Alain Rapaport
email: alain.rapaport@inrae.fr

Denis Dochain
email: denis.dochain@uclouvain.be

An asymptotic observer to reconstruct parameters of systems that are not asymptotically identifiable

Keywords: Parameter estimation, identifiability, detectability, observability singularity, asymptotic observers

We propose the derivation of asymptotic observers for the estimation of parameters of systems whose solutions converge to a set of steadystates that are not identifiable, under some hypotheses. The proposed observer generalizes a former work for batch bioprocess. It is illustrated on a two dimensional models, and its performance is compared with the least squares method.

Introduction

System identification has received a great attention in the literature (see e.g. [START_REF] Walter | Identification of Parametric Models from Experimental Data[END_REF]). Once the system is identifiable, the least squares technique is the widelyused method to reconstruct unknown parameters from measurements. Other approaches based on observers (e.g. [START_REF] Krener | Nonlinear observers[END_REF][START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF]) have also some interest in terms of robustness and noise sensitivity, especially when the state is only partially observed (It might important to briefly recall that in that case the unknown parameters are considered as additional state variables with dynamics equal to zero, and therefore the estimation problem is considered as a pure observability problem). In any case, the requirement of identifiability/observability imposes the system to be identifiable at any steady-state that can be reached by the system (according to the definition of identifiability ( [START_REF] Walter | Identification of Parametric Models from Experimental Data[END_REF]) or observablity ([3])). However, in some applications, systems are identifiable only away from steady states, while the trajectory solutions converge asymptotically to one of these steady-state (and therefore the system is not detectable neither). This is typically the case of batch processes, which stop when all the resources are consumed ( [START_REF] Bastin | On-line Estimation and Adaptive Control of Bioreactors[END_REF][START_REF] Rapaport | A robust asymptotic observer for systems that converge to unobservable states -a batch reactor case study[END_REF]). In practice, those situations lead to several issues when applying the usual estimation techniques, especially when the initial condition is not known to be closed or not to an attractive equilibrium. In this context, there is no theoretical ground to ensure that a least squares estimation is reliable. A Luenberger-like observer ( [START_REF] Luenberger | An introduction to observers[END_REF][START_REF] Kazantzis | Nonlinear observer design using lyapunov's auxiliary theorem[END_REF]) is no longer guaranteed to converge with a null error (in the best case one may expect to obtain a "practical" convergence). The gains of the high-gain observer ( [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF]) explode when the estimation is close to the singularities of the observability map (which are precisely the non-identifiable steady-states).

Recently, the authors have a propose an asymptotic observer for the state reconstruction of a bioprocess model, whose solutions converge asymptotically to a set of non-observable states ( [START_REF] Rapaport | A robust asymptotic observer for systems that converge to unobservable states -a batch reactor case study[END_REF]). This observer have be shown to converge asymptotically without any bias, whatever is the initial condition away from steady-state, differently to classical observers. Numerical implementations have shown the benefit of this observer. The aim of the present work is to generalize this approach to the parameters estimation with lack of asymptotic identifiability for a large class of systems. In the next section, we define more precisely the problem to be investigated. in Section 3, we expose the estimator construction. Section 4 is devoted to examples along with numerical illustrations and discussions.

Description of the problem

Consider a dynamics in R n ẋ = f (x, p), x(t 0 ) = x 0 (1)
where x 0 is unknown, and p is an unknown vector of parameters which belongs to a subset P of R m . Along the solutions, we consider an observation variable in R q y(t) = h(x(t), p), t ≥ t 0 ,

The maps f and h are smooth, say C ∞ .

Assumption 2.1. There exists a domain D of R n which contains 0 that is forward invariant by (1) for any p ∈ P . Moreover, for any p ∈ P , one has f (0, p) = 0 and h(0, p) = 0, and any solution of (1) in K converges asymptotically to 0.

This assumption implies that the system is not identifiable at 0 i.e. the observation y does not allow to reconstruct the parameters vector p when the system is at the steady state 0, and moreover its solutions converge locally to this steady state. In such situations, it is well known that even if the system is identifiable and observable everywhere away from steady state, a construction with a classical smooth observer (such as Luenberger, extended Kalman or high gain) cannot guarantee an exact asymptotic estimation of the parameters because of the asymptotic lack of identifiability and observability. Non-smooth observers (such as sliding mode observers ( [START_REF] Spurgeon | Sliding mode observers: A survey[END_REF])) or numerical differentiators ( [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF]) could be alternatives that provide finite time convergence. However, these constructions are known to be poorly robust to noise, especially when the system is close to a singular point of the observability map, which is the case of the steady state.

Our objective is to construct, under some conditions, a smooth estimator that provides an exact asymptotic estimation of x 0 and p from the single measurements y(•), for unknown initial conditions x 0 = 0 in D (even arbitrarily close to 0).

Assumptions and construction of the estimator

We first assume the following condition on the dimension of the observation set, which will plays a crucial role in the method. Assumption 3.1. One has n + m = 2q.

We also assume that each coordinate y i of the observation vector y can be written as a Lie derivative of a certain function F i with respect to the vector field f (•, p), for any p ∈ P , as expressed in the assumption below. Assumption 3.2. There exists a smooth map

F : (x, p) ∈ R n × R m → F (x, p) ∈ R q
with F (0, p) = 0 for any p ∈ P that fulfills

h(x, p) = ∂F ∂x (x, p).f (x, p), (x, p) ∈ D × P. (3)
Under these two assumptions, we define the map

Γ : (x, p) ∈ R n × R m → (h(x, p), F (x, p)) ∈ R 2q (4)
Let us underline that under Assumption (3.1), Γ can be considered as a map from R n+m to itself.

Here is our main result.

Proposition 3.1. Let Q be a subset of D × P such that Γ is onto on Q, and Ψ a continuous map on W ⊂ R 2q such that Γ(Q) ⊂ W with Ψ = Γ -1 on Γ(Q).
For any x 0 ∈ D and t 0 ≥ 0 such that for any p ∈ P , the solution of (1) with

x(t 0 ) = x 0 verifies h(x(t 0 ), p), - t t0 h(x(τ ), p)dτ ∈ W, t > t 0 (5)
and

h(x(t 0 ), p), - +∞ t0 h(x(τ ), p)dτ ∈ Γ(Q) (6) the following system v = y(t), v(t 0 ) = 0 (x 0 (t), p(t)) = Ψ(y(t 0 ), -v(t)), t > t 0 (7)
is an asymptotic exact estimator of x 0 and p i.e. one has

lim t→+∞ x0 (t) -x 0 = 0, (8)
lim t→+∞ p(t) -p = 0. (9)
Proof. Take p ∈ P and an initial condition (t 0 , x 0 ) with x 0 ∈ D. Let x(•) be the corresponding solution of (1) and y(•) the output given by ( 2). Then the solution of ( 7) verifies, from Assumption 3.2

v(t) = t t0 h(x(τ ), p) dτ (10) = t t0 ∂F ∂x (x(τ ), p).f (x(τ ), p) dτ (11) = F (x(t), p) -F (x 0 , p) (12)
for any t ≥ t 0 . Under condition (5), (y(t 0 ), -v(t)) belongs to W for any t > t 0 and thus the pair (x 0 (t), p(t)) = Ψ(y(t 0 ), -v(t)) is well defined for t > t 0 . From Assumptions 2.1 and 3.2, we get

lim t→+∞ F (x(t), p) = 0 and thus v(•) is bounded with lim t→+∞ v(t) = -F (x 0 , p).
By continuity of Ψ on W and condition [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF], one obtains

lim t→+∞ Ψ(y(t 0 ), -v(t)) = Ψ(y(t 0 ), F (x 0 , p)) = Γ -1 (x 0 , p)
that is the desired convergence

lim t→+∞ (x 0 (t), p(t)) = (x 0 , p).
In the next section, we present examples for which the observation has the structure (3) and show how to define the set W and construct the map Ψ that fulfill conditions ( 5) and (6) of Proposition 3.1.

Examples

We first show that the model treated in [START_REF] Rapaport | A robust asymptotic observer for systems that converge to unobservable states -a batch reactor case study[END_REF] is a particular case of application of Proposition 3.1. Then, we develop a new example in dimension two with two unknown parameters and an observation of dimension two.

A bioreactor model

We revisit the state estimation problem in the batch bioreactor model when measuring the biogas:

   ẋ = µ(s)x ṡ = -µ(s)x (13)
y = µ(s)x (14)
Here the function µ, which is null at 0 only, is assumed to be known, while the initial condition is not known. We show that the asymptotic observer proposed in [START_REF] Rapaport | A robust asymptotic observer for systems that converge to unobservable states -a batch reactor case study[END_REF] for this system is a particular case of the methodology that we propose here. Let us pose

z = x + s (15)
one has ż = 0 (16) Therefore, the system can be expressed equivalently as the one dimensional dynamics ṡ = f (s, z) = -µ(s)(z -s)

with observation

y(t) = h(s(t), z) = µ(s(t))(z -s(t)), t ≥ 0
where z is an unknown parameter. Clearly all positive solutions with s < z converges asymptotically to the steady state s = 0, where parameter z is not identifiable. Here one has h(s, z) = -f (s, z)

which leads to

F (s, z) = -s (18)
as a function that verifies

h(s, z) = ∂F ∂s F (s, z)f (s, z) (19)
with F (0, z) = 0 whatever is z. Then we consider the map

Γ(s, z) = µ(s)(z -s) -s
and the set

Q = {(s, z) ∈ R 2 ; z > s > 0} (20)
For any ξ ∈ Γ(Q), one has

s = -ξ 2 , z = s + ξ 1 µ(s)
which shows that Γ is invertible on Γ(Q). We define the function

Ψ(ξ) = -ξ 2 ξ1 µ(-ξ2) -ξ 2
that is well-defined and continuous on the set

W = {ξ ∈ R 2 ; ξ 2 < 0} (21)
which contains Γ(Q), and coincides with Γ -1 on Γ(Q). Note that for any initial condition (t 0 , s 0 ) with s 0 ∈ (0, z), one hast t0 y(τ )dτ < 0 for any t > t 0 .

Conditions of Proposition 3.1 are thus fulfilled for any initial condition (t 0 , s 0 ) such that s 0 > 0 and s 0 < z.

