domain: spi phys

text:
Alice Odier

Rodwane Chtouki
email: rodwane.chtouki@onera.fr

Pierre Bourdon

Jean-Michel Melkonian

Laurent Lombard

Michel Lefebvre

Anne Durécu

Coherent combining of mid-infrared difference frequency generators

We report what is, to the best of our knowledge, the first experimental demonstration of coherent combining of two mid-infrared difference frequency generators by active phase control in continuous-wave regime. Using the phase relation that is inherent to the nonlinear process, we are able to phase-lock and combine the idler waves by the sole phase control of one of the pump waves. This control is done by an all fiber electro-optic modulator. Combining is achieved with an excellent efficiency with a residual phase error of 𝝀/𝟐𝟖.

Coherent beam combining (CBC) through active phase control is a well-known solution to overcome limitations to laser power scaling such as thermal effects or damage threshold for solid-state lasers, or nonlinear effects for fiber lasers [1].

Nonlinear frequency converters such as parametric generators used in gas sensing [2][3][4] could benefit from CBC power scaling, as it would improve the detection range or the sensitivity of the instrument. CBC has been used in conjunction with frequency converters, but only to increase the power of the pump beam, for instance in the case of second harmonic generation (SHG) [5,6]. The same idea has been used for optical parametric amplification (OPA) with incoherent combination [7]. However, the inherent limit due to the use of a single nonlinear crystal with limited damage threshold [8] remains. To overcome this limitation, we proposed an indirect phase control approach to combine multiple frequency converters [9], increasing the delivered power without exceeding the damage threshold limit, as the conversion is distributed among multiple nonlinear crystals. Another interest of this indirect phase control technique is the capability to use fast standard all-fiber 1-µm or 1.5-µm wavelength electro-optic phase modulator (EOM) rather than having to resort to phase modulators operating at much higher wavelengths, which are often less performing or even not commercially available.

Using this approach, we previously reported the experimental demonstration of CBC by active phase control of continuous wave SHG, achieving the combination of two 532-nm waves from a 1064-nm fundamental wave in the case of perfect phase matching in birefringent lithium triborate (LBO) crystals [10]. The CBC of CW SHG from a 1.55 µm wave has also been demonstrated experimentally in the case of quasi-phase matching using periodically poled lithium niobate (PPLN). In both these experiments, phase control operated on the fundamental wave using a standard all-fiber EOM with high bandwidth, and CBC was very efficient with a low residual phase error of 𝜆/30 [11].

In this paper, we extend the demonstration to higher wavelength in the mid-infrared range, investigating the combination of difference frequency generators (DFG) emitting at 3.4-µm. While SHG is a two-wave degenerate process, we are dealing here with a more complex process involving three non-degenerate waves.

We experimentally demonstrate that CBC with active phase control can nevertheless be implemented using a similar indirect phase control approach as in the case of SHG i.e. with the pump wave phase controlled using standard all-fiber EOM at 1.064 µm. We observe efficient coherent combining of the idler waves at 3.4µm. These results are a first step toward experimental proof of CBC of mid-infrared optical parametric oscillators (OPO) which has already been suggested theoretically [12], to achieve tunable high power mid-infrared sources.

We use the following notations to express the electric field for the pump wave (p), the idler wave (i) and the signal wave (s) with respective angular frequencies  p ,  i and  s . To simplify this proof of concept, we limit our analysis to continuous wave monochromatic plane waves and we assume the beams to be linearly polarized, collinear, and that we are dealing with a single three-wave-mixing process. We then use scalar fields in the equations. The electric field corresponding to each nonlinearly coupled wave can be written as: 𝑬 ̃𝒎 (𝒛, 𝒕) = 𝑨 𝒎 (𝒛)𝒆 𝒋𝒌 𝒎 𝒛 𝒆 -𝐣𝝎 𝒎 𝒕 + 𝒄. 𝒄 (1) with 𝑚 = 𝑠, 𝑖 or 𝑝 and 𝐴 𝑚 is the complex amplitude of the angular frequency 𝜔 𝑚 .

If we assume that the pump wave is not depleted when passing through the crystal, which is true on the first millimeters of the nonlinear crystal where parametric amplification is still weak, the coupled-amplitude equations describing difference frequency generation are [13]

where ∆𝑘 = 𝑘 p -𝑘 s -𝑘 i

In the case of perfect phase matching (i.e. ∆𝑘 = 0), we differentiate the second equation of (2) with respect to z and introduce the complex conjugate of the first equation in (2) to eliminate the

𝑑𝐴 𝑠 * 𝑑𝑧 term.
We obtain the following equation:

𝒅 𝟐 𝑨 𝐢 (𝒛) 𝒅𝒛 𝟐 = 𝜿 𝐬 * 𝜿 𝒊 |𝑨 𝐩 | 𝟐 𝑨 𝐢 (𝒛) (3)
with:

𝜿 𝒎 = 𝟖𝝅𝒋𝝎 𝒎 𝟐 𝒅 𝒆𝒇𝒇 𝒌 𝒎 𝒄 𝟐 (4)
The idler wave being not seeded, we can consider that 𝐴 𝑖 (0) = 0. The initial value for the signal wave amplitude is noted 𝐴 𝑠 (0). Writing down the complex field amplitudes using their phase and modulus parts 𝐴 𝑚 (𝑧) = 𝑎 𝑚 (𝑧) 𝑒 𝑗𝜑 𝑚 (𝑧) , one can verify that the signal field 𝐴 𝑠 (𝑧) retains its initial phase and is simply amplified by the interaction, while the generated idler wave 𝐴 𝑖 (𝑧) has a phase that depends both on that of the pump wave and on that of the signal wave. Noticing from equation ( 4) that 𝜅 i = |𝜅 i |𝑒 𝑗𝜋/2 , we obtain the following relation:

𝝋 𝐢 = 𝝋 𝐩 -𝝋 𝐬 + 𝝅 𝟐 (6)
Similarly, in the QPM case, high conversion efficiency can be achieved in a perfect periodically poled crystal. The relation between phases of the pump, signal and idler waves can be obtained propagating the waves in the crystal domain by domain and step by step in a same way done for SHG [10,14]:

𝝋 𝐢 = 𝝋 𝐩 -𝝋 𝐬 (7)
For higher power levels, depletion of the pump wave can become significant and the propagation equations are solved using Jacobi elliptic functions [15]. However, even in this case of strong depletion, equation ( 5) can be solved step-by-step or domain by domain for QPM, considering there's no pump depletion over one thin slice of the crystal. The initial phase of the signal is thus preserved and seeded step-by-step during propagation, even in long crystals. Consequently, the phase relation "propagates" stepby-step. Periodical poling in QPM crystals compensates for the π phase-offset generated between the three coupled waves after propagation through one domain. Thus, the phase relation should remain valid even in the strong depletion regime.

Note that this approximation can be done as long as the gain in one domain or in one thin slice of the medium is weak. This will be the case in continuous wave or nanosecond regimes, but may not be true in picosecond or femtosecond regimes. There is another condition for CBC operation: the combined idler waves must share the same wavelength. To fulfill this condition, it is required to seed the same 1553 nm signal into both nonlinear crystals. The error signal used here for frequency tagging comes from the idler interference signal following the equation [16]:

𝑺 𝒆𝒓𝒓 = 𝑹 𝑷𝑫 √𝑷 𝒊𝟏 √𝑷 𝒊𝟐 𝐬𝐢𝐧(𝝋 𝒊𝟏 -𝝋 𝒊𝟐 ) 𝑱 𝟏 (𝜷 𝒊𝟐 ) (8)
where 𝑅 𝑃𝐷 is the photodetectors responsivity, 𝑃 𝑖 the power of each idler wave, 𝐽 represents a Bessel function of the first kind and 𝛽 𝑖2 is the phase modulation amplitude. Idler phases are linked to the pump and signal ones by relations previously exposed.

An experimental set-up was built to demonstrate such indirect phase control and to validate the theoretical principle presented before. The setup is based on two PPLN crystals pumped by 1.064-μm Yb:fiber sources. The signal seed is provided by 1553 nm Er:fiber sources. Given the wavelength of pump waves, standard high bandwidth all-fiber electro-optic phase modulators operating at the pump wavelength can be used.

