fishing wire is cut on only one of three air bellows, if maximal intra-lung pressure is limited to 15 mbar and volume required is lower than 0.9L, no alarm is generated because normal operation can be performed using only 2 air bellows instead of 3. Fault detection sensitivity can be tuned by adjusting cluster belonging tolerance D M ax .

VIII. CONCLUSION In this paper, a less than $200 artificial ventilator that can help COVID-19 patients has been proposed. Using lowcost easy-to-find materials, it has been designed for helping developing countries where ventilators and medical supplies are hard to find. Despite its very low cost, it is compliant with most important medical requirements and has been tested by IRBA, the French Biomedical Defence Institute. Most parameters such as inspired volume, maximum intra-lung pressure and breath rate can be adjusted and monitored.

Moreover, considering that it is a life equipment built with low-cost easy-to-find and recycled materials having an unknown quality, an intelligent embedded hardware failure detector has been implemented and embedded in the microcontroller. In case of abnormal functioning an alarm is generated to warn the medical staff.

Its mechanical, electrical and software architectures are open source and published in [START_REF] Gies | Smiot: Scientific microsystems for the internet of things, technological platform[END_REF] and on GitHub: https://github.com/iutgeiitoulon/ArtificialVentilator.

Fig. 1 :

 1 Fig. 1: Artificial ventilator being tested with an artificial lung

Fig. 2 :

 2 Fig. 2: Mechanical architecture of the artificial ventilator.

Fig. 3 :

 3 Fig. 3: Electronics block diagram.

Fig. 4 :

 4 Fig. 4: Volume sensor implemented using home-made Pitot tube assembled on the Omron D6F-PH0505AD3 sensor.

Fig. 6 :

 6 Fig. 6: Volume and pressure features extraction.

  (a) K-Means training algorithm. (b) K-Means anomalies detection algorithm.

Fig. 7 :

 7 Fig. 7: Unsupervised learning algorithms.

Fig. 8 :

 8 Fig. 8: Interface during monitoring phase

ACKNOWLEDGEMENT

The authors would like to thank the Embedded Electronics Technology Platform of Toulon University named SMIoT [9] for its support in the achievement of this work.


Ce Zheng

Malcolm Egan

Laurent Clavier

Anders E Kalør

Petar Popovski

Stochastic Resource Allocation for Outage Minimization in Random Access with Correlated Activation

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

I. INTRODUCTION

To reduce energy consumption in industrial fault detection and monitoring of physical phenomena, it is highly desirable for sensor networks to be event triggered. That is, transmissions occur only when a sensor observes important information. As faults and physical phenomena typically form stochastic processes, the corresponding sensor network relies on random access communications.

A fundamental problem in random access is to design efficient resource allocation strategies in networks synchronized via a beacon sent by an access point. In the typical case that control signaling from the access point is limited, each sensor must independently determine which time and frequency slots should be utilized. One approach is for each sensor to transmit data whenever it is obtained, corresponding to slotted ALOHA [START_REF] Roberts | Aloha packet system with and without slots and capture[END_REF]. An alternative method is for sensors to use a randomly selected code over the time and frequency resources that are utilized within each frame, as developed in coded slotted ALOHA [START_REF] Paolini | Coded slotted aloha: a graphbased method for uncoordinated massive access[END_REF], [START_REF] Stefanovic | Aloha random access that operates as a rateless code[END_REF].

A key assumption in standard ALOHA-based schemes is that each sensor is activated independently. However, the processes observed by different sensors are often correlated (e.g. due to alarm event [START_REF] Thomsen | A traffic model for machine-type communications using spatial point processes[END_REF]) . For example, in earthquake or flood detection [START_REF] Werner-Allen | Monitoring volcanic eruptions with a wireless sensor network[END_REF], [START_REF] Meyer | Event-triggered natural hazard monitoring with convolutional neural networks on the edge[END_REF], nearby sensors will be more likely to transmit than sensors further away. In industrial fault detection, sensors for components on the same machine are more likely to simultaneously detect a fault than sensors on different C. Zheng is with R&D Center Beijing Lab, Sony, China. L. Clavier is with Université de Lille, IRCICA USR CNRS 3380, France. M. Egan is with Univ Lyon, INSA Lyon, Inria, France. A. Kalør and P. Popovski are with the Department of Electronic Systems, Aalborg University, Denmark. machines. As a consequence, activity of the sensors will also be correlated.

In order to improve decoding of sensor data at the access point, it is desirable to incorporate the presence of correlation in the design of the access policy. In many protocols, such as NB-IoT, a frame is divided into multiple time slots. In this case, the design of the access policy corresponds to the rule that each active sensor uses to decide which time slot to utilize.

In this paper, we design a new random access policy tailored to sensor networks with correlated activation. In our scheme, we limit transmission of control information for access policy updates from the access point to the sensors to every L frames, where L is typically large resulting in low levels of control signaling. Moreover, the objective is to optimize the random access policy so that the average number of active sensors can transmit their data with a sufficiently high signal to interference and noise ratio, accounting for successive interference cancellation. While an access policy has been developed in [START_REF] Zheng | Stochastic resource optimization of random access for transmitters with correlated activation[END_REF] for correlated activity, the scheme required control information to be sent within each frame.

To design the random access policy, we optimize the probability each active sensor transmits within a given time slot. A key challenge in solving this problem is that the objective function is not available in closed form due to the impact of random fading, a random set of active sensors, and correlation in the activity of sensors [START_REF] Kalør | Random access schemes in wireless systems with correlated user activity[END_REF]. In order to solve the optimization problem, we observe that it is in the form of a stochastic optimization problem and apply methods of stochastic approximation [START_REF] Kushner | Stochastic approximation and recursive algorithms and applications[END_REF]. A key feature of this approach is that the only required data can be obtained from channel and activity estimates, which are given by standard multi-user detection algorithms (e.g., [START_REF] Chetot | Joint identification and channel estimation for fault detection in industrial IoT with correlated sensors[END_REF] tailored for correlated activity). Updated probabilities for each sensor to utilize a given time slot are then sent from the access point to each sensor every L frames.