Finally the estimator of z

     v(t) = y(t), v(t 0 ) = 0 ẑ(t) = y(t 0 ) µ(v(t)) + v(t), t > t 0
is exactly the one proposed in [START_REF] Rapaport | A robust asymptotic observer for systems that converge to unobservable states -a batch reactor case study[END_REF]. In this last reference, the benefits of the estimator are discussed in comparison with classical Luenberger or high gain observers for various growth functions µ.

A mechanical model

Consider the classical harmonic oscillator with damping

mz + kz + c ż = 0
where the mass m and the damping coefficient c are unknown positive parameters (the spring constant k is assumed to be known). We assume that the position and the damping force are measured. The system with x = (z, ż) in

R 2 writes ẋ = f (x, p) = 0 1 -k m -c m A(p) x, p = m c y = h(x, p) = x 1 cx 2
One can check that the matrix A(p) is Hurwitz and thus all solutions converges to the steady state 0, whatever is the positive vector p. At steady state, one has f (0, p) = 0 and h(0, p) = 0 and the system is thus not identifiable at 0. However, let us show that the system is infinitesimally identifiable away from 0.

• We show now how to apply the methodology exposed in Section 3. The map h can be written as

h(x, p) = -c k -m k c 0 J(p) f (x, p)
and then

F (x, p) = -c k x 1 -m k x 2 cx 1 verifies ∂F ∂x (x, p) = J(p) (22)
with F (0, p) = 0 for any p. The next step is to define

Γ : (x, p) →     x 1 cx 2 -c k x 1 -m k x 2 cx 1    
and consider the set

Q = {(x, p) ∈ R 2 × R 2 ; x 1 = 0, x 2 = 0, m > 0, c > 0}.
For ξ = Γ(x, p) in Γ(Q), one obtains

x 1 = ξ 1 , x 2 = ξ 2 c , c = ξ 4 ξ 1 , m = - kξ 3 + ξ 4 x 2
showing that Γ is invertible on Γ(Q). However, the map

Ψ(ξ) =            ξ 1 ξ 2 ξ 4 ξ 1 - (kξ 3 + ξ 4 )ξ 4 ξ 1 ξ 2 ξ 4 ξ 1           
is well-defined and continuous on the set

W = {ξ ∈ R 4 ; ξ 1 = 0, ξ 2 = 0}
which contains Γ(Q) and coincides with Γ -1 on Γ(Q). Conditions of Proposition 3.1 are thus verified, provided that the initial condition verifies x 1 (t 0 ) = 0 and x 2 (0) = 0. However, note that for any solution x(•) of (1) that is not at steady state, x 1 (t) and x 2 (t) are both non null for almost any t. Therefore, one can initialize the system (7) with t 0 as closed as desired to the initial time, such that the observation at t 0 verifies y 1 (t 0 ) = 0 and y 2 (t 0 ) = 0.

Finally, the estimator is as follows

           vi (t) = y i (t), v i (t 0 ) = 0 (i = 1, 2) m(t) = - (kv 1 (t) + v 2 (t))v 2 (t) y 1 (t 0 )y 2 (t 0 ) ĉ(t) = - v 2 (t) y 1 (t 0 )
We have compared this asymptotic observer with a least-square estimation of the parameters, provided from 1000 measurement points over the same time window [0, T ] using the Levenberg-Marquardt algorithm (lsqrsolve function in Scilab software). Initial condition and parameters values used for the simulations are given in Table 4.2. We have compared the estimations on different time horizons T , without and with measurement noise (normal distribution with 0 mean and 0.01 standard deviation).

x 1 (0) x 2 (0) m k c 0.2 -2 1 1 0.4
Table 1: Initial condition and parameter values used for the simulations Time-varying estimations provided by the asymptotic observer are depicted on Fig. 1 and2. The final estimation errors are given on Tables 4.2 and 4.2.

The comparisons leads to the following comments.

• the asymptotic observer has a relatively slow convergence: it needs T > 30 to give an accurate estimation which improves with larger time windows even with presence of noise. Table 3: Estimation errors on parameters m and c (with noise)

Figure 1: Simulation of the system and the estimation provided by the observer (without noise)

• the least square estimator does not provide an estimation as good as the asymptotic observer on large time horizons.

• the least square estimator is less affected by noise, but the quality of its estimation is deteriorating over large time horizon, when the state get where we pose θ = 1 m . Therefore, one may consider the "naive" Luenbeger observer

       d dt x1 = x2 + G 1 (x 1 -y 1 (t)) d dt x2 = -θ(ky 1 (t) + y 2 (t)) + G 2 (x 1 -y 1 (t)) d dt θ = -G 3 (x 1 -y 1 (t)) with m = 1 θ , ĉ = y 2 x2
to estimate parameters m and c. However, the error dynamics

d dt e =   G 1 1 0 G 2 0 -ky 1 (t) -y 2 (t) G 3 0 0   M (t)
e has a matrix M (•) that is periodically singular and that converges to a singular matrix, whatever is the choice of the gains G 1 , G 2 , G 3 . Then, the convergence of the error cannot be obtained. As an illustration, we have simulated this observer without measurement noise, for the gains G 1 = -13, G 2 = 32, G 3 = -20 which gave the best results for the estimation of m. As one can see on Fig. 3, the estimation of m is biased. While the innovation x1 -x 1 is rapidly small, we did not obtain a fast enough convergence of the error on x 2 with this observer, which explains the bad behavior of the estimation of c. We have proposed the derivation of an asymptotic observer for the estimation of parameters of systems that are not asymptotically identifiable. Although it takes the form of an observer, its philosophy is quite different from the conventional observers, in the sense that it can be considered as an expression that combines initial observation and integrals of the observed variables, without innovation (and gain) terms. As a consequence, its convergence speed cannot be tuned. We believe that it is the price to pay to obtain an asymptotic estimation without bias, in this precise context of lack of asymptotic identifiability. Numerical simulations show that this approach is more reliable than a classical least-squares method, especially on the long term, with or without noise. However, the structure of the proposed observer relies strongly on the initial observation which is prone to noise measurement. A future work will deal with robustness issues of this observer under stronger noises that the ones considered in this preliminary work.

If y 2 •

 2 = 0, ẏ1 = y 2 /c is non null and one has c = y 2 ẏ1 . If moreover ẏ2 = 0, one gets m = -(ky 1 + y 2 )y 2 ẏ1 ẏ2 . If ẏ2 = 0, then ẏ1 = x 2 is non null and ÿ2 = -kc m ẏ1 as well. We get m = -ky 2 ÿ2 . If y 2 = 0, y 1 is non null away from steady state. then ẏ2 = -kc m y 1 is non null and ÿ1 = -ky 1 m as well. We get m = -ky 1 ÿ1 and c = ẏ2 ÿ1 .

Table 2 :

 2 m , e c ) = (3.3%, 1.7%) (e m , e c ) = (9.7%, 11.1%) 30 (e m , e c ) = (7.0%, 3.2%) (e m , e c ) = (2.0%, 2.1%) 40 (e m , e c ) = (1.75%, 0.67%) (e m , e c ) = (0.33%, 0.33%) 50 (e m , e c ) = (4.3%, 2.6%) (e m , e c ) = (0.05%, 0.04%) 70 (e m , e c ) = (3.7%, 2.3%) (e m , e c ) = (0.0006%, 0.0005%) Estimation errors on parameters m and c (without noise) T least square observer 20 (e m , e c ) = (3.1%, 1.9%) (e m , e c ) = (8.2%, 9.6%) 30 (e m , e c ) = (2.0%, 1.9%) (e m , e c ) = (0.69%, 0.81%) 40 (e m , e c ) = (2.7%, 0.60%) (e m , e c ) = (0.057%, 0.13%) 50 (e m , e c ) = (4.2%, 2.3%) (e m , e c ) = (1.7%, 1.6%) 70 (e m , e c ) = (4.2%, 1.5%) (e m , e c ) = (0.5%, 0.5%)

Figure 2 :Remark 4 . 1 .

 241 Figure 2: Simulation of the system and the estimation provided by the observer (with noise)

Figure

  Figure Simulation with the Luenberger observer (without noise). Estimations are in blue

Acknowledgments

The authors tanks the bilateral research program PHC (Hubert Curien Programs) "TOURNESOL" between France and Belgium.


M Montagnac

G Agez

A Patoux

A Arbouet

V Paillard
email: vincent.paillard@cemes.fr

Engineered Near-and Far-Field Optical Response of Dielectric Nanostructures using Focused Cylindrical Vector Beams

Near-and far-field optical properties of silicon nanostructures under linear polarization (Gaussian beam), and azimuthally or radially focused cylindrical vector beams are investigated by finite-difference time-domain method (FDTD) in Meep open-source software. A python toolkit allowing FDTD simulations in Meep for using those excitation sources is provided. In addition to the preferential excitation of specific electric or magnetic resonance modes as function of the excitation beam polarization, it is shown in the case of spheroids that shape anisotropy affects the resonance wavelength and the dipole orientation of the magnetic or electric dipole mode. For radial or linear polarization, the electric dipole resonance is split by an anapole mode depending on the spheroid symmetry axis with respect to the electric field orientation. Finally, the optical properties in both far-field (scattering pattern) and near-field (electric and magnetic field hot spots) can be tuned by changing the excitation polarization at a fixed wavelength and selecting properly the spheroid shape and dimensions. These numerical simulations can be extended to more complex shapes, or fabrication-friendly nanostructures such as nanocylinders with circular or elliptic sections.