A schematic of the experimental set-up for DFG combining is presented in Figure 1. Fig. 1. Schematic of the experimental set-up for demonstrating coherent combining of two 3.4-µm idler beams generated through DFG in PPLN crystals. Coherent combining is achieved by active phasecontrol of one of the pump waves. [17] Two 1.06 µm Yb:fiber amplifiers are seeded by a master oscillator (MO), a laser module (Orion laser module from RIO) at 1.06 µm. These two amplifiers are homemade ytterbium-doped fiber amplifiers (the same we used in our previous work [11]), each of them capable of delivering up to 15 W power.

A standard fiber-coupled electro-optic modulator (Photline) controls the phase of one of the 1.06 μm fiber amplifiers. This EOM provides both the phase modulation for frequency-tagging and the proper phase shifts to compensate for the phase fluctuations.

To generate the signal seed at 1553 nm, a laser diode is used as a master oscillator. Amplification of this oscillator is achieved using a commercial erbium-doped fiber amplifier (Keopsys, 10 W) split in two to seed both DFG channels with a common signal wavelength.

Signal and pump beams are collimated to the same diameter and overlapped. Half-wave plates are used to align the linear polarizations of the 4 beams along the z-axis of each PPLN crystal.

These PPLN DFG crystals were provided by Covesion and their length are L = 20 mm. With a 1064-nm pump wavelength and a seed signal at 1553-nm, the quasi-phase matching in the PPLN crystal is obtained with a poling period of 30.49 µm at an operating temperature of 60 °C for both crystals, generating an idler wave at 3.4 µm. The pump beams are focused down to a 100 µm 1/𝑒 2 radius waist in the PPLN crystals.

At the output of each nonlinear crystal, a dichroic mirror separates the residual signal and pump beams from the 3.4-µm idler beam. The two idler beams are then overlapped using a 50/50 pellicle beam splitter. A coated germanium window filters off the residual signal and pump photons, so that the infrared detector used for phase control only receives the 3.4 µm beams interference pattern. This photodetector is an InSb cooled detector. The far-field interference signal at 3.4 µm provides the feedback needed to close the phase-locking loop.

Frequency tagging operates at 56 kHz, and phase error signal generation is performed using a commercial lock-in amplifier. This error signal is driven to 0 by a proportional integral (PI) controller, used to send the proper command in terms of phase shift on the EOM.

The goal of these preliminary experiments is to demonstrate the feasibility of indirect phase control, not to establish records in terms of DFG efficiency, the DFG efficiency is indeed low, with only a few milliwatts at 3.4 µm, as the lasers are continuous-wave operating. DFG efficiency maximization would have required some enhancement: higher pump power or longer crystals [18].

Before performing indirect phase control, we first verified the phase relation between the pump, signal and idler waves. To measure phase differences, pump, signal and idler beams are overlapped to create interferences pattern. An In-phase and Quadrature (I/Q) detection is implemented, placing photodetectors at different places of the interferences pattern for the signals to be shifted in phase quadrature. 2 presents an example of free phase fluctuation measurements (i.e. when the phase-locking control loop is inactive) for each wave. We measured the phase difference between each wave at the output of the two nonlinear crystals ∆𝜑 = 𝜑 2 -𝜑 1 for the pump, signal and idler. The difference ∆𝜑 𝑝 -∆𝜑 𝑠 -∆𝜑 𝑖 is also plotted. According to equation (7), this difference should be equal to zero, but here it is not the case due to optical path differences between the two channels. This constitutes the experimental proof of the theoretically expected relation between the phase of pump, signal and idler waves.

As DFG is a three-wave nonlinear process, one could expect the need to actively control simultaneously the phases of two of the coupled waves to achieve efficient CBC. In practice, it is not the case as active phase control of the sole pump wave is sufficient to efficiently combine the idler waves. The reason for that is explained thereafter.

In terms of phase offsets of the different waves, using the indices 1 and 2 for each optical channel, the phase relation for each channel can be written 𝜑 𝑖1 = 𝜑 𝑝1 -𝜑 𝑠1 + 𝑐 1 and 𝜑 𝑖2 = 𝜑 𝑝2 -𝜑 𝑠2 + 𝑐 2 respectively. The constants 𝑐 1 and 𝑐 2 are introduced as a general expression of the constant phase offset between the 3 waves, due to potential optical path difference that can appear as both optical channels are not perfectly equidistant in the experimental configuration.

In order to obtain constructive interference between the two idler beams, their phase offsets must be equal: 𝜑 𝑖1 = 𝜑 𝑖2 . This condition corresponds to: 𝜑 𝑝1 -𝜑 𝑠1 + 𝑐 1 = 𝜑 𝑝2 -𝜑 𝑠2 + 𝑐 2 . This relation can be achieved by controlling the phase of the pump beam on the second channel and only this phase, in order to maintain in real time:

𝝋 𝒑𝟐 = 𝝋 𝒑𝟏 -𝝋 𝒔𝟏 + 𝝋 𝒔𝟐 -∆𝒄 (9)
Proper value of the second channel pump beam will be found automatically by the feedback loop controlling it, in order to compensate for the slow thermal fluctuations of phase of both signal channels and of the first channel pump As a consequence, in order to coherently combine the idler waves, there is no need to control both pump and signal phases. Idler CBC can be achieved through controlling the phase of the sole pump wavelength on the second channel, as long as the signal waves on both channels are seeded by the same wavelength from a single-frequency laser source.

As the fixed phase relation has been demonstrated both theoretically and in practice, indirect active phase control can be used to perform coherent beam combining. Figure 3 presents the intensity measured in the center of the idler interference pattern by an InSb cooled detector, when the control loop is open and then closed. As long as the phase-control loop is open, the idler interference signal fluctuates freely between destructive and constructive states. When the feedback-loop is closed, the idler interference signal is locked on a constructive interference and the signal measured by the photodiode is kept near its maximum value.

The time-averaged combining efficiency is excellent: the residual phase error is λ/28 rms for the idler wave. This experiment is the proof that CBC of CW DFG using indirect phase control is possible.

Since we are able to phase-lock the idler wave interferences in a constructive state, one can wonder how the phase difference of the pump and signal wave evolve. Figure 4 presents an example of phase fluctuations of the pump and the signal waves when the phase control loop is active. Due to the low level of efficiency of the DFG process and the very low power of the idler beams, we could not use fast mid-infrared detectors as they did not have the required sensitivity. We had to use slower InSb cooled detection with a higher sensitivity, and a slower lock-in amplifier that induced a strong electronic noise. As we did not have more than 2 InSb detectors, we also used one of them for simultaneous I/Q detection for phase measurement and phase-locking detection. This explains the lower quality of the phase measurement, as the I/Q detection accuracy suffered a lot from the electronic noise induced by sharing one detector with the phase-locking feedback loop.

Consequently, the phase measurement of the idler wave was irrelevant and hence not plotted in figure 4. This electronic noise also slowed down the voltage shift used to bring back the EOM voltage to the center of its operating range when it reaches its boundaries. One consequence is a much slower shift visible on figure 4 on the blue and black graphs: the spikes visible on these curves correspond to the longer than normal voltage shifts.

In figure 4, the signal phase difference is varying freely. The pump phase is controlled to compensate those fluctuations according to equation ( 9). Thus, it will be impossible, by indirect control of one single wave, to phase-lock simultaneously the idler and the signal waves. We demonstrated previously, in the case of SHG CBC [11], that simultaneous phase-locking of both the fundamental and second harmonic waves was feasible, but it will be impossible in the non-degenerate case of DFG CBC. The difference ∆𝜙 𝑝 -∆𝜙 𝑠 is also plotted. We know from Figure 3 that ∆𝜙 𝑖 = 0 since we were able to lock the interferences in a constructive state. Therefore, the black line in Figure 4 should be equal to -∆𝑐 = 𝑐 1 -𝑐 2 which is related to the real optical path differences of the experimental configuration. The black curve is slowly varying around a constant average value, following the thermal fluctuations of the optical path differences.

In this paper, the first experimental demonstration of coherent combining of difference frequency generators by active phase control using all-fiber components is presented and investigated.

Direct measurements of the residual phase difference between the pump beams and between the idler beams experimentally confirm the well-known fixed relation existing between the pump, signal and idler phases.

Efficient combining of a pair of difference frequency generators is achieved with a residual phase error lower than λ/28 rms in this DFG CBC experiment. In a three wave non-degenerated process, we were able to phase lock the idler beams using only one control loop.

This important result opens the way to expand this technique of indirect phase control to other nonlinear devices with higher parametric gain and efficiency, such as CW OPOs.