A key question is whether our algorithm yields locally optimal solutions. This is a non-trivial problem as it is often the case that sensors can transmit for more than one consecutive frame, leading to time-correlation in the data obtained from multi-user detection. We show that the algorithm converges to a locally optimal solution with probability one as L → ∞ when the activity of each sensor in each frame forms an irreducible Markov chain.

To evaluate our policy, we compare its performance to policies obtained via maximization of the throughput and the expected sum-rate. These objectives naturally arise when control information is sent every frame (rather than every L frames) [START_REF] Zheng | Stochastic resource optimization of random access for transmitters with correlated activation[END_REF]. We show that for sufficiently high signal to interference and noise ratio (SINR) decoding thresholds, our algorithm can lead to significant performance improvements over the algorithms proposed in [START_REF] Zheng | Stochastic resource optimization of random access for transmitters with correlated activation[END_REF], [START_REF] Kalør | Random access schemes in wireless systems with correlated user activity[END_REF], even when activity of sensors is correlated between each frame.

II. SYSTEM MODEL

Consider N sensors that aim to transmit data to a common access point utilizing a single common subcarrier. In each frame, a sensor may transmit within at most one of K time slots, with the possibility that a given sensor may not seek to transmit any signal. The activity vector is defined by X = (X 1 , . . . , X N ) ∈ {0, 1} N , where sensor i is active if X i = 1 and inactive otherwise.

It is often the case that elements of X, namely X i , i = 1, . . . , N , are assumed to be mutually independent; however, in many scenarios dependence naturally arises. For example, consider the case where sensors detect earthquakes. In this setting, sensors closest to the epicenter of the earthquake are more likely to detect its presence. As a consequence, we should expect in general that the elements of X are statistically dependent.

Moreover, sensor data transmission may require more than one consecutive frame. As a consequence, the activity random vectors in each frame, denoted by X 1 , X 2 , . . ., will in general be also statistically dependent.

To capture such dependence between sensors and over frames, we assume that the stochastic process (X n , n = 1, 2, . . .) forms an irreducible Markov chain admitting a stationary distribution P X . As individual sensor activations are also not independent, the stationary distribution does not factor as P X (x) = N i=1 P Xi (x i ), x ∈ {0, 1} N . We consider the transmission protocol given in Algorithm 1. Note that Algorithm 1 differs from the protocol in [START_REF] Zheng | Stochastic resource optimization of random access for transmitters with correlated activation[END_REF] as no control signaling between the access point and sensors is available, precluding the possibility of rate-adaptation (illustrated in Fig. 1). In Step 1, sensors randomly select a single slot within the frame according to an allocation matrix A ∈ R N ×K , where A ij is the probability that sensor i transmits in slot j conditioned on activation, and

K j=1 A ij = 1.
In the sequel, we address how the allocation matrix A can be optimized. For each slot j ∈ {1, . . . , K}, detailed in Step 2 for the first slot in Algorithm 1, each sensor that has selected slot j to transmit sends a pilot signal and data at a fixed rate R and transmit power P . At this point, as detailed in Step 3, the access point performs multi-user detection-assumed to be error free-in order to detect active sensors in slot j and their corresponding channel gains for the common subcarrier utilized by all sensors, denoted by {|g i | 2 } i∈Sj , where S j is the set of active sensors in slot j.

Once sensors have been identified and the channel estimated, the access point decodes the transmitted data. To perform data decoding, the access point exploits successive interference cancellation (SIC). In more detail, suppose that each slot j consists of T channel uses and the received signal on the common subcarrier is given by

y j = D j g + w j , (1)
where D j ∈ C T ×N is the data to be transmitted by all sensors. The coefficients

g i = √ P r -η/2 i h i , i ∈ {1, . . . , N } with h i ∈
C, and r -η/2 i denoting the block fading coefficient, and path loss for the i-th sensor, respectively, assumed to be constant for each sensor over the T channel uses. For i ∈ S j , we set the i-th column of D j to zero. The term w j ∼ CN (0, σ 2 I T ×T ) is additive white Gaussian noise.

Under SIC and sufficiently large T , the achievable rate for sensor m ∈ S j with the m-th largest channel gain in

{|g i | 2 } i∈Sj , denoted by |g m | 2 , is well approximated by R j,m = W log 1+ |g m | 2 σ 2 + m<l≤|Sj | |g l | 2 , m = 1, . . . , |S j |, ( 2
)
where W is available bandwidth and assumed in the sequel, without loss of generality, to be W = 1. The data from sensor m ∈ S j can therefore only be reliable decoded if R j,m > R, where R is the fixed rate used by each sensor.

III. OPTIMIZATION CRITERIA

The goal of this paper is to reduce the outages in sensor data transmissions by optimizing the allocation matrix A, which assigns the probability each sensor accesses each time slot. A natural criterion for optimization of A is the expected number of active sensors that transmit without an outage, given by

T 1 (A) = E X [T X 1 (A)], (3)
with

T X 1 (A) = K k=1 S∈P(N ) Q k (S|X) • |S| m=1 1 log 1 + |g S(m) | 2 σ 2 + m<l≤|S| |g S(l) | 2 > R , (4)
where P(N ) denotes the power set of {1, . . . , N } and 1{•} is an indicator function and

Q k (S|X) = i∈S X i A ik j∈S c (1 -X j A jk ), (5)
which corresponds to the probability the sensors in S transmit in slot k conditioned on the activity vector X. The expectation with respect to the activity vector X corresponds to expectation with respect to the stationary distribution, P X , of the Markov chain X 1 , X 2 , . . . (detailed in Sec. II). An alternative criterion is the expected fraction of slots in which exactly one sensor transmits, or the expected throughput, which was considered in [START_REF] Zheng | Stochastic resource optimization of random access for transmitters with correlated activation[END_REF], [START_REF] Kalør | Random access schemes in wireless systems with correlated user activity[END_REF] when the activity process X 1 , X 2 , . . . is independent. Formally, the expected throughput is given by

T 2 (A) = E X T X 2 (A) (6)
where

T X 2 (A) = N n=1 K k=1 X n A nk N m=1 m =n (1 -X m A mk ). (7)
As for the criterion (3), the expectation with respect to the activity vector X corresponds to expectation with respect to the stationary distribution, P X , of the Markov chain X 1 , X 2 , . . .. A further criterion that may be considered is