I. INTRODUCTION

A plane wave can interact strongly with high refractive index dielectric particles of subwavelength dimension supporting optical resonances [START_REF] Cao | Tuning the Color of Silicon Nanostructures[END_REF][START_REF] Kuznetsov | Optically resonant dielectric nanostructures[END_REF] . Because both electric and magnetic multipolar modes of comparable strengths can be excited in such nanostructures, several degrees of freedom can then be used to engineer the optical response of the nanoantenna [START_REF] Kuznetsov | Optically resonant dielectric nanostructures[END_REF][START_REF] Staude | Tailoring Directional Scattering through Magnetic and Electric Resonances in Subwavelength Silicon Nanodisks[END_REF][START_REF] Liu | Multipole and multimode engineering in Mie resonance-based metastructures[END_REF] , as well as its near-field coupling to quantum emitters [START_REF] Krasnok | Spectroscopy and Biosensing with Optically Resonant Dielectric Nanostructures[END_REF][START_REF] Bidault | Dielectric nanoantennas to manipulate solid-state light emission[END_REF][START_REF] Poumirol | Unveiling the Optical Emission Channels of Monolayer Semiconductors Coupled to Silicon Nanoantennas[END_REF] . Many works have been dedicated to improving the directivity of enhanced light scattering by studying the overlap between electric and magnetic multipole modes [START_REF] Fu | Directional visible light scattering by silicon nanoparticles[END_REF][START_REF] Wiecha | Strongly Directional Scattering from Dielectric Nanowires[END_REF] , or creating a Huygens source out of dielectric nanoparticles [START_REF] Bag | Transverse kerker scattering for angstrom localization of nanoparticles[END_REF] .

The optical behavior is usually controlled by modifying the material [START_REF] Baranov | All-dielectric nanophotonics: the quest for better materials and fabrication techniques[END_REF] , size [START_REF] Cao | Tuning the Color of Silicon Nanostructures[END_REF][START_REF] Fu | Directional visible light scattering by silicon nanoparticles[END_REF] , shape [START_REF] Wiecha | Strongly Directional Scattering from Dielectric Nanowires[END_REF][START_REF] Reena | Tunable unidirectional scattering of ellipsoidal single nanoparticle[END_REF] , and coupling [START_REF] Yan | Magnetically induced forward scattering at visible wavelengths in silicon nanosphere oligomers[END_REF][START_REF] Shibanuma | Experimental Demonstration of Tunable Directional Scattering of Visible Light from All-Dielectric Asymmetric Dimers[END_REF] of the nanoparticles, but alternative approaches involving focused cylindrical vector beams (CVB), such as higher-order Laguerre-Gaussian singular beams [START_REF] Youngworth | Focusing of high numerical aperture cylindrical-vector beams[END_REF][START_REF]Cylindrical vector beams: from mathematical concepts to cations[END_REF] , are being considered. Contrary to a linearly polarized excitation (plane wave or focused Gaussian beam), an azimuthally or radially polarized beam was used to selectively excite either magnetic or electric multipolar modes, leading to the modification of the scattering spectrum and radiation pattern of a given individual nanoantenna [START_REF] Banzer | On the experimental investigation of the electric and magnetic response of a single nanostructure[END_REF][START_REF] Woźniak | Selective switching of individual multipole resonances in single dielectric nanoparticles: Selective switching of individual multipole resonances in single dielectric nanoparticles[END_REF][START_REF] Manna | Selective excitation and enhancement of multipolar resonances in dielectric nanospheres using cylindrical vector beams[END_REF][START_REF] Parker | Excitation of Nonradiating Anapoles in Dielectric Nanospheres[END_REF] .

Recent studies deal with various dielectric or plasmonic nanostructures, from single nano-objects [START_REF] Bautista | Second-Harmonic Generation Imaging of Metal Nano-Objects with Cylindrical Vector Beams[END_REF][START_REF] Manna | Selective Induction of Optical Magnetism[END_REF] to an assembly of a few nanoparticles [START_REF] Yanai | Near-and Far-Field Properties of Plasmonic Oligomers under Radially and Azimuthally Polarized Light Excitation[END_REF][START_REF] Reich | Selection Rules for Structured Light in Nanooligomers and Other Nanosystems[END_REF][START_REF] Kroychuk | Enhanced Nonlinear Light Generation in Oligomers of Silicon Nanoparticles under Vector Beam Illumination[END_REF] . In this paper, we present electrodynamical simulations applied to silicon nanospheres and nanospheroids (Si NS) to investigate the effect of shape anisotropy and the kind of CVB excitation on the behavior of the electric and magnetic dipole modes, as well as the anapole. We thus show that, by selecting the Si NS and CVB, it is possible to engineer the far-field properties, such as modifying the radiation pattern and scattering directivity or switching from a strongly radiating to a nonradiating mode at a fixed wavelength, as well as the near-field properties such as controlling, separating spatially, and enhancing electric and magnetic hotspots.

The numerical simulations used for this present study were performed with the finite-difference time-domain (FDTD) method, using the open-source Meep software package [START_REF] Oskooi | Meep: A flexible free-software package for electromagnetic simulations by the FDTD method[END_REF] . We remind hereafter a brief description of two methods for generating radially and azimuthally polarized doughnut excitation in Meep (i) from Laguerre-Gaussian (LG) modes in the paraxial approximation, and (ii) from the calculation of a beam focused by a high-numerical-aperture (NA) aplanatic lens. We discuss their field of applications and provide for both methods python script packages available in our Github repository [START_REF] Agez | Focused cylindrical vector beam[END_REF] , allowing a straightforward implementation in Meep. Note that commercial software has already been used to perform FDTD simulations with CVB excitation [START_REF] Manna | Selective Induction of Optical Magnetism[END_REF][START_REF] Das | Dark modes and field enhancements in dielectric dimers illuminated by cylindrical vector beams[END_REF] but, to the best of our knowledge, no open-source library was available for a direct utilization. We also point out that a purely analytical approach (generalized Lorenz-Mie theory) has been used in the case of high symmetry nanoparticles [START_REF] Parker | Excitation of Nonradiating Anapoles in Dielectric Nanospheres[END_REF][START_REF] Orlov | Analytical expansion of highly focused vector beams into vector spherical harmonics and its application to Mie scattering[END_REF][START_REF] Zambrana-Puyalto | The role of the angular momentum of light in Mie scattering. Excitation of dielectric spheres with Laguerre-Gaussian modes[END_REF][START_REF] Luk ′ Yanchuk | Optimum Forward Light Scattering by Spherical and Spheroidal Dielectric Nanoparticles with High Refractive Index[END_REF] . In this paper, the numerical approach was chosen in the perspective of investigating any kind of nanostructures, even of complex shape, that can be obtained through top-down fabrication techniques [START_REF] Kuznetsov | Optically resonant dielectric nanostructures[END_REF][START_REF] Wiecha | Evolutionary multi-objective optimization of colour pixels based on dielectric nanoantennas[END_REF][START_REF] Patoux | Challenges in nanofabrication for efficient optical metasurfaces[END_REF] .

II. FOCUSED CYLINDRICAL VECTOR BEAMS EXCITING A DIELECTRIC NANOPARTICLE

A. Focused cylindrical vector beam sources in Meep/FDTD

The cylindrical vector beams considered in the present study are characterized by both an azimuthally or radially distributed polarization and an intensity distribution having cylindrical symmetry. They can be seen as linear combination of a right-handed and a left-handed circularly polarized vortex beam with opposite orbital angular momentum l = ±1 34 . Higher-order modes like Laguerre-Gaussian (LG) modes can be used to define analytical expressions in the paraxial regime and then be implemented in FDTD script to define the complex amplitude of the electric and/or magnetic source in a transverse plane before or after the waist position. From this initial condition, the beam propagation is calculated by solving the time-dependent Maxwell's equations. However, for many applications, it is interesting to tightly focus a CVB in order to generate a strong longitudinal electric or magnetic field component and to spatially separate the electric and magnetic energy density [START_REF] Kasperczyk | Excitation of Magnetic Dipole Transitions at Optical Frequencies[END_REF] . When the beam divergence angle becomes too large, the fields computed using LG modes deviate from the expected behavior at the focus point and a more general (nonparaxial) approach has to be considered. For this purpose, several methods have been proposed based on n-order corrected expression of the field components [START_REF] Cerjan | Orbital angular momentum of Laguerre-Gaussian beams beyond the paraxial approximation[END_REF][START_REF] Yan | Description of a radially polarized Laguerre-Gauss beam beyond the paraxial approximation[END_REF][START_REF] Barnett | Orbital angular momentum and nonparaxial light beams[END_REF] or using the scalar-complex-source model [START_REF] Orlov | Complex source beam: A tool to describe highly focused vector beams analytically[END_REF] . In practice, tightly focused beams are obtained using high numerical aperture (NA) optical devices. Thus, we chose to implement a theoretical approach based-on the angular spectrum representation of an optical field focused by an aplanatic lens, proposed by Novotny and Hecht [START_REF] Novotny | Principles of Nano-Optics[END_REF] . This model takes into account the filling factor that describes how much the incoming beam is expanded relative to the size of the lens. Using this expression as initial condition in FDTD is time consuming due to numerous numerical integrations compared to the LG paraxial model, but is very effective for extremely tightly focused beams (i.e. NA > 0.9).

Several CVB examples for NA ranging from 0.3 to 1.3 and generated by the two methods can be found in our Github repository [START_REF] Agez | Focused cylindrical vector beam[END_REF] . All simulations in the following are performed taking NA = 0.9 while keeping n = 1 for the nanoparticle environment refractive index. Finally, note that the expressions of the magnetic H and electric E fields for the radially and the azimuthally polarized cases are related as follows:

E Azi = Z H Rad (1) E Rad = Z H Azi ( 2
)
where Z = µ/ε is the wave impedance, µ and ε are the magnetic permeability and the electric permittivity of the medium, respectively. Since in our numerical method the wave impedance is normalized to 1, the magnetic and the electric amplitudes can be directly compared in all the results presented hereafter.