Funding. Direction Générale de l'Armement.

  The solutions become: { 𝑨 𝐬 (𝒛) = 𝑨 𝐬 (𝟎)𝒄𝒐𝒔𝒉 ( √ 𝜿 𝐬 𝜿 𝐢 *

Fig. 2 .

 2 Fig. 2. Phase measurements when the control loop is open.

Figure

  Figure 2 presents an example of free phase fluctuation measurements (i.e. when the phase-locking control loop is inactive) for each wave. We measured the phase difference between each wave at the output of the two nonlinear crystals ∆𝜑 = 𝜑 2 -𝜑 1 for the pump, signal and idler. The difference ∆𝜑 𝑝 -∆𝜑 𝑠 -∆𝜑 𝑖 is also plotted. According to equation(7), this difference should be equal to zero, but here it is not the case due to optical path differences between the two channels. This constitutes

Fig. 3 .

 3 Fig. 3. Time evolution of the interference signals of the idler beams when the phase control loop is open and then closed.

Fig. 4 .

 4 Fig. 4. Phase measurements when the control loop is closed.

Acknowledgements. The authors are grateful to Antoine Godard for critical reading of the manuscript and valuable suggestions.


text:
Jean-Baptiste Volatier
email: jean-baptiste.volatier@onera.fr

Guillaume Druart

Differential method for freeform optics applied to two-mirror off-axis telescope design

In this letter we present a method to design the freeform surfaces of an off-axis unobscured two-mirror telescope by integration of a system of differential equations. The system is derived from the differentiation of the Fermat path's principle and is integrated as an ordinary differential equation problem. The method is used to design the freeform surfaces of a telescope whose performance is verified in off-the-shelf optical design software (Zemax).

Freeform optics are optical surfaces without axial symmetry which have been shown to improve optical systems performance. State-of-the-art freeform optical design methods rely on numerical optimization to find the optimum values of freeform description coefficients (Zernike, Forbes, RBF, splines), as such they suffer from the difficulties of inverse problem solving. For example, numerical optimization algorithms may converge to a local optimum of the performance merit function and the final result depends on the design chosen as a starting point. Global optimization methods are often not practical in freeform optical design because of the large solution space. Nonetheless, methods have been propose recently to mitigate these difficulties with strategies to find smart starting points [START_REF] Bauer | Starting geometry creation and design method for freeform optics[END_REF] or analytic differentiation of optical design merit functions [START_REF] Volatier | Generalization of differential ray tracing by automatic differentiation of computational graphs[END_REF].

Other methods to design freform systems exist which take a direct approach by constructing the freeform surfaces from discrete rays. In illumination, if the light source can be considered punctual, each point on the optical surface can be uniquely associated with one ray, therefore it is possible to deduce for each point an optimum location and surface normal and to construct the freeform in that way [START_REF] Ries | Tailoring freeform lenses for illumination[END_REF]. Though in imaging (our problem of interest), the source cannot be considered punctual (extended object) so it is challenging to find the optimum location and surface normal because a dense set of rays will intersect the surface at the same point. Inconsistent results may occur, for example multiple surface normal orientations at the same point.

We broadly classify imaging construction methods by how they ensure the consistency of the freeform surfaces. First, simultaneous multiple surfaces (SMS) based methods [START_REF] Juan C Miñano | An application of the sms method for imaging designs[END_REF][START_REF] Duerr | Analytic free-form lens design in 3d: coupling three ray sets using two lens surfaces[END_REF][START_REF] Hou | Sms2d designs as starting points for lens optimization[END_REF] use a small bundle of construction rays which are cleverly coupled to design consistent surfaces. Secondly, point-by-point methods [START_REF] Yang | Automated design of freeform imaging systems[END_REF][START_REF] Gong | Point-by-point design method for mixed-surface-type off-axis reflective imaging systems with spherical, aspheric, and freeform surfaces[END_REF] use a dense batch of feature rays and fitting methods to recalculate iteratively surfaces normals. Third, methods have been developed [START_REF] Gd Wassermann | On the theory of aplanatic aspheric systems[END_REF][START_REF] Lynden-Bell | Exact optics: a unification of optical telescope design[END_REF][START_REF] Rv Willstrop | Exact optics-ii. exploration of designs on-and off-axis[END_REF] which use differential information to extrapolate from a known ray to the optical system, they are however not freeform, since they only consider axial symmetric two-mirror systems and construct aspherical profiles (with axial symmetry).

In this letter we are particularly interested in the method described by Wassermann and Wolf [START_REF] Gd Wassermann | On the theory of aplanatic aspheric systems[END_REF]. A distinctive point is that it guarantees simultaneously axial stigmatism and exact satisfaction of the sine condition by designing two aspherical profiles, therefore subsequent numerical optimization of the system is not always necessary. Other distinctive characteristics are that both surfaces are simultaneously designed without iteration (contrary to point-by-point methods) and that rays are considered one-by-one (SMS considers small bundles of rays, while point-by-point use a large batch of feature rays).

This method deduces differential information by imposing that the optical system fulfills the sine condition, therefore even if all the constructed rays originate from the on-axis field point the degradation of performance in a field of view (FoV) close to the axis is controlled. From the on-axis ray a dense sampling of rays is constructed by numerical integration from the center of the pupil to its edge. The surfaces profiles are defined by the rays' intersection and are obtained as series of tabulated sag values. Another derivation of these equations, allowing analytical integration, was described later in literature [START_REF] Lynden-Bell | Exact optics: a unification of optical telescope design[END_REF][START_REF] Rv Willstrop | Exact optics-ii. exploration of designs on-and off-axis[END_REF].

This method may not have seen widespread adoption by the optical design community due to two factors: the limitation to two optical surfaces and to axial symmetric systems. In this letter we will address the last point; future work will consider multiple surfaces. In our case, we control the performance in an extended field-of-view by imposing constraints derived from a differentiation of Fermat's path principle. These differential constraints are advantageous compared to the sine condition because they are suitable for the representation of a system without symmetry. The freeform surfaces obtained as tabulated values are then interpolated to model the surface in off-the-shelf optical design software as a mean of cross-validation by classical numerical ray tracing.

We first introduce a vector φ φ φ defined on R 4 uniquely identifying rays. Since we consider a reflective infinite-to-finite conjugate system in this letter, we ignore the wavelength dependency and decompose φ φ φ in four scalars (k x , k y , p x , p y ). The orientation of the ray is determined by the vector k = (k x , k y ) and its intersection with the pupil of the system at a point identified by p = (p x , p y ). We also define P i the intersection point (X i , Y i , Z i ) of a ray with an optical surface i in a global coordinate system. A ray is then represented by a sequence of segments between the points P 0 , ..., P N depending on φ φ φ.

The goal of the following derivation is to compute the Taylor expansion coefficients of the functions characterizing the dependency of the rays as a function of φ φ φ. Doing so will allow us to extrapolate by numerical integration from the parabasal ray (defined for φ φ φ = (0, 0, 0, 0)) to the entire pupil and then to construct the freeform profiles. We set the propagation of the parabasal ray according to Figure 1 and consequently the origin of the pupil and field coordinates. These values are defined by the designer and the freeform profiles are derived from them, this is a limitation of the method described here: it is dependent on the parabasal ray configuration chosen by the designer. In our example we chose these values to compute a very compact telescope whose form factor would be appropriate for a micro-satellite. We choose k x = tan θ x and k y = tan θ y where θ x and θ y are the angles between the parabasal ray and the incoming ray projected respectively on the x, z and y, z planes.

P 0 P 1 P 2 P 3 M1 M2 z y x θ y # Y [mm] Z [mm] P 0 0 0 P 1 0 80 P 2 60 20 P 3 -50 60
Figure 1: Parabasal ray φ φ φ = (0, 0, 0, 0) is drawn as a dashed line. A ray φ φ φ = (0, tan θ y , 0, 0) is drawn as a solid line, showing that the system pupil is a plane intersecting M1 and parallel to the x, y plane, this is a design choice suitable for this particular system. P 0 is on a dummy plane Z 0 = 0 and P 3 is on the image plane, X values are all zero.

The optical system requirements are a focal length of 150 mm and an entrance pupil diameter of 70 mm with earth observation from micro satellites in the infrared (8 -12 µm) as intended application. Our requirement is a root-mean-square (RMS) geometrical spot-size better than 15 µm over a field of 4 × 0.4 • .