Pr   min k=1,...,K m=1,...,|S k | R k,m > R   , (8)
which is the probability that the data sent by all active sensors can be decoded error free. Note that (8) can be bounded by a criterion based on the sum-rate. Indeed,

Pr   min k=1,...,K m=1,...,|S k | R k,m > R   ≤ Pr   K k=1 |S k | m=1 R k,m > R   ≤ E X,h K k=1 |S k | m=1 R k,m R , (9)
by the Markov inequality. This suggests a useful criterion is given by the expected sum-rate

T 3 (A) = E h,X T h,X 3 (A) (10)
where

T h,X 3 (A) = K k=1 S∈P(N ) Q k (S|X) • |S| m=1 log 1+ |g S(m) | 2 σ 2 + m<l≤|S| |g S(l) | 2 (11)
with Q k (S|X) given in [START_REF] Werner-Allen | Monitoring volcanic eruptions with a wireless sensor network[END_REF]. As for the previous criteria, the expectation with respect to the activity vector X corresponds to expectation with respect to the stationary distribution, P X , of the Markov chain X 1 , X 2 , . . .. We remark that the expected sum-rate was also considered in [START_REF] Zheng | Stochastic resource optimization of random access for transmitters with correlated activation[END_REF], albeit in the case that the access point could perform rate-adaptation and the activity process X 1 , X 2 , . . . is independent.

The criteria in (3), ( 6) and [START_REF] Chetot | Joint identification and channel estimation for fault detection in industrial IoT with correlated sensors[END_REF] do not in general admit closed-form expressions-partly due to the fact that P X is not known to the access point-and are non-convex functions of the continuous variables in A. In the following section, we propose new algorithms for the optimization of the allocation matrix A with respect to these criteria, with provable convergence guarantees. In particular, we seek solutions

A * 1 = arg max A∈R N ×K + : j Aij =1, i=1,...,N T 1 (A), (12)
A * 2 = arg max

A∈R N ×K + : j Aij =1, i=1,...,N T 2 (A). (13)
and

A * 3 = arg max A∈R N ×K + : j Aij =1, i=1,...,N T 3 (A). (14)
At present, solutions based on tractable bounds of (6) have been obtained for the problem (12) in [START_REF] Kalør | Random access schemes in wireless systems with correlated user activity[END_REF]. We have recently introduced stochastic optimization methods to solve (13) and ( 14) under the assumption that the sequence of activity vectors X 1 , X 2 , . . . is independent. In the following section, we develop a new algorithm to optimize (12) and show that this algorithm along with those proposed in [START_REF] Zheng | Stochastic resource optimization of random access for transmitters with correlated activation[END_REF] to solve (13) and (14) all converge with probability one, even when the activity vector sequence forms a Markov process.

IV. OPTIMIZATION OF THE ALLOCATION MATRIX

A key observation is that the optimization problems in Sec. III are stochastic optimization problems with P X unknown. As a consequence, (12), ( 13) and ( 14) can be solved via stochastic approximation algorithms [START_REF] Kushner | Stochastic approximation and recursive algorithms and applications[END_REF], which exploit samples X 1 , X 2 , . . .. In particular, since the functions inside the expectations arising in the objectives are continuously differentiable in A, stochastic gradient ascent (SGA) can be applied. In this section, we derive the SGA algorithms for the problems in Sec. III and show that they converge almost surely to a local maximum.

Suppose that for a sequence of training frames, sensors access slots according to an initial allocation matrix A 1 . The access point can then obtain knowledge of sensor activity via multi-user detection; namely, in each frame t, the activity vector X t , t = 1, 2, . . .. Note that data can be transmitted during the training frames, as the same multi-user detection algorithm is required for data decoding.

A. SGA Algorithms for (13) and (14)

We first recall the algorithms proposed in [START_REF] Zheng | Stochastic resource optimization of random access for transmitters with correlated activation[END_REF] to optimize the allocation matrix A for the criterion in ( 13) and ( 14). We highlight that these algorithms were developed for random access with feedback allowing for rate-adaptation, which is not the case in the scenario studied in this paper. Moreover, convergence of the algorithms was restricted to independent activity vector sequences X 1 , X 2 , . . ., which will be generalized in Sec. IV-C for the Markov case.

Algorithm 2 provides details of the throughput maximization algorithm to solve (13), where Π H (A) denotes the closest point (w.r.t.

A F = N i=1 K j=1 A 2 ij is the Forbenius Norm) in the constraint set H = {A ∈ R N ×K + : j A ij = 1, i = 1, . . . , N } from A. The term Y t 2 (A t 2
) is an unbiased estimate of the gradient of the throughput T 2 at A t 2 . To compute the estimate, first observe that

∂T 2 ∂A ql (A) =E X       X q N m=1 m =q (1-X m A ml )- N n=1 n =q X q X n A nl N m=1 m =n m =q (1-X m A ml )       . ( 15
)
Contrary to gradient ascent, SGA need only use one training sample in each iteration. Given a realization X i , the unbiased estimate of the gradient in the t-th iteration is then given by

Y t 2,ql (A) =X t q N m=1 m =q (1-X t m A ml )- N n=1 n =q X t q X t n A nl N m=1 m =n m =q (1-X t m A ml ). ( 16
)
Algorithm 2 SGA Algorithm for (13) 1: Input: Choose an initial iterate A 1 2 and step-size α 1 2 > 0.

2: t ← 0. 3: While not converged 4: Using X t , compute an unbiased estimate Y t 2 (A t 2 ) of ∇T 2 (A t ), given in (16). 5: Set A t+1 2 ← Π H [A t 2 + α t 2 Y t 2 (A t 2 )]. 6: t ← t + 1. 7: End While 8: Output: A t 2 .
We now turn to the expected sum-rate maximization problem in ( 14), for which the allocation A is obtained via Algorithm 3.