B. Silicon nanosphere excited by focused cylindrical vector beams

Despite their simple shape and high symmetry, highrefractive index nanospheres such as Si NS exhibit more or less complex resonance spectra as function of size (radius R) when excited by a nonpolarized plane wave. The lowest order resonance, occurring at a wavelength in the material λ /n Si commensurate with the sphere perimeter 2π R, corresponds to a magnetic dipole (MD) resulting from a curl-like electric displacement field (See Fig. 1). With increasing size, higher order modes may appear starting with the electric dipole (ED), then the magnetic quadrupole (MQ), etc. Several authors [START_REF] Woźniak | Selective switching of individual multipole resonances in single dielectric nanoparticles: Selective switching of individual multipole resonances in single dielectric nanoparticles[END_REF][START_REF] Manna | Selective excitation and enhancement of multipolar resonances in dielectric nanospheres using cylindrical vector beams[END_REF][START_REF] Das | Dark modes and field enhancements in dielectric dimers illuminated by cylindrical vector beams[END_REF] have reported that using focused beams with different polarization states yields to very different resonance spectra. A linearly polarized Gaussian beam gives a spectrum that is similar to that of a linearly polarized plane wave (assuming a spot size at waist larger than the particle diameter) [START_REF] Zambrana-Puyalto | Excitation of single multipolar modes with engineered cylindrically symmetric fields[END_REF] . The scattering spectrum of a Si NS illuminated by an azimuthally (resp. radially) polarized focused CVB, on the the other hand, is very different as only the magnetic (resp. electric) multipolar modes are excited (See the scattering spectra in the center panel of Fig. 1). By changing the beam shape and polarization, which is fairly simple experimentally, it is thus possible to switch on and off different optical resonances of the nanosphere [START_REF] Woźniak | Selective switching of individual multipole resonances in single dielectric nanoparticles: Selective switching of individual multipole resonances in single dielectric nanoparticles[END_REF] .

In addition, it must be emphasized that the magnetic (resp. electric) multipolar modes excited by the azimuthally (resp. radially) polarized focused CVB are not the same as the corresponding modes observed under linearly polarized Gaussian beam excitation (lin-GB). This is obvious for instance by comparing the MD resonance in the case of azimuthally polarized focused CVB (azi-CVB) and lin-GB excitations. The electric displacement current lies in different planes (Fig. 1(d,j)), and the resulting magnetic dipole and magnetic vector field have a different orientation (Fig. 1(e,k)). Under lin-GB excitation, the displacement current lies in the xz-plane (Fig. 1(j)) and the magnetic dipole is oriented along the y-axis, perpendicularly to both the propagation direction and polarization of the incident wave (Fig. 1(k)). On the other hand, under azi-CVB illumination, the displacement current lies in the xyplane, and the magnetic dipole is parallel to the z-axis due to the strong magnetic field longitudinal component at the beam waist (Fig. 1(e-f)) [START_REF] Novotny | Principles of Nano-Optics[END_REF] . The same analysis can be made for the ED resonance. With lin-GB, the electric dipole is roughly aligned with the incident polarization along the x-axis (Fig. 1(h)). For the radially polarized CVB (rad-CVB), it is found along the z-axis due to the strong electric field longitudinal component at the beam waist (Fig. 1(a-c)) [START_REF] Novotny | Principles of Nano-Optics[END_REF] . The dipole orientation as function of the excitation beam is also illustrated in Fig. 1(c,f,g,l), where the torus-like far-field scattering patterns correspond to electric or magnetic dipoles parallel to the z-axis (azi-and rad-CVB, Fig. 1(c,f)), or along the x-axis (LG-beam, Fig. 1(g,l)). It is also interesting to note that, due to a full cylindrical symmetry around the optical axis z, the MD (resp. ED) radiation pattern for the azi-CVB (resp. rad-CVB) is closer to a perfect dipole than the one obtained by lin-GB. All these points have to be taken into account when interpreting experimental and theoretical results. Hence, although the same resonance modes are found at the same wavelength in the case of highly symmetric structures such as the sphere, scattering patterns cannot be extrapolated from the lin-GB case to the other configurations. This will be also the case for the near-field maps as shown latter. C. Silicon nanospheroid excited by focused cylindrical vector beams: influence of shape anisotropy

The shape influence is investigated by considering oblate and prolate nanospheroids. Starting from a sphere, a deformation is applied along one of the axis (referred to as the sym-metry axis) at constant volume. The spheroid shape (oblate or prolate) is defined by the aspect ratio r, equal to the symmetry axis half-length divided by the sphere radius R. r > 1 (resp. r < 1) corresponds to a prolate (resp. oblate) spheroid. An example of deformation along the z-axis is given in Fig. 2(a-c).

Magnetic dipole resonance behavior as function of excitation

In the case of the example of Fig. 2(a-c), a wavelength shift is expected for the magnetic dipole resonance when changing the excitation from lin-GB to azi-CVB, except in the case of the sphere r = 1. For the azi-CVB excitation, the expected behavior is a redshift (resp. blueshift) of the MD resonance wavelength in the case of the oblate (resp. prolate) spheroid compared to the sphere case. Indeed, as seen in Fig. 2 (lower orange frame) the disk radius enclosing the doughnut shape of the electric displacement current in the xy-plane is larger (resp. smaller) than the disk ra-dius in the sphere. At the same time, the wavelength of the MD mode excited by the x-axis-polarized lin-GB follows a shift in the opposite sense: blueshift in the oblate case, due to the smaller dimension in the xz-plane, and redshift in the prolate case, due to the larger dimension in the same plane (see Fig. 2, upper blue frame). The scattering spectra showing the MD mode wavelength shift as function of the aspect ratio r for each polarization are given in Fig. 2(e-f), as well as the crossover of the curves giving the MD resonance position (Fig. 2(d)).

In summary, in the case of spheroids, switching between lin-GB and azi-CVB leads simultaneously to a change of the magnetic dipole orientation, hence the far-field scattering pat- tern, and to a shift in the opposite way of the MD resonance wavelength as function of the aspect ratio r.

Considering the deformation applied along the y-axis, the discussion is very similar to the case described above, except that the MD resonance wavelength shift between the lin-GB and the azi-CVB is reversed (See Supplemental Material, Fig. S1). Finally, for a deformation at constant volume along the x-axis, the MD resonance wavelength for the two excitation beams follows the same shift with r as the electric displacement current describes the same elliptic trajectory (See Supplemental Material, Fig. S1), but still in different planes (xz-plane for lin-GB and xy-plane for azi-CVB) leading to different far-field scattering patterns.

Electric dipole resonance behavior as function of excitation

Intuitively, a similar behavior could be expected for the ED resonance when changing the excitation from lin-GB to rad-CVB. Taking the case previously described of a deformation along the z-axis (Fig. 2(a)-(c)), the ED resonance of the oblate spheroid should be at shorter wavelength for the rad-CVB illumination (dipole along the short z-axis) than for the lin-GB (dipole along the long x-axis). In the prolate spheroid case, an opposite behavior could be expected, with a redshift of the ED resonance wavelength for the rad-CVB illumination (dipole along the long z-axis) compared to the lin-GB.

Such scenario is however not true, as illustrated in Fig. 3(a)-(b), where no significant ED mode wavelength shift is evi-denced. while a more complex behavior occurs in the lin-GB case (Fig. 2(e)). In that case, and more generally when the incident light polarization is perpendicular to the spheroid symmetry axis (Fig. 4), the ED mode is split in two peaks separate by a dip corresponding to an anapole, as discussed latter.

As for the MD mode previously described, the electric vector field and the scattering pattern at ED resonance have different orientations depending on the excitation polarization Fig. 3(a)-(b). Furthermore, the scattering pattern is distorted in the case of the lin-GB (Fig. 3(c)-(d)), due to the transverse character of the polarization and the propagation direction, as well as the fact that other modes such as the MQ are simultaneously excited. Contrary, the scattering pattern in the rad-CVB case corresponds rather well to a perfect electric dipole oriented along the z-axis (Fig. 3(f)-(g)).

The zero scattering point at the wavelength just below the ED peak (480-490 nm) (marked (e) and (h) in the scattering spectra in Fig. 3(a)-(b)) corresponds to an anapole, a nonradiating mode resulting from the destructive interference of electric and toroidal dipole modes (See also the far-field radiationless behavior in Fig. 3(e)-(h)). This toroidal dipole is a consequence of the circulating magnetic flux along a closed loop [START_REF] Parker | Excitation of Nonradiating Anapoles in Dielectric Nanospheres[END_REF][START_REF] Miroshnichenko | Nonradiating anapole modes in dielectric nanoparticles[END_REF][START_REF] Wei | Excitation of the radiationless anapole mode[END_REF][START_REF] Yang | Nonradiating anapole states in nanophotonics: from fundamentals to applications[END_REF] . In highly anisotropic nanostructures, such as the spheroids considered here, the loop discontinues reducing the system to a transverse magnetic quadrupole formed by two anti-parallel magnetic moments, as reported by Miroshnichenko et al. [START_REF] Miroshnichenko | Nonradiating anapole modes in dielectric nanoparticles[END_REF] . This does not prevent the formation of an anapole. Examples showing either the closed loop or the transverse MQ as function of the deformation ratio are given in Supplemental Material. In any case, the phase difference between the toroidal dipole (or the transverse magnetic quadrupole) and the electric dipole drives the far-field behavior (See also Appendix A, Fig. 7). The toroidal dipole and the electric dipole of the ED mode are out-of-phase in the anapole case. They are in phase in the case of the ED mode.

Compared to the previous case, when the symmetry axis is perpendicular to the excitation polarization (Fig. 4(a)-(b)), the ED mode behavior is totally different and is split, giving a dip between two peaks, hereafter referred to as ED ⊥ and ED ∥ . The antiresonance mode between ED ⊥ and ED ∥ can be attributed to an anapole, in agreement with the near-zero scattering cross-section in the spectra and the nonradiating patterns in Fig. 4(d)-(g). The far-field behavior of ED ⊥ and ED ∥ resonances is also striking, as they exhibit highly directional scattering patterns along a specific axis, very different from the isotropic scattering pattern of an oscillating dipole (Fig. 4(c),(e),(f),(h)). The scattered intensity is along the symmetry axis in the case of ED ∥ , and perpendicular to the symmetry axis in the case of ED ⊥ . The ED ⊥ and ED ∥ peak wavelengths exhibit a crossover as function of the deformation ratio r (the peaks are merged in the sphere case, see Fig. 4(a)-(b)). Thus, for an x-symmetry axis nanospheroid excited by rad-CVB, the ED ∥ , whose scattered intensity is mainly directed along the x-axis in the case of a deformation ratio r = 0.8, becomes a ED ⊥ whose scattered intensity is mainly directed along the y-axis in the case of a deformation ratio r = 1.2 (Fig. 4(c),(f)), and reciproqually (Fig. 4(e),(h)). Contrary to the anapole, both the resonance and the highly directional scattering pattern at the wavelength corresponding to ED ⊥ and ED ∥ resonances are due to the constructive interference of the electric dipole mode and either the transverse MQ or the toroidal dipole mode. The resonance at shorter (resp. longer) wavelength corresponds to the electric field vortices positioned symmetrically to the z-axis-oriented electric dipole in the smaller (resp. larger) section of the spheroid. The behavior will be similar in the other cases where the same behavior of ED mode split by the anapole is encountered, including the Gaussian beam linearly polarized along the x-axis illuminating spheroids with either the yor z-symmetry axis (See for instance Fig. 4a). The difference with the rad-CVB is that the scattering will not be totally cancelled at the anapole wavelength between ED ⊥ and ED ∥ due to the presence of the MQ mode in the same wavelength range, which is excited simultaneously with the other modes. This can be seen in Fig. 4, where the scattering intensity drops to zero between ED ⊥ and ED ∥ in the rad-CVB case (Fig. 4(b)), but lies slightly above zero in the lin-GB case (Fig. 4(a)).