Since P i is a point on a surface, there also exists a vector w i in R 2 such that a function w i → P i exist which in turn imposes restrictions on the partial derivatives of P i with respect to φ φ φ. Taking as an example the jacobian matrix, by application of the chain rule, the following decomposition holds:

∂P i ∂φ φ φ = ∂P i ∂w i ∂w i ∂φ φ φ (1)
As a consequence of Eq. ( 1), since for two matrices A, B, the rank of the dot product is such that rank(AB) ≤ min(rank(A), rank(B)), the jacobian matrix ∂P i /∂φ φ φ is degenerated and of rank 2.

If we do not fulfill the constraint rank(∂P i /∂φ φ φ) ≤ 2 the optical surfaces properties will be inconsistent. Introducing the vector w i allows us to take into account correctly the jacobian rank constraint and other constraints applicable to the hessian tensor and higher order derivatives of P i . The vector w i does not need to correspond to a particular vector in a physical plane as its sole function is to express the fact that there is a R 2 space which acts as a dimensional bottleneck between R 4 the space of definition of φ φ φ and R 3 the space of definition of P i . Nonetheless, in this letter we will choose w i to be expressed in the global coordinate plane x, y to simplify the calculus.

We recall that the Fermat path principle states that rays travel along a stationary path throughout the system. Let L be the optical path throughout an optical system and n i,i+1 the refractive index between surface i and i + 1 and P i P i+1 is the Euclidean distance between P i and P i+1 .

L = N i=0 n i,i+1 P i P i+1 (2)
As a consequence of Fermat's path principle we have:

∂L ∂w i = 0 ∀i ∈ [1, N -1] (3)
We define in Eq. ( 4), the vector F composed of the numerators of the ∂L/∂w i terms so given Eq. ( 3), F = 0.

F i = ( P i-1 P i (X i -X i+1 ) + P i P i+1 (X i -X i-1 )) ∂X i ∂w i + ( P i-1 P i (Y i -Y i+1 ) + P i P i+1 (Y i -Y i-1 )) ∂Y i ∂w i + ( P i-1 P i (Z i -Z i+1 ) + P i P i+1 (Z i -Z i-1 )) ∂Z i ∂w i F = [F 1 , . . . , F i , . . . , F N-1 ] (4)
In the considered geometry, N = 3 so F is of dimension 4. The vectors and matrices introduced and their dependencies are represented in Figure 2.

φ φ φ w 0 P 0 w 1 P 1 ∂P 1 w 2 P 2 F 1 ∂P 2 w 3 P 3 F 2
Figure 2: Vectors and matrices introduced and their dependencies. By ∂P i we imply the matrix ∂P i /∂w i Since F = 0, the derivatives of F with respect to φ φ φ are also zero. By application of the chain rule, we can set equations linking the derivatives of F and the derivatives of the functions representing the dependencies summarized in Figure 2. Solving these systems of equations allows us to compute the partial derivatives, some of the partial derivatives being known already.

The functions w i → P i are known partially. We define u i , v i subject to w i = (u i , v i ) we deduce Eq. ( 5) from our earlier assumption that w 0 , w 1 , w 2 are expressed in the x, y plane of Figure 1.

∂X i ∂u i = ∂Y i ∂v i = 1 ∂X i ∂v i = ∂Y i ∂u i = 0 ∀i ∈ [0, 1, 2] (5)
The terms ∂Z i /∂w i are undetermined but can be derived according to Snell-Descartes law given that they are linked to the normal at the point of intersection since the matrix ∂P i /∂w i is a basis for the tangential plane at the same point.

The function φ φ φ → w 1 is known since the surface 1 is the optical system pupil therefore w 1 = p.

The function φ φ φ → w 3 is known at low order because we want to constraint paraxial properties and to correct for a set of aberrations. We set f the focal length and we impose axial stigmatism and aplanetism to obtain Eq. ( 6).

∂u 3 ∂k x = ∂v 3 ∂k y = f and ∂u 3 ∂k y = ∂v 3 ∂k x = 0 ∂w 3 ∂p = 0 and ∂ 2 w 3 ∂k∂p = 0 (6)
Higher order aberration terms like ∂ 3 w 3 /∂k 2 ∂p are not set to zero otherwise the system will be overconstrained. By allowing multiple surfaces and higher order differentiation, future work will allow to constrain these orders and to design anastigmats. There are similarities between our approach and the Taylor expansions used to derive aberration coefficients of freeform systems [START_REF] Bryan | Low-order aberration coefficients of systems with freeform surfaces[END_REF]. However we also establish relationships between aberrations and the derivatives of the freeform surfaces.

Finally, φ φ φ → w 0 can be deduced from freespace propagation as a function of Z 1 , surface 0 being a plane at Z 0 = 0.

The system of equations is set from ∂F/∂φ φ φ = 0 giving 4 × 4 equations and from ∂ 2 F/∂φ φ φ∂p = 0 giving 4 × 4 × 2 equations. These two differentiations are necessary to take into account all the constraints set in Eq. ( 6). We thus obtain a system of 48 equations with 37 unknowns according to Table 1, column 1. The derivatives ∂Y 3 /∂v 3 , ∂Z 3 /∂v 3 are left undetermined by Eq. ( 5), so they are included in the 37 variables (line ∂P 3 in Table 1). This way, we adjust the detector orientation while maintaining the vector ∂P 3 /∂v 3 normalized.

A software was developed leveraging the SymPy library [START_REF] Meurer | Sympy: symbolic computing in python[END_REF], to perform the above differentiation automatically as required by the large operations count of the differentiated expressions (typically around 250 for one equation of ∂F/∂φ φ φ). Expressions of F are constructed from applications of UndefinedFunction SymPy objects. Differentiation of F produces new expressions containing Derivative objects. A merit function and its jacobian is then generated to evaluate the residuals of the system of equations and operates by substituting a vector of candidate values to the Derivative and UndefinedFunction applications objects, more details are given in Algorithm 1.

Once the optical surfaces are known at their intersection with the parabasal ray, we construct the full optical system by integration over the pupil. To maintain the consistency of the surfaces we use high order derivatives at rays already known to deduce lower order derivatives for new rays, see Eq. ( 7) for an example.

∂ 2 Z 1 ∂w 1 2 px= ≈ ∂ 2 Z 1 ∂w 1 2 px=0 + ∂ 3 Z 1 ∂w 1 3 ∂w 1 ∂p x px=0 (7)
Applying the same reasoning to the variables marked integrated in the Table 1, column 2, we create an ordinary differential equation integration problem. At each integration step the 48 equations are solved variable parabasal off-axis P 0,...,3

set integrated ∂Z 1,2 derived derived ∂P 3 set or solved constant ∂ 2 Z 1,2 solved integrated ∂ 3 Z 1,2 solved solved ∂ 1,2 w 0,1 derived derived ∂w 1 solved integrated ∂ 2 w 1 solved solved ∂w 3 set integrated ∂ 2 w 3 set set or solved
set defined as part of the system definition integrated calculated by the ordinary differential equation solver from the higher order derivatives at the previous point derived solved in closed form solved determined by the system of equations and numerical root-finding set or solved like solved but some terms of the vector are set, see text for details constant solved once on-axis and kept constant afterwards. Denominators are omitted, we imply that the derivative is with respect to the entire vector w i or φ φ φ.