In this case, we have

∂T 3 ∂A ql (A) = E X,h   S∈P(N ) ∂Q q (S|X) ∂A ql • |S| m=1 log 1+ |g S(m) | 2 σ 2 + m<p≤|S| |g S(p) | 2   , (17)
where

∂Q q (S|X) ∂A ql = X q i∈S\{q} X i A il j∈S c (1-X j A jl ), q ∈ S -X q i∈S X i A il j∈S c \{q} (1-X j A jl ), q ∈ S c . (18)
Given realizations of X t and h t , an unbiased estimate of the gradient in the t-th iteration is then given by

Y t 3,ql (A) = S∈P(N) ∂Q q (S|X t ) ∂A ql |S| m=1 log 1+ |g t S(m) | 2 σ 2 + m<p≤|S| |g t S(p) | 2 . ( 19
)
Algorithm 3 SGA Algorithm for (14)

1: Input: Choose an initial iterate A 1 3 and step-size

α 1 3 > 0. 2: t ← 0. 3: While not converged 4: Using X t , h t , compute an unbiased estimate Y t 3 (A t 3 ) of ∇T 3 (A t 3 ), given in (21). 5: Set A t+1 3 ← Π H [A t 3 + α t 3 Y t 3 (A t 3 )]. 6: t ← t + 1. 7: End While 8: Output: A t 3 .

B. New SGA Algorithm for (12)

We now introduce a stochastic gradient ascent algorithm for the problem (12), detailed in Algorithm 4. Convergence is established in Sec. IV-C.

For the criterion in (3), the gradient is given by

∂T 1 ∂A ql (A) = E X,h   S∈P(N ) ∂Q q (S|X) ∂A ql • |S| m=1 1 log 1 + |g S(m) | 2 σ 2 + m<p≤|S| |g S(p) | 2 > R   , (20)
where

∂Qq(S|X)
∂A ql is given in (18). Given realizations of X t and h t , an unbiased estimate of the gradient in the t-th iteration is then given by

Y t 1,ql (A) = S∈P(N ) ∂Q q (S|X t ) ∂A ql • |S| m=1 1 log 1+ |g S(m) | 2 σ 2 + m<p≤|S| |g S(p) | 2 > R . (21)
Algorithm 4 SGA Algorithm for (12) 1: Input: Choose an initial iterate A 1 1 and step-size

α 1 1 > 0. 2: t ← 0. 3: While not converged 4: Using X t , h t , compute an unbiased estimate Y t 1 (A t 1 ) of ∇T 1 (A t 1 ), given in (21). 5: Set A t+1 1 ← Π H [A t 1 + α t 1 Y t 1 (A t 1 )]. 6: t ← t + 1. 7: End While 8: Output: A t 1 .

C. Convergence

We now show that Algorithm 4 converges with probability one to a local maximum of the problem (12).

Theorem 1. Suppose that {X i } forms an irreducible Markov chain admitting a stationary distribution P X and α t = α 1 /t, t = 1, 2, . . . with 0 < α 1 < ∞. Then, the iterates A t in Algorithm 4 converge almost surely as t → ∞ to a local maximum of the problem (12).

Proof. (Sketch) We use the notation defined in [START_REF] Kushner | Stochastic approximation and recursive algorithms and applications[END_REF]Chap 6]. To establish the theorem, it is necessary to verify (A6.1.1)-(A6.1.7) and (A4.3.3) in [START_REF] Kushner | Stochastic approximation and recursive algorithms and applications[END_REF] and apply [START_REF] Kushner | Stochastic approximation and recursive algorithms and applications[END_REF]Theorem 6.1.1]. Due to the fact that (3) is bounded and Lipschitz continuous on H, Assumptions (A6.1.1)-(A6.1.7) immediately hold with the exception of (A6. 1.3). Assumption (A4.3.3) holds since the constraint set H forms a linear connected compact surface and hence has a continuously differentiable outer normal.

All that remains is to prove that (A6.1.3) holds. We first identify the "effective memory" variables ξ n in [9, Sec. 6.1.1] with X n-1 . It is straightforward to show that the bias variables β n are zero with probability one and hence the conditional mean (defined in [START_REF] Kushner | Stochastic approximation and recursive algorithms and applications[END_REF]Chap 6]) satisfies g n (θ n , ξ n ) = g(θ n , ξ n ) for all n; i.e., independent of n. Using the remark in [9, Sec. 6.2] and applying the strong law of large numbers for irreducible Markov chains on countable state spaces [START_REF] Durrett | Probability: Theory and Examples[END_REF], it follows that (A6.1.3) holds.

A similar argument also establishes that Algorithm 2 and Algorithm 3 also converge almost surely to a locally optimal solution of their respective problems. This generalizes the convergence results proven in [START_REF] Zheng | Stochastic resource optimization of random access for transmitters with correlated activation[END_REF] to the case where the activity vectors form a Markov process.

V. SIMULATION RESULTS

To evaluate our algorithm, we consider the following eventtriggered scenario based on the model introduced in [START_REF] Kalør | Random access schemes in wireless systems with correlated user activity[END_REF]. In particular, events are generated according to a homogeneous Poisson point process (HPPP) on [0, L] 2 with rate λ and a given sensor transmits if an event occurs within a radius R of the sensor. Throughout our numerical study, we assume that L = 50, λ = 10/(L + 2R) 2 and σ 2 = 1. In this model, each sensor has the same activity probability given by

E[X i ] = 1 -exp{-λπR 2 }. ( 22
)
For the purpose of evaluating the relative performance of each of the criteria, the activity vectors X 1 , X 2 , . . . form a Markov chain induced by active devices transmitting in the subsequent frame with probability 0.1. In [START_REF] Zheng | Stochastic resource optimization of random access for transmitters with correlated activation[END_REF], it was observed that the throughput and expected sum-rate maximization algorithms were sensitive to initial conditions. We therefore consider five choices of the initial value of the allocation matrix A 1 :

A 1 =      1/K 1/K . . . 1/K 1/K 1/K . . . 1/K . . . . . . . . . . . . 1/K 1/K . . . 1/K      N ×K A 2 =      1 0 . . . 0 1 0 . . . 0 . . . . . . . . . . . . 1 0 . . . 0      N ×K , A 3 =           1 0 . . . 0 0 1 . . . 0 . . . . . . . . . 0 0 0 1 1 0 . . . 0 . . . . . . . . . . . .           N ×K (23)
and A 4 and A 5 , which correspond to the allocations obtained via Alg upper and Alg lower in [START_REF] Kalør | Random access schemes in wireless systems with correlated user activity[END_REF], respectively. In the following results, we select the initial value of the allocation matrix leading to the highest performance. We first compare Algorithm 4 based on (3) with the algorithms for throughput maximization developed in [START_REF] Kalør | Random access schemes in wireless systems with correlated user activity[END_REF]. In order to establish reasonable rate thresholds, we express the rate R in (3) in terms of the SINR τ = e R -1. We consider N = 10 sensors in the network, uniformly distributed on the circles with distances from the access point given by d =[0.1743, 0.8433, 1.2320, 1.5303, 5.8221, 6.5653, 7.1598, 8.0840, 8.4006, 9.7600].