III. CYLINDERS WITH CIRCULAR OR ELLIPTIC SECTIONS

For applications, a top-down approach involving electron lithography and reactive ion etching is commonly used to produce Si NS from Silicon on Insulator or similar substrates [START_REF] Baranov | All-dielectric nanophotonics: the quest for better materials and fabrication techniques[END_REF][START_REF] Wiecha | Evolutionary multi-objective optimization of colour pixels based on dielectric nanoantennas[END_REF] . Such approach is however not very suited to obtain spheroidal NS, but instead wires with cubic or rectangular section or cylinders with circular or elliptic section. To compare with spheres and spheroids, we take the example of cylinders with circular and elliptic sections, respectively. Starting from a nanocylinder of height H and a circular section of radius R=H, a deformation at constant volume is applied along either the cylinder axis (z-axis) or along a diameter (defined as either the xor y-axis). The general behavior for the cylinders is very similar to the one of the spheres and spheroids of similar dimensions, with only a slight shift of the resonances wavelengths. The observed shift is usually a redshift due to a larger volume of the cylinder compared to the spheroid (the sphere or the spheroid is circumvented in the corresponding cylinder with the circular or elliptic section, as shown in Fig. 5). The scattering intensity spectra for various nanocylinders and excitations are given in Supplemental Material (Fig. S3).

The chosen example of Fig. 5 corresponds to a cylinder with a deformation ratio r = 1.2 along the x-axis illuminated by a radially polarized beam. The elliptic section dimensions are thus R.r (long axis), and R/ √ r (short axis), the height being H/ √ r. This configuration can be compared to the sphere of radius R = 165 nm and the prolate spheroid shown in Fig. 4(c)-(e). The electric and magnetic vector fields are very similar (including the phase change between the electric dipole and the transverse magnetic quadrupole or the toroidal dipole, when looking at ED ⊥ , ED ∥ , and the anapole), as well as the patterns of the far field scattered intensity.

IV. ENGINEERING NEAR AND FAR FIELD OPTICAL RESPONSE OF SILICON NANOSPHEROIDS USING CYLINDRICAL FOCUSED BEAMS

As shown in the previous sections, it is possible to tune the optical response of a Si sphere or spheroid by changing the polarization of the incident light. This includes triggering either electric or magnetic modes using rad-CVB or azi-CVB excitation. In addition, it is possible by choosing properly the spheroid dimensions, to tune the intensity and the directivity of the scattered light at a given wavelength. To illustrate, we consider a spheroid with a deformation ratio r = 1.15 along the x-axis illuminated by a 516 nm wavelength radiation. Fig. 3 and Fig. 6 reveal that this wavelength corresponds to the ED resonance in the lin-GB case, to the MQ mode in the azi-CVB case, and to an anapole in the rad-CVB case. The first one is strongly radiating in the far-field while the others are considered dark or nonradiating. The spheroid should thus be invisible under CVB excitation at the 516 nm wavelength, and not under lin-GB. Another interest is to modify the electric and magnetic nearfield intensity distribution, which is an important parameter controlling the excitation of quantum emitters positioned in the vicinity of the nanoantenna [START_REF] Bidault | Dielectric nanoantennas to manipulate solid-state light emission[END_REF][START_REF] Poumirol | Unveiling the Optical Emission Channels of Monolayer Semiconductors Coupled to Silicon Nanoantennas[END_REF][START_REF] Albella | Low-Loss Electric and Magnetic Field-Enhanced Spectroscopy with Subwavelength Silicon Dimers[END_REF] .

The example of Fig. 6 shows that the maximum intensity of the electric near field has roughly the same value, but is located at different spatial positions around the spheroid. As a consequence, optimizing the nanoantenna -emitter couple needs to spatially position precisely the emitting dipoles in the nanostructure near field, as well as their orientation. Taking the example of Transition Metal Dichalcogenide (TMD) monolayer wrapped around Si nanostructures [START_REF] Poumirol | Unveiling the Optical Emission Channels of Monolayer Semiconductors Coupled to Silicon Nanoantennas[END_REF][START_REF] Sortino | Dielectric Nanoantennas for Strain Engineering in Atomically Thin Two-Dimensional Semiconductors[END_REF] , the near field hot-spot leading to an increased number of excitons would be located at different positions according to the excitation polarization. Thus, in the case of the lin-GB, more excitons would be generated in the TMD monolayer on each side of the spheroid along the x-axis, while they would be created on each side of the spheroid along the y-axis in the rad-CVB case. In addition to this process at the excitation wavelength, the nanostructure shape could be calculated in order to make a resonance mode wavelength coincide with the exciton emission one. Furthermore, since the emitting dipoles are preferentially generated at different positions around the nanoantenna, the photoluminescence (PL) intensity pattern is expected to be very different depending on the excitation beam polarization. Note also that the PL emission directivity could be further modified by the orientation of the emitting dipoles as, depending on the kind of TMD, the emitting dipole can be parallel (neutral exciton), or perpendicular (dark exciton in specific materials such as WSe 2 ) to the TMD layer [START_REF] Poumirol | Unveiling the Optical Emission Channels of Monolayer Semiconductors Coupled to Silicon Nanoantennas[END_REF] .

The spatial separation of electric and magnetic near field is important for controlling emitters that support both electric and magnetic dipolar transitions, such as rare-earth ions [START_REF] Bidault | Dielectric nanoantennas to manipulate solid-state light emission[END_REF][START_REF] Chacon | Measuring the Magnetic Dipole Transition of Single Nanorods by Spectroscopy and Fourier Microscopy[END_REF] . Selective excitation of the magnetic dipole transition can be obtained thanks to resonant azi-CVB excitation [START_REF] Kasperczyk | Excitation of Magnetic Dipole Transitions at Optical Frequencies[END_REF] . By combining azi-CVB excitation and locally enhanced value of the magnetic near-field, this effect could be further improved. In Fig. 6, the maps of the electric and magnetic energy density at 5 nm outside the surface of the spheroid show the electric and magnetic hot spots that can be manipulated by changing the excitation beam polarization at the same wavelength. In addition, the position and the nature (electric or magnetic) of the emitting dipole exciting a resonance mode (for instance the magnetic dipole resonance, at longer wavelength than the excitation tuned to ED or MQ resonance, see Fig. 1) would lead to specific emission patterns [START_REF] Bidault | Dielectric nanoantennas to manipulate solid-state light emission[END_REF][START_REF] Sugimoto | Magnetic purcell enhancement by magnetic quadrupole resonance of dielectric nanosphere antenna[END_REF] . The electromagnetic local density of states (LDOS) is an important parameter for enhanced emission rate, which could be investigated simulta-neously for either enhanced magnetic or electric Purcell effect. Optimizing several parameters at both the excitation and emission wavelengths, the nature of the emitting dipole and its orientation with respect to the antenna, will require an evolutionary algorithm coupled to the optical simulation tools [START_REF] Wiecha | Evolutionary multi-objective optimization of colour pixels based on dielectric nanoantennas[END_REF] .

V. CONCLUSION

To summarize, we provide a Python toolkit that describes the various CVB sources and allows FDTD simulations under CVB excitation to be performed using the Meep open-source software. We demonstrate that, by properly selecting the shape of a silicon nanospheroid and the polarization of the incident cylindrical vector beam, the optical response of the nanoantenna can be adjusted, either in the near-field or in the far-field. An example of selective excitation at a fixed wavelength of an electric dipole, magnetic quadrupole, or anapole mode as a function of CVB polarization is described (linear, azimuthal, radial). At last, as spheres and spheroids are certainly not the best fabrication-friendly candidates for large scale reproducible production of nanostructures, we show that the trends in nanospheroids also found in nanostructures obtained by techniques on beam lithography, such as nanocylinders with circular or elliptic section. Fig. 7 shows the coupling of an electric dipole and a transverse magnetic quadrupole as function of the phase difference between the electric and magnetic fields radiated by the electric and magnetic dipoles, respectively. The -π/2 phase difference corresponds to the anapole mode and the +π/2 phase difference corresponds to an ED-like resonance mode, illustrating the nonradiating behavior of the anapole and the high directivity of the ED ⊥ and ED ∥ scattered intensity.

The example in Fig. 7 is analogous to the ED ∥ peak at 480 nm shown in Fig. 4(e). Aligning the magnetic dipoles along the x-axis also rotates the scattering pattern by 90 • to give a system analogous to the ED ⊥ peak at 497 nm in Fig. 4(c).

Supplemental Material:

The online version provides Supplemental Material at [URL will be inserted by publisher].

Fig. S1 summarizes all configurations such as presented in Fig. 3 and Fig. 4, i.e. the evolution of the scattering intensity as function of the aspect ratio r for a deformation along the three different symmetry axis and for each polarization (linear, azimuthal and radial). It evidences the cases where the ED mode is split in two peaks separated by an anapole,when the excitation polarization is perpendicular to the spheroid symmetry axis.

Fig. S2 gives more insight of the nanospheroid deformed along the x-symmetry axis excited by rad-CVB given in Fig. 4(b)-(h). In particular, the magnetic vector fields are shown. Depending on the deformation ratio, the magnetic vector field loop or the tranverse magnetic quadrupole can be seen.