Table 1: Variables and their determination numerically for new values supplied by the ordinary differential equation solver for the variables marked integrated. The degrees of freedom are the higher order derivatives as explained in Table 1. Compared to the on-axis resolution we add supplementary degrees of freedom (DoF) given in Eq. ( 8) for the off-axis resolution to compensate for the fewer overall DoF available. These DoFs allow more optical performance degradation in the field along the k y direction. We select the k y direction to take advantage of the asymmetric field of view specification. Physically, the system will not be aplanetic along all axes in the k y field direction.

additional DoF = ∂ 2 u 3 ∂k y ∂p x , ∂ 2 u 3 ∂k y ∂p y , ∂ 2 v 3 ∂k y ∂p x (8)
We need an entrance pupil of 70 mm but since we are only integrating for the 0 • field we need to work on an oversized pupil to anticipate the effective clear aperture required by the full field of view of the system. Therefore we integrate r from 0 to 40 mm. To obtain tridimensional surfaces we introduce pupil polar coordinates r and α given Eq. ( 9) and we perform the integration repetitively, for values of θ equally spaced between 0 and 2π according Algorithm 2. p x = r cos α and p y = r sin α

Function SolveParabasal: Solve variables in the parabasal case (Table 1, col. 1). The solver used is SciPy [START_REF] Jones | SciPy: Open source scientific tools for Python[END_REF] implementation of Levenberg-Marquardt [START_REF] Moré | The levenberg-marquardt algorithm: implementation and theory[END_REF]. Variables marked derived are solved separately in closed-form. Input: Variables marked set Output: Variables marked solved or derived Function Integrate:

The integration uses SciPy [START_REF] Jones | SciPy: Open source scientific tools for Python[END_REF] implementation of a Runge-Kutta method [START_REF] Hairer | Solving ordinary differential equations. i[END_REF] and calls a solve function similar to SolveParabasal with variables according Table 1, col. After integration the profiles are interpolated on a grid (200 × 200 points) for importation in Zemax (Figure 3a). The optical system is refocalized and the FoV centered on the optimum direction by adjusting its orientation, this is necessary to overcome a shortfall of the algorithm which constructs a system with perfect axial stigmatism. By evaluating the RMS spot size performance in the field, we show that we obtain an asymmetric performance profile that allows to meet the requirement on a field with a triangular shape that encompass the rectangular specification (Figure 3b). The distortion evaluated on the same grid is below 0.48 % for an achieved focal length of 149.7 mm. We expect that a system with a larger field would exhibit significant distortion since no constraints were applied to this effect. The departures from a best fit sphere are shown in Figure 3c, both surfaces are concave which is an unusual combination for a telescope. No high-frequency ripples are observed, additionally the peak-to-valley departures are reasonable (< 200µm) so we expect that these surfaces can be manufactured using single point diamond turning methods for infrared applications.

In this letter we have demonstrated a new method capable of constructing the freeform surfaces of a high-performance two-mirror freeform telescope without axial symmetry from only 5 degrees of freedom indicated in Figure 1, paving the way for a systematic exploration of the solution space. Future work on the method will concentrate on expanding to higher orders with the following expected improvements: suitability to systems with more than two freeform surfaces and the possibility to balance low and high order aberration terms, sacrificing if necessary axial stigmatism for an improved field performance. Additionally the method will be studied as a complement to classical optimization methods, generating starting points for problems with tight packaging constraints.

2

 2 Input: α and variables marked integrated or solved at r Output: Variables marked integrated or solved at r + ∆R Algorithm 1: Functions definition M1 M2 50 mm (a) Zemax view of the system, refocusing of -517 µm and FoV tilt of ∆θy = 0.168 • Geometrical spot size, 50 × 50 fields, 120 rays per field in gaussian quadrature pattern, angles in degrees. Solid black: spot size < 15µm, dashed: Departure in the normal direction from a best fit sphere, M1 has a best fit radius of curvature of 799.21 mm and M2 of 377.18 mm, both surfaces are concave.

Figure 3 : 2 , 3 Z 1 , 2 , ∂ 1 ,2 w 1 , 3 )

 32312113 Figure 3: Performance verification in Zemax and freeform surfaces maps.


Manon Kervella
email: m.kervella@groupeonepoint.com

Alexandra Delmas
email: a.delmas@groupeonepoint.com

Alexandra Delmas -Onepoint

Improving employees' well-being by integrating human factors into work situations:The case of a digital service company

Improving employees' well-being by integrating human factors

Introduction

State of the art

Method

Performance and competitiveness well-being at work

• Interest of the UCD approach to foster acceptability

• Perspectives: optimizing employee well-being by adapting their activities and teams to their transversal skills and personality traits. Auto-evaluation survey

  , quite stable in time that may influence motivation as well as close variables such as perception of control Expected results 1 Tracy Kantrowitz, « Development and Construct Validation of a Measure of Soft Skills Performance », 2005, 198. 2 Dr Jessy John, « Study on the Nature of Impact of Soft Skills Training Programme on the Soft Skills Development of Management Students », s. d., 9. 3 R. J. Vallerand et al., « Construction et validation de l'échelle de motiavation en éducation (EME) », Canadian Journal of Behavioral Science/Revue canadienne des sciences du comportement, 1989, 323. 4 Aimao Zhang, « Peer Assessment of Soft Skills and Hard Skills », Journal of Information Technology Education: Research 11 (2012): 155-68, https://doi.org/10.28945/1634. 5 L. Muller et E. Spitz, « Multidimensional assessment of coping: validation of the Brief COPE among French population », Encephale 29, no 6 (2003): 507-18. 6 Jerry M. Burger et Harris M. Cooper, « The Desirability of Control », Motivation and Emotion 3, no 4 (décembre 1979): 381-93, https://doi.org/10.1007/BF00994052. 7 Hassenzahl, M. (2003). The thing and I: Understanding the relationship between user and product. In M. A. Blyth, A. F. Monk, K. Overbeeke, & P. C. Wright (Eds.), Funology: From usability to enjoyment, 1-12 (chap. 3). Kluwer Academic Publishers.= the personality traits, noncognitive skills and competencies that inform how an individual relates


Valentina Di

Martin Heusse

Bernard Tourancheau

Improving Downlink Scalability in LoRaWAN

Although LoRaWAN offers many advantages, one of its main limitation is the insufficient downlink capacity in large-scale scenarios. This makes reliable communication impractical as, in LoRaWAN, reliability is achieved through the acknowledgement of confirmed messages. The contribution of this paper is to provide a comprehensive description of the problems associated with the presence of downlink traffic and to propose solutions to enable the use of acknowledgments for low to medium downlink load. The proposed mechanisms build on a more adequate and flexible choice of the gateway for each downlink transmission by the network server. Thus, they are entirely compatible with existing LoRaWAN technologies and deployments. Our simulation results show that the proposed solutions enable the successful realization of a higher percentage of LoRaWAN confirmed messages. Thus, the common practice of strongly discouraging the use of confirmed messages could be reconsidered to allow applications that require reliable communication.

I. INTRODUCTION

Low Power Wide Area Network (LPWAN) solutions fill the gap between multi-hop sensor networks based on short range radio technologies (BLE or IEEE 802.15.4) [START_REF] Centenaro | Long-range communications in unlicensed bands: the rising stars in the IoT and smart city scenarios[END_REF] and cellular networks which are currently evolving to be able to serve massive amounts of clients [START_REF] Boccardi | Five disruptive technology directions for 5G[END_REF] [START_REF] Lien | Toward ubiquitous massive accesses in 3GPP machine-to-machine communications[END_REF]. LoRaWAN is one of the most prominent LPWAN technology operating in the Industrial, Scientific and Medical (ISM) band, alongside with SigFOX and Weightless, among others. It allows long range communication between thousands of low power battery driven devices and always-on gateways in a cellular kind of architecture. In LoRaWAN, pure uplink traffic, from enddevices to gateways, is strongly favored. For this reason the existing literature focuses mostly on IoT best-effort applications, exhibiting scenarios with only unconfirmed messages, without taking into account downlink traffic even when it would have been necessary (e.g. [START_REF] Bor | Do LoRa lowpower wide-area networks scale[END_REF] [START_REF] Gupta | Modelling of IoT traffic and its impact on LoRaWAN[END_REF]). Indeed, sending sporadic data back to devices is an intrinsic necessity of LoRaWAN functioning, making downlink traffic a crucial and significant component for a proper scalability analysis of the technology. Besides, Maalel et al. [START_REF] Maalel | Reliability for emergency applications in internet of things[END_REF] point out that missioncritical applications require a reliable data transport system. LoRaWAN implements a data retransmission scheme in order to improve transmission reliability by using acknowledgement messages (ACK) to guarantee data reception. Currently, network operators strongly restrict the generation of these kind of messages since the ACKs drain the network capacity, as shown in various studies (see Section VII). Thus, improving downlink communication efficiency and scalability seems to be an important need for LoRaWAN. After an overview in Section II of the LoRa technology aspects that are relevant to our problematics, Section III identifies the challenges linked to the presence of downlink traffic. Section IV proposes three possible solutions to improve scalability issues. Section V addresses our simulation approach and the results are presented and discussed in Section VI. Before the conclusion, Section VII reviews the related works.

II. LORA OVERVIEW

LoRa can refer to two different layers:

• LoRa is the physical layer, it defines the physical coding of data symbols. It exploits unlicensed ISM frequency bands and uses a Chirp Spreading Spectrum (CSS) radio modulation technique [START_REF] Springer | Spread spectrum communications using chirp signals[END_REF]. • LoRaWAN is the protocol stack (at the Media Access Control (MAC) and network layers) which exploits the LoRa physical layer. It is an open standard developed by the LoRa Alliance [START_REF] Sornin | LoRaWAN TM 1.1 specification[END_REF].