(24) Fig. 2 and Fig. 3 show the percentage improvement in the expected number of successful transmissions (i.e., criterion (3)) using Algorithm 3 compared with the methods in [START_REF] Kalør | Random access schemes in wireless systems with correlated user activity[END_REF], with τ = 0 dB and τ = 20 dB, respectively. For both τ = 0 dB and τ = 20 dB significant improvements over the algorithms in [START_REF] Kalør | Random access schemes in wireless systems with correlated user activity[END_REF] on the order of 20% are evident for low numbers of available time slots. We now compare Algorithm 4 based on the criterion in (3) with Algorithm 3. Fig. 4 shows that performance improvement with τ = 0 dB is limited. Fig. 5 shows the performance improvement with τ = 20 dB. In this case, we observe performance improvements of over 20% for low numbers of slots (on the order of 20% -30%).

VI. CONCLUSIONS

A key problem in the design of random access communications for event-triggered sensor networks is minimizing outages in the presence of correlated activation. In this paper, we developed a new stochastic resource optimization algorithm, which can outperform existing methods when a limited number of slots are available per frame. The proposed algorithm is also shown to converge with probability one, even when activity vectors in each frame form a Markov process. A further benefit of our approach is that parameters of the distribution governing activation do not in general need to be directly estimated.

Algorithm 1 1 Fig. 1 :

 111 Fig. 1: Comparison of protocols.

Fig. 2 :

 2 Fig.2: Improvement of Algorithm 4 over Alg lower and Alg upper[START_REF] Kalør | Random access schemes in wireless systems with correlated user activity[END_REF] for τ = 0 dB with N = 10 sensors for varying K, where λ = 10/(50 + 2R) 2 and p = 0.1.

Fig. 3 :

 3 Fig.3: Improvement of Algorithm 4 over Alg lower and Alg upper[START_REF] Kalør | Random access schemes in wireless systems with correlated user activity[END_REF] for τ = 20 dB with N = 10 sensors for varying K, where λ = 10/(50 + 2R) 2 and p = 0.1.

Fig. 4 :

 4 Fig. 4: Improvement of Algorithm 4 over Algorithm 3 based on the expected sum-rate for τ = 0 dB with N = 10 for varying K, where λ = 10/(50 + 2R) 2 and p = 0.1.

Fig. 5 :

 5 Fig. 5: Improvement of Algorithm 4 over Algorithm 3 based on the expected sum-rate for τ = 20 dB with N = 10 for varying K, where λ = 10/(50 + 2R) 2 and p = 0.1.

ACKNOWLEDGEMENTS

This work has been (partly) funded by the French National Agency for Research (ANR) under grant ANR-16-CE25-0001 -ARBURST. It has also been supported by IRCICA, USR CNRS 3380, Lille, France and the COST Action CA15104 IRACON.

It has been in part supported by Danish Council for Independent Research, Grant Nr. 8022-00284B SEMIOTIC as well.


Eva Kempf

M Calvo

Florian Domengie

Stéphane Monfray

Frederic Boeuf

Paul Charette

Régis Orobtchouk

Low temperature SiN waveguides optimization for photonic platform

Keywords: Si photonics, SiN waveguide, MMI splitter, asymmetric Mach Zehnder interferometer

.

 coupled with the Mode Matching method. This method, used to simulate the propagation of light, also allows to evaluate side wall roughness contributions, Rayleigh scattering and absorption due to 2 nd harmonic vibrations of Si-OH and N-H bonds.

INTRODUCTION

Development of SiN platforms have been the subject of a large development this last decade to overcome some limitation of silicon such as lower thermal conductivity, band gap exceeding 4 eV and a lower refractive index contrast allowing the possibility of realization of athermal devices, avoiding two photon absorption for the transmission of high power and reduction of propagation losses. Different multilayer Silicon Nitride-on-Silicon Photonic Integrated Platform are now available in the world using LPCVD and PECVD deposition processes. Low-Pressure Chemical Vapour Deposition (LPCVD) at high temperature (>700°C) is required to avoid the N-H and Si-H bonds absorption existing in the C-band. Ligentec's thick SiN platform has been used for the realization of an optical-frequency synthetizer [START_REF] Spencer | An optical-frequency synthesizer using integrated photonics[END_REF] and a photonic chip based on optical frequency comb [START_REF] Brasch | Photonic chip based optical frequency comb using soliton Cherenkow radiation[END_REF]. Low loss optical waveguide [START_REF] Roeloffzen | Low-loss Si3N4 TriPleX optical waveguides: Technology and applications overview[END_REF] and basic building blocks for optical routing [START_REF] Worhoff | TriPleX: a versatile dielectric photonic platform[END_REF][9] are developed on the LioniX International TriPleX platform and optical sensors with the AMO's Silicon Nitride Photonic Platform: Full Custom Pilot Line Services [START_REF] Munoz | Silicon nitride photonic integration platforms for visible, near-infrared and mid-infrared applications[END_REF][START_REF] Zinoviev | Integrated Bimodal Waveguide Interferometric Biosensor for Label-Free Analysis[END_REF][START_REF] Tinguely | Silicon Nitride Waveguide Platform for Fluores-cence Microscopy of Living Cells[END_REF]. More fully CMOS compatible Plasma Enhanced Chemical Vapour Deposition (PECVD) at low temperature (<400 °C) are also available at the BioPIX SiN platform of IMEC [9]. This paper presents design and optical characterization of basic building blocks for optical routing made with the 300mm PECVD/PVD DAPHNE SiN platform of STMicroelectronics.