Fig. S3 shows additional scattering spectra of nanocylinders of circular or elliptic section with various deformation ratio r and excitation polarization. This illustrates the same trend developed in the case of nanospheroids, but for nanostructures easier to produce by top-down fabrication techniques.

Figure 1 .

 1 Figure 1. Center panel: Scattering spectra of a 165 nm diameter Si nanosphere as function of excitation polarization. The ED (rad-CVB) and MD (azi-CVB) modes are normalized with respect to the corresponding modes of the lin-GB case. Electric ((a), (d), (h), (j)) and magnetic ((b), (e), (i), (k)) vector fields for the different resonance modes and excitation polarizations (ED with rad-CVB (green panel), MD with azi-CVB (orange panel), ED and MD with Gaussian beam (blue lower panels)); (c), (f), (g), and (l) show the far-field scattering patterns characteristic of a dipole resonance, with different dipole orientations given by the yellow arrows in (a), (e), (h) and (k). The incident light propagates from the bottom to the top; the incident beam polarization is represented in the xy-plane using red arrows.

  intensity (arb. u.)Scattering intensity (arb. u.)

Figure 2 .

 2 Figure 2. Si nanospheroids from (a) oblate (compressed along z-symmetry axis) to (c) prolate (elongated along z-symmetry axis). The sphere has a diameter of 165 nm. Vector fields and far-field scattering patterns of the MD mode with Gaussian beam (upper blue panel) or azi-CVB (lower orange panel) excitation. The incident light propagates from the bottom to top; the beam polarization is represented in the xy-plane using red arrows. (e) (resp. (f)) shows the scattering spectra as function of the deformation ratio r under lin-GB (resp. azi-CVB) excitation. The spectra are translated for better clarity, and the dashed horizontal line gives the zero scattering for the extreme r values. (d) shows the opposite shift of the MD resonance wavelength as function of r and excitation polarization.

Figure 3 .

 3 Figure 3. Scattering spectra as function of the aspect ratio r for polarization parallel to the symmetry axis: (a) lin-GB and x-symmetry axis, and (b) rad-CVB and z-symmetry axis. When indicated, the dashed horizontal line gives the zero scattering. ((c)-(h)) Far-field scattering patterns and vector field maps giving to the comparison of the ED resonance and the anapole excited by either lin-GB or rad-CVB for r=1.2 and for r=0.8. The far-field patterns are normalized to the maximum value within a given kind of excitation. The intensity and vector field maps are normalized independently. Green arrows indicate the deformation axis.

Figure 4 .

 4 Figure 4. Scattering spectra as function of the aspect ratio r for a polarization perpendicular to the symmetry axis: (a) lin-GB and z-symmetry axis, and (b) rad-CVB and x-symmetry axis. When indicated, the dashed horizontal line gives the zero scattering. ((c)-(h)) Far-field scattering patterns and vector field maps corresponding to ED ⊥ , ED ∥ , and the anapole excited by rad-CVB for r=1.2 and for r=0.8. The far-field patterns are normalized to the maximum value. The intensity and vector field maps are normalized independently. Green arrows indicate the deformation axis.

Figure 5 .

 5 Figure 5. ED mode split in ED ⊥ and ED ∥ peaks separated by the anapole mode in the case of a Si nanocylinder (r = 1.2 with respect to the x-axis), illuminated by a rad-CVB propagating from bottom to top along the z-axis. The vector field maps show either the toroidal dipole or the transverse quadrupole mode canceling totally (anapole at 560 nm) or partially (ED ⊥ at 529 nm, ED ∥ at 593 nm) the electric dipole parallel to the z-axis. The far-field patterns are normalized showing the quasi-uniaxial strong scattering of ED ⊥ and ED ∥ , and the nonradiating behavior of the anapole (zero scattering intensity).

Figure 6 .

 6 Figure 6. Different near-fields maps obtained by switching the polarization (azi-CVB, rad-CVB and x-axis-polarized lin-GB) of a 516 nm wavelength excitation. The Si nanostructure is a prolate spheroid with a deformation ratio of r=1.15 along the x-symmetry axis. The colored maps in the left and right columns show the electric and magnetic near-field, respectively, at 5 nm outside the spheroid surface. Images with black background depict the 3D distribution of electric and magnetic energy density inside the spheroid for each kind of excitation. They are characteristic of a MQ mode (upper row), an anapole (middle row), and a ED mode (lower row) (See also the scattering spectra in Fig.3(b-c)). Each color bar is normalized by the local maximum energy density. For clarity, the values corresponding to the hatched part in the scale bar are not plotted.

Figures

  Figures/2021_09_09_figure6_ED-TMQ.pdf

Figure 7 .

 7 Figure 7. Scheme showing the electric dipole and the two magnetic dipoles forming the transverse magnetic quadrupole, and far field scattering patterns of the coupled ED and transverse MQ as function of phase difference ∆Φ E/H between E and H.

ACKNOWLEDGMENTS

The authors acknowledge funding from Agence Nationale de la Recherche under projects HiLight (ANR-19-CE24-0020-01) and EUR NanoX 2DLight (ANR-17-EURE-0009) in the framework of the Programme des Investissements d'Avenir, and CALMIP for the access to the supercomputing facility Olympe under project P19042.


Samir M Perlaza
email: samir.medinaperlaza@orange-ftgroup.com

Mérouane Debbah
email: merouane.debbah@supelec.fr

Samson Lasaulce
email: samson.lasaulce@lss.supelec.fr

Hanna Bogucka
email: hbogucka@et.put.poznan.pl

On the Benefits of Bandwidth Limiting in Decentralized Vector Multiple Access Channels

We study the network spectral efficiency of decentralized vector multiple access channels (MACs) when the number of accessible dimensions per transmitter is strategically limited. Considering each dimension as a frequency band, we call this limiting process bandwidth limiting (BL). Assuming that each transmitter maximizes its own data rate by water-filling over the available frequency bands, we consider two scenarios. In the first scenario, transmitters use non-intersecting sets of bands (spectral resource partition), and in the second one, they freely exploit all the available frequency bands (spectral resource sharing). In the latter case, successive interference cancelation (SIC) is used. We show the existence of an optimal number of dimensions that a transmitter must use in order to maximize the network performance measured in terms of spectral efficiency. We provide a closed form expression for the optimal number of accessible bands in the first scenario. Such an optimum point, depends on the number of active transmitters, the number of available frequency bands and the different signal-to-noise ratios. In the second scenario, we show that BL does not bring a significant improvement on the network spectral efficiency, when all transmitters use the same BL policy. For both scenarios, we provide simulation results to validate our conclusions.

I. INTRODUCTION

In a vector multiple access channel (MAC), a large set of transmitters share a limited set of frequency bands (channels) to communicate with a unique receiver [START_REF] Cover | Elements of information theory[END_REF]. When there exists a central controller (normally the receiver) a capacity achieving power allocation can be implemented by using an iterative water-filling algorithm [START_REF] Yu | Iterative water-filling for gaussian vector multiple-access channels[END_REF], [START_REF] Viswanath | Asymptotically optimal water-filling in vector multiple-access channels[END_REF]. In this case, the central controller knows the transmission parameters and instantaneous channel realizations of each transmitter over each channel. Thus, it is able to solve the global optimization problem and feed back the optimal power levels to each transmitter. However, in the absence of a central controller or the impossibility to interchange signaling messages between the transmitters to obtain a complete information of the network, achieving capacity becomes a non-trivial task [START_REF] Ghasemi | Capacity of fading channels under spectrumsharing constraints[END_REF]. Here, game theory has played a remarkable role, but solutions remain being highly suboptimal due to the lack of global information [START_REF] Perlaza | Game theory for dynamic spectrum sharing[END_REF].

To overcome this sub optimality, imposing orthogonal communications between transmitters using only a single channel has been a well-accepted solution, e.g., IEEE802.11 networks.

In this case, transmitters reduce the mutual interference and only overcome the interference of transmitters sharing the same channel. Nonetheless, up to the knowledge of the authors, the choices of the total number of available channels [START_REF] Jindal | Bandwidth partitioning in decentralized wireless networks[END_REF] as well as limiting the bandwidth to a single channel have been done in an ad hoc manner. This paper provides an analysis of the benefits of bandwidth limiting (BL), i.e., reducing the number of channels each transmitter can use in vector MAC. More specifically, we provide an answer to the following question: is it worth to limit the number of channels each transmitter might use regarding the network spectral efficiency?

We consider two scenarios. In the first scenario, transmitters have to use non-intersecting sets of channels. In the second one, transmitters can freely exploit all the available channels. In the second case, the receiver implements multiuser decoding and successive interference cancelation (SIC). Here, each transmitter is aware of both the decoding order and its respective noise plus interference levels. Under these conditions, the optimal decentralized policy for each transmitter to maximize its own data rate is to use a water-filling power allocation scheme considering the multiple access interference as noise [START_REF] Yu | Iterative water-filling for gaussian vector multiple-access channels[END_REF].

Our work is motivated by the following reasoning. In the first scenario, the fact that a transmitter uses several channels significantly reduces the total number of active transmitters. For instance, in the high signal to noise ratio (SNR) regime and considering a finite set of channels, few transmitters might occupy all the available channels. Following a water-filling power allocation, a given transmitter allocates the highest power levels to the channels with the highest gains. Then, the channels being used with low powers might have a negligible impact on its individual data rate. However, no other transmitter can access the spectrum, even though, a higher rate can be obtained by another transmitter. In the second scenario, having a transmitter using several channels does not reduce the number of active transmitters since they can co-exist in the same channels. However, the multiple access interference produced by the transmitters decoded in the last places might significantly reduce the data rates of those decoded in the first places. More specifically, the gain in data rate obtained by transmitters decoded in last places on certain channels, might not compensate the loss of the transmitters decoded in first places.

In both cases, this effect stems from the fact that transmitters maximize its own spectral efficiency independently of the others. This resulting suboptimal usage of the spectrum recall us the dilemma presented in [START_REF] Hardin | The tragedy of the commons[END_REF], known as the tragedy of the commons. Therein, it is shown how multiple individuals acting independently in their own self-interest can ultimately destroy a shared limited resource even when it is clearly not in anyone's long term interest. One of the solutions for these dilemma is to introduce regulation by an authority. In this paper, we analyze this point of view and try to find the rules (in terms of BL) each transmitter must follow to maximize the network spectral efficiency.