A. LoRa modulation

In LoRa, the frequency spreading is achieved by coding the information into the time offset applied to a base signal composed of a simple frequency ramp (or chirp) sweeping the entire bandwidth (BW). Indeed, the frequency bandwidth of the chirp is equivalent to the spectral bandwidth of the signal. Each chirp spans 2 SF samples at the BW sampling rate and there are 2 SF different time offsets, which allows to convey SF bits. As a result, LoRa trades data rate (DR) for sensitivity within a fixed channel BW, as shown in Table I for the 125 kHz bandwidth modulations. Notice that an error correcting code (generally of coding rate (CR) 4/5) introduces some redundancy at the bit encoding level. The chirps at different DR are orthogonal with each other. This allows the receiver to simultaneously receive multiple transmissions on different SFs as described in the specifications [START_REF]LoRa modulation basics[END_REF].

B. LoRaWAN

LoRaWAN presents a cellular star-of-stars topology [START_REF] Vangelista | Long-range IoT technologies: The dawn of LoRa TM[END_REF] with three main entities:

• The end-device emits LoRa packets towards a server each time a specific event triggers the transmission. • Gateways receive LoRa packets from all LoRa enddevices sending traffic in their reception range. Each gateway is registered to a network-server to which it transparently forwards all the received traffic through an IP connection. • Network-Server is in charge of all the decisions and parameters adaptation. Several gateways can be registered to the same server and they can receive the same packet more than once. After a de-duplication operation, the packet is forwarded to the application. In order to allow end-devices to receive downlink frames, three different classes are defined based on energy consumption:

• Class A -All end-devices are booted in this mode.

It allows the lowest energy consumption possible since the end-device only gets to receive downlink traffic in one of two reception slots after each transmission. Every time an uplink frame is sent, a first receive window, RX1, is opened by the end-device RECEIVE DELAY1 seconds after the end of the transmission (default value 1 second) with the same SF used for the uplink frame increased by a fixed amount (default 0) and using the same carrier frequency. If a preamble is detected, the radio receiver stays active for the downlink frame reception. If no downlink frame is received in RX1, the end-device opens a second receive window, RX2, after RECEIVE DELAY2 seconds (default value RE-CEIVE DELAY1 + 1 seconds) with preset parameters (default values 869.5MHz/SF12/125kHz).

Fig. 1: Class A end-devices receive slot timing.

• Class B -Beacons. A Class A end-device can decide to switch to Class B. The purpose is to make an end-device available to receive downlink frames at other predictable times, in addition to RX1 and RX2. • Class C -Continuous listening allows devices to constantly be in receive mode, except when an uplink frame has to be sent. Obviously, this mode of operation is not suitable for devices with energy constraints. The most relevant LoRa message fields for this work relate to the generation and management of downlink traffic:

• Adaptive Data Rate: LoRaWAN allows end-devices to adapt the DR in order to increase the battery life and maximize the network capacity. When the ADR bit is set, the network can control the DR through the associated MAC commands LinkADRReq and LinkADRAns. If an end-device uses a DR which is greater than its default one, it needs to periodically validate that its uplink frames are still received by the network server. Thus, if the device has not received a downlink frame after ADR ACK LIMIT (typically 63) uplink frames, the ADRACKReq bit is set. This informs the network server to send a downlink frame as soon as possible. If after ADR ACK DELAY (e.g. 32) more transmissions, the end-device has still not received a downlink, it switches to the next lower DR to try regaining connectivity. • ACK. A CONF is a frame with a bit set in the header that requires an acknowledgement from the receiver. If a CONF is sent by the end-device, the network server must send a downlink frame with the ACK bit set in one of the end-device RX windows. If an acknowledgement frame is not received after a number of retransmissions (default value 0), the end-device may try to regain connectivity by lowering its DR to send the message again, until eventually discarding it. It is worth noticing that acknowledgments are sent by the server only in response to the latest CONF received and are never retransmitted. Moreover, LoRaWAN offers several MAC commands to the network server and the end-devices, in order to adapt various physical layer parameters and to assess the status of devices or network links. Each time such MAC command is sent by the end-device, it must be answered/acknowledged by the server and thus, a downlink frame is involved.

Also, in order to participate in a LoRaWAN network, each end-device has to be activated in one of the two following manners:

• Over-The-Air-Activation (OTAA) requires an enddevice join-request uplink frame and a network server join-accept downlink frame if the end-device is permitted to join the network. This type of activation is the most secure. • Activation By Personalization (ABP) directly ties an end-device to a specific network, bypassing the join procedure. The end-device is assumed to have the required information for participating in the network when started.

III. DOWNLINK TRAFFIC: PROBLEMATICS

In a nutshell, downlink frames arise for a variety of reasons: (a) Reception of CONF and subsequent ACKs; (b) Management functions required by LoRaWAN, e.g. the OTAA join procedure for devices; (c) MAC commands for parameters adaptation and link connectivity testing, e.g. LinkADRReq, LinkCheckAns, etc.; (d) In response to frames with ADRACKReq bit set; (e) Events notification from IoT applications (e.g. software update, sampling interval change, . . . ). Even if CONFs are currently not recommended due to their negative impact on the network, there will be downlink traffic for reasons (b), (c), (d), and possibly (e). Hence, it is important to understand which are the problems related to downlink frames along with the normal uplink traffic and how this affects network capacity.

Below, we analyze three limiting factors for downlink frames transmission.

A. Duty-Cycle Saturation

LoRaWAN operates in the ISM band which is regulated by the ETSI standard [START_REF]Electromagnetic compatibility and radio spectrum matters (ERM); short range devices (SRD); radio equipment in the 25 MHz to 1000 MHz with power levels up to 500 mW[END_REF]. In order to be compliant, the LoRaWAN specification exclusively uses duty-cycled transmissions with an ALOHA access method [START_REF]LoRaWAN TM 1.1 specification[END_REF]. In Europe, LoRaWAN operates in the 863-870 MHz band, for which the standard defines five sub-bands with per sub-band limitations. In those bands, the duty cycle ratios (DutyCycle subband ) range from 0.1% to 10%. Each time a frame is transmitted, its emission timestamp and its Time On Air (ToA) are recorded by the emitter for each transmission sub-band. The ToA defines the transmission time of a frame and it depends on the frame size, SF, BW and CR [20]. Then the same sub-band cannot be used again for the subsequent time off (TOff subband ) seconds, where

TOff subband = ToA DutyCycle subband ToA (1)
Both end-devices and gateways comply to this limitation. Therefore it is evident that, in scenarios with downlink traffic, gateways can potentially become the bottleneck of the network by saturating their DutyCycle subband and hence exhausting their ability to forward downlink frames to the numerous enddevices. Indeed, taking into consideration class A end-devices (the most used), the gateway must send the downlink frame exactly at the beginning of one of the two RX windows, in order to allow the radio receiver of the end-device to detect the downlink preamble. If the duty-cycle is saturated for the sub-band corresponding to the downlink transmission channel, the gateway will not be able to forward the downlink frame. Next, the end-device will perform the retransmission of the CONF and possibly end up switching to a higher SF, with a greater and greater radio channel footprint.

B. The gateway radios are Half-Duplex

Due to the lack of frequency band separation between uplink and downlink communication, LoRa gateways operate in halfduplex mode, i.e. they cannot receive and transmit at the same time. When a frame is sent, all ongoing receptions are aborted and no uplink frame can be received for all the duration of the downlink frame ToA, called the no-reception window. The presence of downlink traffic in a large network will consequently cause an overall decrease of the end-devices PDR. We thus identify two different modes related to the gateway's ability to receive frames:

• "free" when the channel is used for receiving uplink frames. In this mode the gateway is available for reception and all received uplink frames are forwarded to the network server. • "busy" when the transmission direction is switched and the gateway sends a downlink frame. When in this mode, the gateway is unable to receive any uplink frame.

C. Downlink frames sequential sending

The modulation profile (SF, CR, BW, ..) and the emission timestamp of a downlink frame are computed by the network server at the downlink frame scheduling phase. A LoRa gateway can only have one frame programmed for departure at a time, when the internal clock reaches the downlink timestamp value. Thus, downlink frames are buffered in a Just In Time queue and sent exclusively sequentially. This hardware constraint exacerbates the problem caused by the gateway's half-duplex mode and speeds up the duty-cycle saturation process for downlink traffic.