SiN deposition process

The PECVD SiN deposition is carried out at a temperature above 400°C. One drawback of the vapourphase deposition is the creation of -NH bonds which absorb light around 1550nm, a widely used wavelength in the field of telecommunications. Therefore, PVD depositions with different concentrations of silicon were made in order to attempt to decrease absorption loss. The PVD deposition at 200°C was carried out by the sputtering of a Si target. A radiofrequency (RF) plasma of Argon and Nitrogen with a variable stoechimoetry was used in order to control the optical index of the deposited material. These depositions were made on 300mm wafers with a silicon substrate and a 2µm layer of thermal SiO2. Waveguides were patterned using DeepUV lithography and Reactive Ion Etching (RIE) process, and were encapsulated in TEOS SiO2.

The table 2.1 gives the thickness of the fabricated SiN films as well as their optical index. The values of the optical indexes were computed applying Cauchy dispersion laws to ellipsometry measurements.

Buiding blocks.

Modelings are performed with Full Vectorial Finite Difference mode solvers in Cartesian and cylindrical coordinates respectively for the design of straight and bend waveguide [START_REF] Hu | Modeling the anisotropic electro-optic interaction in hybrid silicon-ferroelectric optical modulator[END_REF][START_REF] Masi | Towards a realistic modelling of ultra-compact racetrack resonators[END_REF]. Transparent boundary conditions are added for the calculation of propagation losses induces by the leakage through the buried silica layer and the radiation effect inherent to the bend. A waveguide width of about 700 nm is chosen to preserve a single mode condition for a spectral range between 1.2 and 1.7 µm and for all the materials involve in this study. A 2 µm thick buried silica layer and 40 µm bend radius are enough to obtain bending losses lower than 0. Mode matching method (MMM) is used to calculate propagation losses induced by the sidewall roughness. Firstly, two independant random surfaces with Gaussian statistic ( = 2 nm and Lc = 50 nm) are generated and added on both lateral sides of the waveguide. Segmented approach is then applied to calculate a set of guided and radiative modes in all of the local waveguides generated. Mode Matching algorithm gives propagation losses thanks to the reduction of power transmission after the propagation in a 20 µm long waveguide with a discretization of 1 nm (corresponding to 20 000 local waveguides). We can notice that losses of 1 dB/cm require an accuracy on the transmission coefficient of about 10 -7 , the accuracy on the orthogonality relations between the different modes. It can be evaluated with the mode solvers of commercial softwares, which are calculating an averaging of the refractive indexes at each boundaries to overcome instabilities due to the discontinuity of the normal components of the electrical field. This limitation is overcome with a nine points stencil discretization scheme [START_REF] Hu | Modeling the anisotropic electro-optic interaction in hybrid silicon-ferroelectric optical modulator[END_REF] giving an accuracy of 10 -17 . The propagation losses induced by the sidewall roughness are respectively of 0.04 and 0.16 dB/cm at  = 1.31 and 1.55 µm.

Design of MMI splitter is also performed with MMM. As the broadening of the spectral bandwidth increases with the reduction of the MMI footprint, the minimum width of 3.2 µm is require to have 5 different guided modes in both polarizations providing good images of the input waveguide at the output and a reduction of scattering losses lower than 0.1 dB (see fig. 1), resulting on a 7.7 µm long splitter. All of the basic building blocks are merged to design a set of optical test devices for an accurate characterization of propagation losses and dispersion curves of the SiN waveguides.

Optical test devices

The characterization of the deposited SiN was carried out for the TE mode both around 1310 and 1550nm where the NH and OH absorption pics are located. Light injection was performed with butt coupling and polarization maintaining fibers, and the data were collected using an Optical Spectrum Analyzer (OSA). Different structures, presented in fig. 2, were fabricated.

In order to determine optical propagation losses, the light was injected into three spirals of respective length 4.2cm, 7cm and 9.8cm with 142 bends of constant radius 40µm. The waveguides width is equal to 700nm ± 6nm. For each wavelength, a linear regression was done using the optical power detected at the end of each spiral and error bars were computed with the standard error of the linear regression. This set up was then improved by using three waveguides of respective length 1mm, 2mm and 3mm, with 3 bends of constant radius and, for each waveguide, a reference arm. The measurement arm and the reference arm are separated by a 1x2 Multi-mode interferometer (MMI), allowing a more accurate measurement. An asymmetric Mach-Zehnder interferometer (MZI) was also designed in order to extract the effective index dependance with the wavelength. Identical structures with different waveguide width, respectively 500nm ± 6nm and 600nm ± 6nm were fabricated. The input light is split into two with a 1x2 MMI, it propagates into the two arms of different length and is recombined at the end of the dispositive with a 2x1 MMI. The optical power at the end of the interferometer can be expressed as :

P=P max .cos 2 ( π. n eff .ΔL λ ) (1)
Where P max is the optical power at the input of the interferometer, n eff is the optical effective index of the waveguide, ΔL is the difference of length between the two arms of the interferometer ΔL =100µm, and λ is the optical wavelength. The output power at the end of the interferometer can be modelled as a function of the wavelength as: n eff (𝜆 𝑚 )=𝑎 1 +𝑎 2 𝜆 𝑚 +𝑎 3 𝜆 𝑚 ² (2) where 𝑎 1 , 𝑎 2 and 𝑎 3 are coefficients that can be experimentally obtained with non-linear regression routine.

Experimental results

Figure 3 shows the propagation losses of the three different characterized SiN films. While the film grow by PECVD presents low losses at 1310nm (-0.81 ± 0.23 dB/cm), the -NH absorption peak causes the losses to drop at 1550nm (-2.42 ± 0.97 dB/cm). On the contrary, the Si-rich PVD film displays losses as high as -3.68 dB/cm at 1310nm. As the signal was so weak at the output of the longest spiral, only the two shortest spirals could be used for this measurement. However, at 1550nm, the losses are much weaker as for the PECVD film, with only 1.68 ± 0.023 dB/cm.

Fig 1 :

 1 Fig 1 : Mapping of light power in the SiN MMI splitter (WMMI = 3.2 µm, LMMI = 7.7 µm).

Figure 2 .

 2 Figure 2. Fabricated structures for optical characterization : (a) Spirals, (b) Waveguides with 1x2 MMI, measurement and reference arms, (c) Asymmetric Mach-Zehnder interferometer.