We show the existence of an optimal BL point in the first scenario. Such optimal number of channels is a function of the total number of active transmitters, the total number of channels, and the different signal to noise ratios (SNR). We present simulations where we observe a significant gain in terms of spectral efficiency. In the second scenario, we show that BL does not bring a significant improvement if all transmitters use the same BL parameter, i.e., all transmitters are limited to use the same number of channels.

II. SYSTEM MODEL

Consider a set K = {1, . . . , K} of transmitters communicating with a unique receiver using a set N = {1, . . . , N } of equally spaced frequency bands (channels) as shown in Fig. 1. In the information theory jargon, this network topology is known as vector MAC or parallel MAC [START_REF] Cover | Elements of information theory[END_REF]. Transmitters arrive sequentially to the network. Their index in the set K shows the order of arrival. All the radio devices are equipped with a unique antenna. Transmitter k ∈ K is able to simultaneously transmit over all the channels subject to a power-limitation,

∀k ∈ K, 1 N N n=1 p k,n p k,max , (1)
where p k,n and N p k,max denote the transmit power over channel n and the maximum transmittable power of transmitter k.

Without any loss of generality, we assume that all transmitters are limited by the same maximum transmittable power level, i.e., ∀k ∈ K and ∀n ∈ N , p k,max = p max . We denote the channel coefficients in the frequency domain between the receiver and transmitter k over channel n by h k,n . We assume that for the whole transmission duration, all the channel realizations remain constant. For all n ∈ N and for all k ∈ K, h k,n is a realization of a complex random variable h with independent and identically distributed (i.i.d) Gaussian real and imaginary parts with zero mean and variance 1 2 . The channel gain is denoted by g k,n = ||h k,n || 2 . Then, the channel gains can be modeled by realizations of a random variable g with exponential distributions with parameter ρ = 1, whose cumulative distribution function (c.d.f) and probability density function (p.d.f) are denoted by F g (λ) = 1e -λ and f g (λ) = e -λ , respectively. The received signals sampled at symbol rate

h 1 h 2 h 3 h K T x 1 T x 2 T x 3 T x K Fig. 1.
Vector multiple access channel with K transmitters and N available channels with h k = `hk,1 , . . . , h k,N ´for all k ∈ K.

can be written as a vector y = (y 1 , . . . , y N ) where the entries y n for all n ∈ N represent the received signal over channel n. Hence,

y = K k=1 H k s k + w, (2)
where

H k is an N -dimensional diagonal matrix with main diagonal (h k,1 , . . . , h k,N ). The N -dimensional vector s k = (s k,1 , . . . , s k,N
) represents the symbols transmitted by transmitter k over each channel. The power allocation profile of transmitter k, the vector (p k,1 , . . . , p k,N ), is the diagonal of the diagonal matrix P k = E s k s H k . The N -dimensional vector w represents the noise at the receiver. Its entries, w n for all n ∈ N , are modeled by a complex circularly symmetric additive white Gaussian noise (AWGN) process with zero mean and variance σ 2 .

Regarding the channel state information (CSI) we assume that each transmitter perfectly knows its own channel coefficients and the noise plus interference level at each channel. This is the case when transmitters are able to sense its environment or the receiver feeds back this parameter as a signaling message to all the transmitters.

We denote the set of channels being used by transmitter k by L k , i.e., ∀k ∈ K and ∀n ∈ L k , p k,n = 0, and ∀m ∈ N \L k , p k,m = 0. Depending on the conditions over each set L k , for all k ∈ K, we consider two different scenarios.

A. Scenario 1: Spectral Resource partition

In this scenario, a given channel cannot be used by more than one transmitter. Thus, this is equivalent to define the sets L k for all k ∈ K as a partition of the set N , i.e.,

• ∀(j, k) ∈ K 2 and j = k, L j ∩ L k = ∅, • ∀(j, k) ∈ K 2 and j = k, L j ∪ L k ⊆ N , • ∀k ∈ K, |L k | > 0.
Due to the asynchronous arrival of the users, we assume that there exists an order to access the set of channels N . We index the transmitters such that transmitter k ∈ K arrives in the k th position.

B. Scenario 2: Spectral Resource Sharing

In this scenario we allow several transmitters to use the same channels. Thus, this is equivalent to define the sets L k for all k ∈ K as a cover of the set N , i.e.,

• ∀k ∈ K, L k ⊆ N , • ∀k ∈ K, |L k | > 0.
We assume that the receiver performs multiuser decoding and successive interference cancelation (SIC) at each channel. The decoding order is the same in all channels and it is related to the arrival order. Here, transmitter k ∈ K, arriving in the k th position, is decoded in the Kk + 1 position.

To make a difference between scenario 1 and 2, we use the super index (i) with i ∈ {1, 2} for all the sets and variables associated with each of them. The noise plus multiple access interference (MAI) for transmitter k over channel n is denoted by α

(i) k,n , where α (1) k,n = σ 2 and α (2) k,n = σ 2 + k-1 j=1 p j,n g j,n ,
where, for all n ∈ N , p 0,n = 0 and g 0,n = 0. The SINR for transmitter k over channel n is denoted by γ

(i) k,n and ∀k ∈ K and ∀n ∈ N , γ (i) k,n = p k,n g k,n α (i) k,n . (3)
In both scenarios each transmitter k ∈ K aims to maximize its own data rate

R k (γ k ) (i) , i.e., R k (γ k ) (i) = N n=1 log 2 1 + γ (i) k,n , (4)
with γ

(i) k = (γ (i) k,1 , . . . , γ (i)
k,N ) subject to its power limitations and independently of the data rate of the other devices. We explain this process in the next section.

III. INDIVIDUAL SPECTRAL EFFICIENCY

Assuming that each channel bandwidth is normalized to 1 Hz, a given transmitter k sets out its transmit power levels p k,n , ∀n ∈ N by solving the optimization problem (OP)

max p k,n , ∀n∈Z (i) k n∈Z (i) k log 2 1 + γ (i) k,n s.t. 1 N n∈Z (i) k p k,n p max , (5)
where, for all k ∈ K, the set Z

(1) k = N \ L 1 ∪ . . . ∪ L k-1 , with L 0 = ∅, and Z (2) k = N . Thus, Z (i)
k is the set of channels available for user k in scenario i.

The solution to the OP in (Eq. 5) is given in [START_REF] Cover | Elements of information theory[END_REF] and thus, we only provide the solution hereafter; ∀k ∈ K and ∀n ∈ Z

(i) k , p k,n = β - α (i) k,n g k,n + , (6)
and, ∀k ∈ K and ∀n ∈ N \ Z

(1)

k , p k,n = 0. (7)
Here, the operator [.] + is the same as max (0, .). Given the sets A, B and the complement of the latter, B ′ , in a given universal set, the operation A \ B = A ∩ B ′ . The term β is a Lagrangian multiplier, known as water-level, chosen to satisfy (Eq. 1). The transmit power levels in (Eq. 6) can be iteratively obtained by using the water-filling algorithm described in [START_REF] Cover | Elements of information theory[END_REF].

From expression (Eq. 6), it can be implied that

L (i) k ⊆ Z (i)
k . Once the OP in (Eq. 5) has been solved, the data rate per channel of transmitter k ∈ K, is

Rk (γ k ) (i) = 1 |Z (i) k | n∈Z (i) k log 2 1 + γ (i) k,n , (8)
and then, its spectral efficiency Φ k is

Φ (i) k = |Z (i) k | N Ω (i) k Rk (γ k ) (i) , (9)
where, Ω

k represents the fraction of spectrum accessible for transmitter k. that due to the decentralized nature of the network, the individual spectral efficiency is maximized independently by each transmitter. As described in [START_REF] Hardin | The tragedy of the commons[END_REF], it might lead to significant losses in the network spectral efficiency. We study this effect in the next section.

IV. NETWORK SPECTRAL EFFICIENCY

We define the network spectral efficiency (NSE) Φ (i) as

Φ (i) = K k=1 Φ (i) k Φ (i) = K k=1 Ω (i) k R(i) k , (10)
for both scenarios, spectral resource partition (scenario 1) and spectral resource sharing (scenario 2). We analyze the NSE in the asymptotic regime, i.e., we assume that both the number of channels (N ) and the number of transmitters (K) grow to infinity at a constant ratio N K = α < ∞. Under these conditions, we determine the NSE in the absence of bandwidth limiting and we provide closed form expressions in both cases. Later, we determine the NSE using BL and provide closed form expressions.

A. NSE Without BL

A first result on the analysis of NSE in the absence of BL for the case of spectral resource partition is presented in [START_REF] Haddad | Spectral efficiency of spectrumpooling systems[END_REF], [START_REF]Spectral efficiency of cognitive radio systems[END_REF]. We revisit those results and extend it to the case of resource sharing.