IV. PROPOSED SOLUTIONS

We propose various strategies to tackle the identified downlink traffic issues.

A. Multi-Gateway

One way of obviating the half-duplex nature of the gateways is to change the network architecture. Indeed, by deploying additional gateways (even with small coverage zones), many uplink frames which overlap in time with downlink transmissions could still be received by free-gateways. In this scenario, the no-reception window of a busy-gateway is covered by all the other free-gateways able to receive uplink frames and forward them to the network server. Note that the base chirps used for uplink and downlink transmissions are complex conjugate of each other. They are orthogonal, so that downlink transmissions from distant gateways have little impact on uplink reception.

B. Parallel sending and downlink frames combination

The current hardware design of LoRa gateways allows the transmission of a single frame at a time. Nevertheless, it is possible to imagine a hardware design able to overcome this restraint, since it is not inherent the technology. The idea here is to not only allow frames emissions in parallel on different channels, but also simultaneously on the same channel with different orthogonal spreading factors, similarly to what is done in UMTS cellular networks. Indeed, if LoRaWAN gateways had the ability to send more downlink frames simultaneously, the no-reception window and the TOff subband would improve considerably. Two factors have to be taken into account in the implementation of the downlink frames parallel sending:

• Downlink frames combination requires that the receive windows of both candidate receivers overlap. This can be made more likely by lengthening the RX duration, as explored by Centerano and Vangelista [START_REF] Centenaro | Boosting network capacity in Lo-RaWAN through time-power multiplexing[END_REF].

• The gateway must consider also the restriction on the maximum Effective Radiated Power (ERP) imposed by the ETSI regulation. All the sub-bands used by LoRa are limited to a maximum radiated power of 14 dBm, except for the default sub-band of RX2, characterized by an ERP limitation of 27 dBm. As a consequence, to stay within the ERP limitation, it is necessary to compute a correct transmission power profile when combining several downlink frames on the same sub-band.

C. Balanced Gateway Selection Algorithm

In a multi-gateway scenario, the network server could (and actually should) possibly receive the same uplink frame more than once, forwarded by different gateways. When the network server needs to send a downlink frame to an end-device, it has to select the gateway that will perform the transmission. The LoRaWAN specifications do not provide any information or recommendation on how the network server should perform such operation. One possible selection algorithm, adopted by an open-source LoRaWAN network-server, the LoRaServer project [START_REF] Brocaar | LoRa server, open-source LoRaWAN network-server[END_REF], is based on the Signal to Noise Ratio (SNR). This is a good way to estimate the wireless link quality [START_REF] Tan | SNR-based link quality estimation[END_REF] assuming the same bidirectional transmission conditions. In this SNR-based algorithm, the selection of the gateway only depends on the information carried by the duplicates received from the targeted end-device. In the de-duplication phase, the server records the gateway that forwarded the frame with the best SNR value.

Although the radio link quality matters, it is clear that if the gateway is in Toff for RX1 and RX2, then it is for sure unable to send the downlink. As emphasized in Section III, the duty-cycle of a gateway which serves a large number of enddevices may quickly reach saturation and the gateway would then miss the RX windows. Moreover, if the selected gateway is consistently receiving a large amount of uplink traffic, the number of missed uplink frames during the downlink transmissions would most probably also be significant. Thus, with such a naive selection algorithm, the gateway load is not taken into consideration and it mechanically introduces notable frame loss. In order to better balance and spread the downlink traffic throughout the network, the gateway selection algorithm should also take into account the effective ability of the gateway to forward the downlink, by checking its dutycycle saturation and its already scheduled downlink frames, in order to avoid overlaps, similarly to what is currently done by The Things Network [START_REF]The Things Network[END_REF]. This multicriteria gateway choice algorithm attempts to strike a balance between trying to use the best radio link and avoiding losses due to duty-cycle limitations and to the downlink frames sequential sending.

V. SIMULATION DATA & EVALUATION APPROACH

In order to validate the proposed solutions experimentally, it would be necessary to deploy not just some gateways and a network server, but also a considerable number of end-devices. Indeed, the network server can only interact (e.g. sending downlink frames) with registered end-devices, whereas all the traffic received from non-registered devices is ignored. We chose the pragmatic approach to implement a simulator to evaluate and quantify the improvements brought by the proposed solutions. But instead of having a traffic generation model which follows a Poisson distribution as in many studies [START_REF] Bankov | Mathematical model of LoRaWAN channel access[END_REF], [START_REF] Georgiou | Low power wide area network analysis: Can LoRa scale[END_REF], our simulations exploit real traffic traces to observe the behavior of the network when it is virtually exposed to various percentages of downlink frames, relatively to the uplink traffic. In order to collect real traffic traces, four Semtech LoRa PicoCell Gateways [START_REF]LoRa PicoCell gateway user guide[END_REF] were set up and attached to Raspberry Pi boards. The WalT platform [START_REF] Brunisholz | WalT: A reproducible testbed for reproducible network experiments[END_REF] greatly facilitated the deployment. As shown in Table II, the gateways positions were meant to produce a configuration where a gateway serves a large number of nodes (GW1) and at least part of the received traffic is shared with other gateways, in order to have some duplicates.

A. Virtual downlink traffic generation

In real traffic, usually most, if not all, uplink frames are unconfirmed frames. In order to analyze the network reaction to various percentages of downlink frames, we vary the proportion of CONF virtually. In practice, the simulator randomly selects in the considered trace an unconfirmed frame (and all its duplicates) and marks it (them) confirmed. This is repeated until a new trace is produced with the targeted percentage of CONFs.

B. Downlink frames scheduling algorithm

In the simulation, a downlink frame can be triggered by the reception of either an unconfirmed message with the ADRACKReq bit set or a CONF. In order to send a downlink, the simulator has to select the gateway that will take care of the downlink transmission: gateways are ordered by the SNR value of the forwarded uplink frame and the "best" gateway is initially selected.

Two conditions are necessary so the downlink frame can actually be scheduled on the selected gateway in any of the two RX windows.

1) Sequential sending: the downlink frame must not overlap for all its duration with the ToA of any other downlink frames already scheduled on this gateway. 2) Duty-cycle: the TOff subband must not overlap with the TOff subband of any message already scheduled in that subband for the gateway.

A first attempt is performed to schedule the downlink frame in RX1, as shown in Figure 2. If one of the two criteria is not met, a second attempt is performed in RX2. If again the downlink frame cannot be scheduled in RX2, then we distinguish two algorithms:

(a) SNR-based algorithm: the downlink frame (and thus the confirmed message associated) is considered lost and we record which unsatisfied criteria caused the loss. (b) Balanced algorithm: other attempts in RX1 and RX2 are performed with the second best gateway in terms of SNR, and so on. Only when the downlink frame cannot be scheduled at any of the gateways that forwarded the corresponding uplink frame, it is considered lost.

VI. SIMULATION RESULTS

A. Single-gateway architecture One of the contributions of this paper is to give clear insights about the problems related to downlink traffic.

In this experiment, the simulator is fed with the real traffic trace of the main gateway (GW1). The percentage of CONF varies virtually from 0% to 100% in order to understand how the network reacts. The captured traffic is intense enough to saturate the downlink channels, as long as enough frames are virtually changed to confirmed.

Each frame loss shown is due to one of the causes exposed in Section III. Thus, these frames are surely lost and would have been received if downlink traffic was not introduced in the network.

In Figure 3, CONF can be considered lost due to:

• ACK lost: If one of the two criteria detailed in Section V is not satisfied for both RX1 and RX2, the ACK is lost. The unmet criterion is registered only for the failure in RX2. Thus, the loss can be caused by: -Failure in RX1 and duty-cycle saturation of RX2 subband (in red) -Failure in RX1 and downlink frames sequential sending for RX2 start time (in yellow)

• Half-duplex mode: the CONF was supposed to be received while the gateway is sending, for this reason it is lost (in cyan). Unconfirmed messages can be lost only because the gateway radio is half-duplex -and busy transmitting (in purple).