Figure 3 :

 3 Figure 3:Propagation losses measurement for the 2 kind of SiN materials in the spectral range around 1.31 and 1.55 µm.

Table 2 .

 2

		n(1.31µm)	n(1.55µm)	SiN thickness
	PVD	1.944	1.942	632 ± 8 nm
	Si-rich PVD	1.989	1.985	627 ± 16 nm
	PECVD	1.913	1.910	600 ± 20 nm

1. Thickness of the deposited SiN films and computed optical indexes.

Table 1 .

 1 Effective indexes of the 700nm wide waveguides at 1310nm and 1550nm.

	1 dB. Effective indexes

ACKNOWLEDGEMENTS

This work is supported by a European IPCEI program.


Michele Calvo

Guillaume Beaudin

Laurence Mercier-Coderre

Pauline Girault

Pedro Rojo Romeo

Romain Stricher

Serge Ecoffey

Dominique Drouin

Frederic Boeuf

Michael Canva

Stephane Monfray

Regis Orobtchouk

Paul G Charette

Improving Silicon Nitride Ring Resonator Performances On 300 mm Industrial Environment For Point of Care Applications

Keywords: Biosensors, lab-on-chip, biophotonics, plasmonics, higher order modes

Point-of-care tests (POCT) are important for detecting illnesses and monitoring patients without the need of a medical laboratory. To be useful, POCT must be sensitive, specific, integrated, and affordable. Since the early 2000s, integrated photonics have offered a possible solution for this problem. In particular, silicon micro-ring resonators represent a compact and sensitive choice known in the industry. This paper details the design, fabrication, and characterization of two methods for improving the performance of ring resonators by engineering their cross section. More precisely, improving devices made out of silicon nitride in an industrial environment to work in the infrared (around 1.31 µm).

INTRODUCTION

To be effective, POCT devices have to be sensitive, compactness, low cost and compatible with functionalization process. This paper shows two alternative approaches to integrated nano-photonic sensing for point of care testing [START_REF] Cetin | Hand-Held plasmonic biosensor for high-throughput sensing for point-of-care applications[END_REF], [START_REF] Chrostowski | Silicon photonic resonator sensors and devices[END_REF]. The proposed structures, demonstrate not only a higher sensitivity, but consider selectivity and manufacturing issues, fundamental for POCT development. Concerning the low cost aspect, the devices were made in an industrial environment Datacom Advanced PHotonic Nanoscale Environment (DAPHNE) which allowed a high throughput and repeatability. The two approaches, Hybrid Plasmonic WaveGuides and the higher order modes, are listed below and each section describes the differences and the common points.

Hybrid Plasmonic waveguides

Hybrid plasmonic waveguides, or HPWG, are partially made of a plasmonic waveguide and partially made of a dielectric waveguide. Picture 1.a shows the cross section of a hybrid plasmonic waveguide. The vertical walls of the waveguide support a plasmonic mode only with a polarization perpendicular to its surface. The dielectric waveguide, although it can support polarization in both directions, it should be polarized in the same direction as the plasmonic waveguide to maximize the coupling between the two. The polarization direction of the field of the separate modes is shown in picture 1. The geometry of the HPWG can be engineered to maximize the power deconfinement from the dielectric waveguide into the target liquid. The HPWG improves designed in this paper improves the sensitivity of the sensor with respect to a dielectric sensor. Furthermore, the sensor is made of a dielectric and a metallic part, which makes the functionalization easier to achieve for analyte detection applications.

The materials making up the waveguide are SiN and SiO2 because they are compatible with the industrial fabrication. The plasmonic waveguide was chosen to be Au because of its plasmonic aspect, its compatibility with biological application and its availability both in the university and industrial cleanroom.

Higher order modes

Higher order mode sensors approach consists in selectively excite the first order mode of the ring resonator's waveguide. An example of cross section is shown in figure 1.b. The higher order mode is designed to be close to its cutoff. Thanks to its greater exposure to the sensing liquid than the fundamental mode, first order modes are more sensitive. Furthermore, the first order waveguide constituting the device is multimode, thus wider than a monomode one (1 µm and 0.5 µm respectively). This greater width increases the exposed surface and reduces the constraints on fabrication especially concerning the lithography resolution. Thus the waveguides are easier to make and have lower impact on fabrication errors.

In the proof of concept in this paper, the first order mode is selectively excited by using a monomode bus waveguide with a fundamental mode having the same effective refractive index. The monomode and the multimode waveguide couple generating even and odd modes as shown in the simulation section.

SIMULATION

Photonic devices studied in this paper were simulated by a custom design mode solver verified with FIMMWAVE simulation software. Once the mode of the waveguide and its effective refractive index was found, the whole ring resonator was simulated thanks to the analytical expression of the transmission of the ring. Next two sections present the simulation results for the proposed geometry: the hybrid plasmonic waveguide or HPWG and the higher order mode ring resonator.

Hybrid plasmonic waveguides

Hybrid plasmonic waveguides cross section geometry and simulation is shown in figure 2.a and 2.b. It can be noticed that simulations confirm that the coupling between the plasmonic walls and the dielectric waveguide takes place. In fact, the field distribution shown in figure 2.b shows a high energy in the gap between the Au and the SiN, in the sensing liquid. The filling factor in the sensing liquid, i.e. the percentage of the field in the sensing liquid, is of about 60% for the geometry described in the picture. This high percentage increases the shift of effective refractive index of the mode and by consequences the resonance peak shift of the ring resonator. Simulations give a sensibility of 550nm/RIU (RIU = refractive index unit, it is the unit of the refractive index which in the international system is dimensionless). However, HPWG also have large losses with respect to a dielectric waveguide due to the high absorption coefficient of the metal. Furthermore, in a ring resonator the bend of the waveguide can cause deconfinement effects on the propagation of the mode. For this reason the shape of the ring resonator is a racetrack where only the straight parts where HPWG. The percentage of the ring to be metalized is from 10% to 30% based on the geometry. The simulated sensitivity considering the dielectric bends is around 270nm/RIU. As expected it is lower than the sensitivity of a full hybrid plasmonic ring, because the bends are dielectric waveguides, thus less sensitive.