1) Scenario 1: Spectral Resource Partition: Following the same line of the analysis presented in [START_REF] Haddad | Spectral efficiency of spectrumpooling systems[END_REF], we have that in the asymptotic regime the data rate per channel for a given transmitter k is

Rk (γ k ) (1) N →∞ -→ ∞ 0 log 2 1 + p k (λ)λ σ 2 dF g (λ) R(1) k,∞ (11)
where the functions p k (λ) for all k ∈ K, satisfy the power constraints,

∞ 0 p k (λ)dF g (λ) = p max . ( 12
)
The function p k (λ), ∀k ∈ K, which maximizes expression (Eq. 11) subject to expression (Eq. 12) is also a water-filling solution, i.e.,

p k (λ) = β k - σ 2 λ + . ( 13
)
Note that since all the channel coefficients are drawn from the same probability distribution f g (λ) described in Sec. II and all the transmitters have the same maximum transmittable power level, we can write that ∀k ∈ K, R

k,∞ = R(1) ∞ . Hence, the water-level β k satisfying the condition (Eq. 13) is the same for all the transmitters. By combining expression (Eq. 13) and (Eq. 12), we obtain the water-level β k = β * , ∀k ∈ K in the asymptotic regime by solving the equation

∞ β * σ 2 β * - σ 2 λ dF g (λ) -p max = 0. (14)
The fraction Ω

k , for all k ∈ K, can be approximated in the asymptotic regime by

Ω (1) k,∞ [8] Ω (1) k,∞ N →∞ -→ Ω (1) ∞ k-1
where,

Ω (1) ∞ = Pr β * < σ 2 λ = β * σ 2 0 dF g (λ) 1. (15)
Then, the NSE (Eq. 10) in the asymptotic regime Φ

∞ is

Φ (1) ∞ = K i=1 Ω (1) k,∞ i-1 R(1) ∞ = 1 -Ω (1) ∞ K 1 -Ω (1) ∞ R(1) ∞ . ( 16
)
2) Scenario 2: Spectral Resource Sharing: In the asymptotic regime, we can approximate the data rate per channel of transmitter k ∈ K as

Rk (γ k ) (2) N →∞ -→ ∞ 0 ... ∞ 0 Γ ′ k (λ k )dF g (λ k )..dF g (λ 1 ) R(2) k,∞ (17)
with λ k = (λ 1 , . . . , λ k ) and

Γ ′ k (λ k ) = log 2        1 + p k (λ k )λ k σ 2 + k-1 j=1 p j (λ j )λ j        , ( 18
)
where p 0 (λ 0 ) = 0 for all λ 0 ∈ R. Additionally, for all k ∈ K, the functions p k (λ k ) satisfy the power constraints shown in (Eq. 12). As shown in the previous section, the maximization of (Eq. 17) is a water-filling solution. For transmitter k = 1, it yields expression (Eq. 13) and for all 1 < k K,

p k (λ k ) = β * k - β * k-1 λ k-1 λ k 2 . ( 19
)
Now, by plugging expression (Eq. 19) in (Eq. 17), we obtain that for all

1 < k K, R(2) k,∞ = ∞ σ 2 β * 1 ∞ β * 1 λ 1 β * 2 ... ∞ β * k-1 λ k-1 β * k Γ k (λ k )dF g (λ k )...dF g (λ 1 ) (20) with Γ k (λ k ) = log 2 β * k λ k β * k-1 λ k-1
, whereas for transmitter k = 1, the asymptotic data rate is given by expression (Eq. 11). The water-levels β * k in (Eq. 20) for all k ∈ K are the solution to expression (Eq. 14) in the case k = 1, and

p max = ∞ σ 2 β * 1 ∞ β * 1 λ 1 β * 2 ... ∞ β * k-1 λ k-1 β * k p k (λ k )dF g (λ k )...dF g (λ 1 ), (21
) in the case k ∈ K \ {1}. Thus, the spectral efficiency of transmitter k ∈ K is equivalent to its data rate per channel, i.e., Φ

k,∞ = R(2) k,∞ . (2)
Here, the factor

Ω (2)
k in (Eq. 10), is Ω

k = 1 for all k ∈ K. This is because each transmitter can access all the channels regardless of its order of arrival.

By developing expression (Eq. 21) and (Eq. 20) using the p.d.f of the channel gains f g (λ) described in Sec. II, we arrive to the following conclusions,

∀k ∈ K \ {1}, β * k-1 β * k , (23)
and

∀k ∈ K \ {1}, R (2)
k-1,∞ R(2) k,∞ , (24)
respectively. Then, the NSE (Eq. 10) in the asymptotic regime Φ

∞ is

Φ (2) ∞ = K k=1 R(2) k,∞ . (25)
In Sec. VI, we compare both asymptotic and non-asymptotic expressions to validate our statements.

B. NSE With BL

Now, we limit the number of channels each transmitter can use. For the ease of calculations, we keep the conditions that both K and N grow to infinity at the same rate, i.e., N → ∞, and K → ∞, and N K = α < ∞. To provide at least one channel to each user, we assume that α 1.

1) Scenario 1: Spectral Resource Partition: When the number of accessible channels for transmitter k ∈ K is limited to L ∈ N channels, the fraction of accessible spectrum Ω (1,BL) k for each transmitter is

Ω (1,BL) k,∞ = min Pr β * < σ 2 λ , L N . (26)
Then, BL has an effect if and only if

L N < Pr β * σ 2
λ . This condition is equivalent to state that we should limit the transmitters to use a smaller number of channels of that used on the absence of BL. Hence, under the asymptotical assumptions, we have that

Ω (1,BL) k,∞ = L N , (27)
and

Φ (1,BL) ∞ = K i=1 Ω (1,BL) k,∞ R∞ = KL N R∞ . ( 28
)
2) Scenario 2: Spectral Resource Sharing: In this scenario, each transmitter can access all the channels and thus, Ω

k,∞ = 1. When we limit the bandwidth for transmitter k we impose that

Ω (2),BL k,∞ Ω (2)
k,∞ . Then, the NSE under BL is

Φ (2),BL ∞ = K k=1 Ω (2),BL k,∞ R(2),BL k,∞ . (29)
We have provided expressions for the NSE in both absence and presence of BL in the asymptotic regime. Now, it remains to determine the conditions over which BL brings benefits to the network in terms of spectral efficiency.

V. OPTIMAL BANDWIDTH LIMITING

In this section, we investigate the existence of an optimal BL point, i.e., optimal values of the fractions

Ω (i,BL) k , with i ∈ {1, 2} such that Φ (i),BL ∞ Φ (i) ∞ .
3) Scenario 1: Spectral Resource Partition: To improve the NSE by introducing BL in the network, the following condition must be met,

Φ (1) ∞ Φ (1),BL ∞ 1 -Ω (1) ∞ K 1 -Ω (1) ∞ R∞ KL * N R∞ L * N K 1 -Ω (1) ∞
K 1 -Ω (1) ∞ (30)
In expression above we show that the optimal BL parameter L * depends mainly on the network load (transmitters per channel, K N ) and the SNR of the transmitters. Note that the factor Ω (1) ∞ is a function of p max , σ 2 and the probability distribution of the channels gains f g (λ) described in (Eq. 15). 4) Scenario 2: Spectral Resource Sharing: Following the same reasoning as in scenario 1, we improve the NSE by using BL, if

Φ (2) k,∞ Φ (2),BL k,∞ . (31)
However, under BL we have that

Ω (2),BL k,∞ Ω (2)
k,∞ and thus,

K k=1 Ω (2) k,∞ R(2) k,∞ K k=1 Ω (2),BL k,∞ R(2),BL k,∞ . (32)
Then, we have shown that in the asymptotic regime, any kind of BL does not bring any improvement on the NSE. On the contrary, it might introduce significant losses of NSE.

VI. SIMULATION RESULTS

In this section, we provide numerical results of our mathematical model. First, we compare the asymtotical expressions of the NSE with those obtained by simulations, for both scenarios. In Fig. 2, we plot the NSE of a network with 2 transmitters. Therein, we observe that our asymptotic model perfectly describes the system in the finite case i.e., when K and N are small numbers.

We present also simulations of the NSE obtained in both scenarios as a function of the BL parameter L for different Fig. 2.

Data rate per channel in bps/Hz as a function of the number of available channels in the scenario 2. Dashed lines are obtained from simulations considering 10 log 10 ( pmax σ 2 ) = 20dB for transmitter 1 and 2. Straight lines are obtained from expression (Eq. 11) for transmitter k = 1 and expression (Eq. 20) for k = 2. network loads. In Fig. 3, we observe the existence of an optimum BL point for scenario 1. Conversely, in the second scenario, the existence of such optimal is not evident, as at a certain point, the NSE is invariant with respect of the BL parameter L.

We compare the optimal BL parameter L obtained from simulations with that obtained from expression (Eq. 30). In Fig. 4 we plot both results. Therein, we show that the asymptotical approximation (Eq. 30) is a precise approximation of the optimal number of channels each transmitter must use to maximize the NSE.

Finally, we show in Fig. 5 the NSE obtained with absence and presence of BL. In the first scenario, we observe a significant gain in NSE when BL is used. This gain is more important for non-overloaded networks, whereas for quasi fullloaded or overloaded networks (K N ), the gain obtained by BL approaches that of limiting the transmitters to use a unique channel. In the same figure, we observe that the NSE appears to be constant for certain intervals. This is due to the fact that inside those intervals the optimal BL parameter remains constant, as shown in Fig. 4. Moreover, the gain in NSE is very significant at high SNR (SNR = pmax σ 2 ) levels. On the contrary, for low SNR levels, small gains in NSE are obtained when the network is low loaded. In Fig. 6 we plot the NSE for several values of SNR in the second scenario. In any case, we observe that there is not significant gain when all transmitters use the same BL parameter L.

VII. CONCLUSIONS

We have shown, that in a decentralized vector MAC where each transmitter aims to maximize its own data rate by using water-filling based power allocation, the network sum-rate can be improved by limiting the number of available channels for each transmitter (bandwidth limiting). We provide closed form expressions for the optimal maximum number of channels each transmitter must access in the case where transmitters use non-intersecting sets of channels. In this case, such an optimum operating point depends mainly on the network load (transmitters per channel) and the different signal to noise ratios. Contrary to the first scenario, in the case of spectrum resource sharing, we show that when all transmitters use the same BL policy, BL does not bring a significant improvement on the network spectral efficiency. Further studies will focus to the case when transmitters have different channel statistics, since it might lead to the usage of different BL policies for each transmitter. Fig. 6. Network Spectral Efficiency (Eq. 10) in bps/Hz for scenario 2 as a function of the network load ( K N ). Total number of available channels N = 50.

Fig. 3 . 2 "

 32 Fig. 3. Network Spectral Efficiency (Eq. 10) in bps/Hz as a function of the maximum number of accessible channels L. Total number of available channels N = 50, and 10 log 10 " pmax σ 2 " = 10dB.

Fig. 4 .

 4 Fig. 4. Optimal BL parameter L (Eq. 30) for scenario 1 as a function of the network load, ( K N ). Total number of available channels N = 50, and 10 log 10 " pmax σ 2 " = 10dB.

Fig. 5 .

 5 Fig.5. Network Spectral Efficiency (Eq. 10) in bps/Hz for scenario 1 as a function of the network load ( K N ). Total number of available channels N = 50.

This work is supported by the European Commission in the framework of the FP7 Network of Excellence in Wireless Communications NEWCOM++.