The sum of all the four loss components gives the total frame loss over the percentage of confirmed messages. Figure 3a highlights three main factors:

1) With a low percentage of CONF, most losses are related to unconfirmed messages. Indeed, if the downlink load is low, it does not congest the gateway and ACKs can be sent. On the other hand, every time a downlink frame is sent, all the uplink frames received during that period are lost. 2) From 20% of CONF and on, the main cause of frame loss is duty-cycle saturation. If all the frames are confirmed (100% of CONF), 8% of them are not received by the gateway because of its half-duplex nature, and 76% of the requested ACKs cannot be transmitted. In this case, the frame loss reaches 86%: almost all frames are lost. Fig. 3: Frame loss due to downlink traffic.

3) The sequential sending of downlink frames causes no more than 2% of losses even with 100% CONF. This can be explained by observing that a heavier traffic load than the one deployed here would correspond to a higher downlink frames demand. With more downlink frames requested, the probability of overlapping with other downlink frames already scheduled in the gateway would surely increase. Since the main ACK loss is due to the duty-cycle saturation, the parallel sending solution will not produce a tangible improvement with these data (its validation is left for future work).

B. Multi-gateway architecture

In order to decrease the frame loss due to the gateway halfduplex mode, a second simulation was performed with a multigateway implementation. The simulator was fed with a real traffic trace file that combines frames received by the main gateway and duplicates received in the same time window by three other gateways. Figure 3b shows the improvement brought by a quad-gateway implementation with the SNRbased selection algorithm. The frame loss is almost halved. Indeed, all the frames that are lost while deploying just one gateway, are received by the other free-gateways. Moreoverwhereas in a single-gateway implementation all the downlink frames are sent through the same gateway-in a multi-gateway scenario, downlink traffic is spread among several gateways, decreasing the congestion. As a result, more downlink frames can be sent in RX1 and the RX2 dedicated sub-band is less (or more gradually) saturated.

C. Gateway selection algorithms

Here, despite the use of more gateways, frame loss for higher proportion of confirmed messages is still important. This is a direct consequence of the behavior of the gateway selection algorithm which does not attempt to spread downlink traffic in any way by always and only selecting the gateway with best reception conditions. This concentration of downlink traffic is clear in our simulations: Figure 4a shows that GW1 is selected at least 60% of the time when an ACK has to be scheduled; in Figure 4b it is possible to see that, as a consequence of such high downlink frame request load, the effective percentage of ACKs sent by the gateway significantly decreases with the increasing load. With 100% of CONF, GW1 is only able to send 24% of the requested ACKs. On the other side, two gateways (GW3 and GW4) are almost never used, while GW2 is complementary to GW1. To better balance the load of the downlink traffic, the balanced gateway selection algorithm is tested on the same trace. We show the improvements introduced by the balanced algorithm compared to the SNR-based one with their respective frames loss. Figure 5 shows that, in a quad-gateway implementation (in green), the balanced algorithm decreases the frame loss by 25% compared to the SNR-based algorithm and by 66% compared to the single-gateway architecture (in red), producing losses that never exceed 20% of the total traffic. The main reason of this improvement is the better distribution of the downlink traffic. ACK loss, which was the main cause of frame loss for high percentage of CONF, is strongly decreased. Unlike before, a downlink frame is not scheduled on the gateway with the best SNR if it is duty-cycle saturated for the given sub-bands or it overlaps with other already scheduled downlink frames. The algorithm tries the second best gateway in terms of SNR, and so on. Obviously, ACKs can still be lost due to other factors, especially if transmitted by a gateway that presents a lower SNR value. But in fact, the downlink channel is less challenging than the uplink as in all cellular networks: nodes are in view of less contending entities than the gateways, which use a specific modulation and are generally placed so as to receive from as many nodes as possible.

VII. RELATED WORK

Downlink traffic is arousing interest only recently. Most of the studies on the topic mostly focus on showing its negative impact on the network, usually identifying the duty-cycle as the main or single problem. Adelantado et al. [START_REF] Adelantado | Understanding the limits of LoRaWAN[END_REF] conduct an analysis of LoRaWAN in order to identify the main problems of the technology. Among other issues, the duty-cycle saturation is recognized as one of the limits of LoRaWAN, especially for the development of ultra-reliable applications which, as such, require acknowledgments.

Pop et al. [START_REF] Pop | Does bidirectional traffic do more harm than good in LoRaWAN based LPWA networks?[END_REF] address the bidirectional traffic problem by introducing LoRaWANSim simulator, an extension of Lo-RaSim that includes downlink frames and retransmissions. Some of the findings are relative to the aggressive data-rate back-off approach during retransmission recommended in the old version of LoRaWAN specification, and hence they are not relevant anymore. On the other hand, they correctly identify the scalability problems that arise when a large number of nodes request ACKs and the duty-cycle limitation issue.

An empirical evaluation of the downlink traffic impact is conducted by Mikhaylov et al. [START_REF] Mikhaylov | Effect of downlink traffic on performance of lorawan lpwa networks: Empirical study[END_REF]. The results show the strong negative effect on the performance of uplink frames, consistent with the previous studies. Nevertheless, the explanation behind this phenomenon does not take into account the half-duplex nature of the gateway and the I/Q inversion performed by gateways when transmitting uplink and downlink frames, in order to avoid their collision.

The aim of Centenaro et al. [START_REF] Centenaro | On the impact of downlink feedback on LoRa performance[END_REF] is to investigate the impact of downlink frame feedback in a multi-gateway architecture using an event-driven simulator. In order to have some scheduling flexibility on the network side, one of the modifications of the model is to have receive windows one second longer than the actual ones. This significant change imposes that each RX window stays open for an unreasonable amount of time and the resulting downlink frame scheduling process is completely different from the real one. However, the obtained results show again the negative impact of downlink frames without providing additional reasons besides the duty-cycle issue.

One of the main contributions to the downlink frame issue is given by Van et al. [START_REF] Den Abeele | Scalability analysis of large-scale LoRaWAN networks in ns-3[END_REF] by performing a downlink traffic analysis using the ns-3 network simulator. The authors clearly explain the reasons of gateways congestion by highlighting the duty-cycle limitation, whereas, the half-duplex problem is merely mentioned. A multi-gateway architecture is proposed and improvements are evaluated in terms of lower duty-cycle saturation and better distribution of the workload through the gateways. Yet, neither the explanation, nor the proposed solution, take into consideration the gateway selection algorithm. In their implementation, the server simply tries to schedule an ACK in the first available gateway among the pertinent gateway(s).

VIII. CONCLUSION

This paper presents a comprehensive analysis of the impact of downlink frames on the network capacity. The roots of the problem are identified and checked on a real trace file: the fact that gateway transmissions are half-duplex (either sending or receiving) causes up to 8% of frame loss; the downlink frame sequential sending at the gateways -by which only one downlink frame can be scheduled at a given timeproduces up to 2% of frame loss due to downlink frame timing overlapping with other already scheduled ones; and the dutycycle limitation that induces up to 76% of frame loss due to the saturation of the downlink frame sub-band. We propose three solutions in order to mitigate such effects: a multigateway implementation, downlink frames parallel sending and a balanced gateway selection algorithm. Two of them are tested by the implementation of an event-driven simulator fed with captured real traces. From an initial 86% of frame loss with a single-gateway architecture, the proposed solutions provide an enhancement of 66% in a quad-gateway scenario in presence of a high downlink communication load corresponding to 100% confirmed messages. Moreover, for low and medium downlink traffic loads, frame loss never exceeded 5% thanks to the balanced gateway selection algorithm, enabling the deployment of applications that require such confirmed data communication.

Fig. 2 :

 2 Fig. 2: Scheduling illustration for a new downlink frame (D or E). Downlink frames A, B and C are already scheduled.

  (a) Percentage of ACK requested per gateway under total number of CONF frames. (b) Percentage of ACK sent on ACK requested per gateway.

Fig. 4 :

 4 Fig. 4: Gateways behavior under SNR-based selection algorithm.

Fig. 5 :

 5 Fig. 5: Frame loss percentage under different algorithms and number of deployed gateways.

TABLE I :

 I LoRa transceiver (Semtech SX1276) parameters.

	SF	7	8	9	10	11	12
	Sensitivity [dBm]	-123	-126	-129	-132	-133	-136
	DR [kbit/s]	3.4-5.5	2.0-3.1	1.1-1.8	0.6-1.0	0.3-0.5	0.2-0.3
	min SNR [dB]	-7.5	-10	-12.5	-15	-17.5	-20

TABLE II :

 II Gateways set up for real traces collection.

	GW	Displacement Received traffic
	1	Outdoor	100%
	2	Outdoor	52%
	3	Indoor	6,5%
	4	Indoor	4,7%