Higher order modes

The modes of monomode and multimode waveguides are shown in figure 3.a and 3.b. The higher order mode has two power peaks in the waveguide and its power in the sensing liquid is more important that the one of a fundamental mode. This shows that the first order mode is more sensitive with respect to the fundamental.

However, simply increasing the width of a waveguide brings up another problem: the first order mode has to be selectively excited so that no power leaks to other modes, especially for the fundamental. The price to pay if the fundamental mode is a decrease in sensitivity, increase incertitude and may cause double resonances in the ring complicating the detection. The system that is used in this paper to avoid excitation of the first order is through asymmetrical directional couplers: one of the waveguide is monomode and the other is multimode. They are designed to have the same refractive index of the modes. In this case, coupling is very low from the two fundamental modes of the waveguides while the preferential one is the one between the fundamental mode of the monomode waveguide and the first order mode of the multimode.

FABRICATION AND SETUP

The fabrication of the SiN waveguides and the SiO2 is performed in an industrial environment called Datacom Advanced PHotonic Nanoscale Environment or DAPHNE. Working in an industrial environment is important because one of the key points of POCT is that the production per device should be cheap. It is then paramount to already work with in the condition of a 300mm SOI industrial platform [START_REF] Calvo | Ring Resonator Designed for Biosensing Applications Manufactured on 300 mm SOI in an Industrial Environment[END_REF]. Once the rings were manufactured at STMicroelectronics, a polydimethylsiloxane (PDMS) ring was glued to the device surface to contain the fluids used during measurements. Figure 6.a show the sample with the empty PDMS reservoir.

The schematics of the setup is shown in picture 6.b. The input laser (Agilent 81682A tunable laser, 1.46 to 1.58 µm wavelength range, 0.1 pm resolution) is followed by a polarization control device which selects TE light in this case. Using a lensed fiber, light is then injected into the bus waveguide. A photodetector (Agilent 81618A with 81624 head, 0.8 to 1.7 µm wavelength range, -90 to +10 dBm power range) measures the optical power from a lensed fiber coupled to the output bus waveguide.

EXPERIMENTAL RESULTS

This section presents the experimental results. These results are compared to a conventional ring resonator, built with the same platform with a rib waveguide. This reference ring has a sensitivity of 120nm/RIU.

Hybrid plasmonic waveguides

The transmission of the HPWG resonator is shown in figure 7.a. The quality factor is low (around 500) because of the presence of the metal. Furthermore, the algorithm to calculate the peak position is a fit with a Lorentzian curve to find the minimum and the associated uncertainty [START_REF] Claes | Experimental characterization of a silicon photonic biosensor consisting of two cascaded ring resonators based on the Vernier-effect and introduction of a curve fitting method for an improved detection limit[END_REF]. As expected, the sensitivity of the HPWG device is around 275nm/RIU. This value is calculated as the slope of the calibration curve.

Higher order modes

The transmission curves of the higher order ring resonator are shown in figure 8.a. The quality factor is higher than in the case of HPWG (around 5000) because there is no metal. The sensitivity of the sensor was found to be around 212nm/RIU. This value is slightly lower than the simulation because of fabrication tolerances.

CONCLUSION

Two approaches to improve the sensitivity of a silicon nitride waveguide are shown. Both approaches are compatible with the cleanroom environment, which is fundamental for a lab-on-chip device. Moreover both devices have been tested with different refractive index liquids and they show a higher sensitivity compared to a rib waveguide (120nm/RIU). The perspective of the HPWG resonator is to improve the sensitivity by increasing the hybrid plasmonic part in the ring.

Figure 1

 1 Figure 1 Cross section of a HPWG and a second order mode waveguide made of silicon nitride with the industrial DAPHNE platform developed at STMicroelectronics

Figure 2

 2 Figure 2 Cross section of the HPWG waveguide and its simulation of the its propagation mode for h = 600nm, W1 = 300nm and d1 = 300nm.

Figure 3 .

 3 c shows how the electric field of the two waveguides is influenced by the presence of the other one. Critical coupling conditions are achieved by choosing the distance between the waveguides.Like with HPWG, a racetrack ring resonators is used to change the phase shift caused by the analyte into intensity. The sensitivity found in the simulation is of 250nm/RIU.

Figure 3 .

 3 Figure 3.a monomode waveguide mode. 3.b two-modes waveguide where the first order mode is simulated. Its effective index is 1.45, close to the cutoff. 3.c shows the two modes close to each other and the coupling between the two modes having similar index of refraction.

Figure 5

 5 Figure 5 Scanning electron microscope image of the higher order ring resonator. A zoom on the coupling section is shown as well as the schematics showing the cross section.

Figure 4 .

 4 Figure 4.a image of the process to build a HPWG, lift-off. 4.b scanning electron microscopy image of the manufactured ring resonator

  Figure 7.c shows the devices during the experiment. The inset shows a zoom on the sample in the adapted sample holder. Different concentrations of glucose monohydrate in H2O were used to test the bulk response of the device: from 2g/100ml to 6g/100ml, in refractive index from 2x10 -3 to 7x10 -3 RIU (refractive index unit).

Figure 5 a

 5 Figure 5 a schematics of the setup. 5.b setup with the chip and the PDMS reservoir. 5.c setup while charging the reservoir with the syringe.

  Error bars are found by using the combined standard uncertainty that links the temperature uncertainty, the refractive index uncertainty and fabrication uncertainty. The results of HPWG calibration curve are shown in picture 7.b. The experimental sensitivity value is different from the simulated one because the alignment of the metal was not perfectly symmetric.

Figure 7 .

 7 Figure 7.a Transmission spectrum of the HPWG device for different glucose monohydrate dilutions in H2O. 7.b Calibration curve to calculate the sensitivity as the slope of the fit line. A represent the reference wavelenght, B the 3g/100mL, C the 5g/100mL and D the 6g/100mL. Sensitivity is of 275nm/RIU.

Figure 8 .

 8 Figure 8.a transmissions for the higher order mode device for different glucose monohydrate dilutions in H2O. 8.b Calibration curve to calculate the sensitivity as the slope of the fit line. A represent the reference wavelenght, B the 2g/100mL, C the 3g/100mL and D the 6g/100mL. Sensitivity is of 212nm/RIU.