Nabil Madali

Antonin Gilles

Patrick Gioia

Luce Morin

Self-Supervised Focus Measure Fusing for Depth Estimation from Computer-Generated Holograms

Keywords: 3D Imaging, Holography, Depth Estimation, Self-supervised learning, Depth from focus

Depth from focus is a simple and effective methodology for retrieving the scene geometry from a hologram when used with the appropriate focus measure and patch size. However, fixing those parameters for every sample may not be the right choice, as different scenes can be composed with various types of textures. In this work, we propose a self-supervised learning methodology for fusing the depth maps produced using different focus measures with variable patch sizes applied to the holographic reconstruction volume. Experimental results show that fusing depth information produces more accurate and smoother depth maps, which can be directly used for alternative tasks such as motion estimation.

Introduction

Designing an efficient holographic video codec is still an open problem in the current state of the art, due to the lack of a fast and accurate motion estimation algorithm. Indeed, traditional video codecs based on blockmatching are not appropriate for holographic data because there is no clear spatial correlation between consecutive frames [START_REF] Blinder | Signal processing challenges for digital holographic video display systems[END_REF]. To accurately estimate motion from one frame to another, the scene geometry must first be recovered to calculate the 3D motion vectors.

In [START_REF] Madali | Automatic depth map retrieval from digital holograms using a depth-from-focus approach[END_REF], the authors demonstrated that the Depth-From-Focus (DFF) [START_REF] Ma | Numerical reconstruction of digital holograms for three-dimensional shape measurement[END_REF][START_REF] Conor | Extraction and reconstruction of shape information from a digital hologram of three-dimensional objects[END_REF][START_REF] Ozgurun | Depth extraction from a single compressive hologram[END_REF] method is a simple and effective technique for retrieving the scene geometry when implemented within a well-defined framework. This technique involves building a reconstruction volume by performing a series of numerical reconstructions at sampled distances. Then, the level of focus for each pixel in the reconstruction volume is evaluated using a Focus Measure (FM) on a patch centered around the pixel, and the distance at which the pixel reaches its optimal focus level is selected as its depth value. Experimental results have shown that the focus measure and patch size significantly impact the precision of the estimated scene geometry. Different focus measures may perform differently depending on whether the input region is uniform or has high texture variation. Furthermore, the authors point out that, due to the unique nature of holographic numerical reconstruction, the focus curve can be subject to polarity change when transitioning from uniform to textured regions, depending on the reconstruction interval and patch size. To address this issue, an automatic switch between the global minimum and maximum of the focus curve is implemented. This allows for the handling of polarity change and ultimately improves the final performance.

Inspired by the latest breakthroughs in selfsupervised image denoising [START_REF] Lehtinen | Noise2noise: Learning image restoration without clean data[END_REF][START_REF] Ulyanov | Deep image prior[END_REF][START_REF] Long Nguyen | Selfsupervised super-resolution for multi-exposure push-frame satellites[END_REF][START_REF] Izadi | Image denoising in the deep learning era[END_REF][START_REF] Ehret | Modelblind video denoising via frame-to-frame training[END_REF], we propose in this paper an enhanced version of the automatic switch process which determines the appropriate depth value using both local and global information. The method involves extracting multiple depth maps from a reconstruction volume using various focus measures and patch sizes, then training a neural network in a selfsupervised manner to identify the depth map with the lowest average deviation. The experimental results show that incorporating multiple focus measures and different patch sizes significantly improves the accuracy and smoothness of the predicted depth map. The remaining of the article is organized as follows: Section 2 details the different components of the proposed method. Then, in Section 3 a series of experiments are conducted to validate the proposed approach. Finally, in Section 4, we discuss the advantages and limitations of the presented method.

Methodology

Overview

The proposed approach is composed of three steps: First, the input hologram H of size L×L is used to compute a reconstruction volume. This step involves several hyperparameters that can affect the final results, so to ensure fair and accurate evaluation, the same methodology as proposed in [START_REF] Madali | Automatic depth map retrieval from digital holograms using a depth-from-focus approach[END_REF] is consistently used. Next, the DFF method is applied to the reconstruction volume using various focus measures and patch sizes, each of which produces a depth map referred to as measurement. All these measurements can be considered as different distorted versions of the ground truth depth map. Finally, a neural network is used to estimate a depth map that is as close as possible to the ground truth by minimizing the average deviation from the measurements.

Reconstruction volume acquisition

Given an input hologram H, a reconstruction volume is computed by performing a set of numerical structions using the Angular Spectrum Method [START_REF] Joseph W Goodman | Introduction to fourier optics[END_REF] at uniformly sampled reconstruction distances inside a manually defined depth interval [z min , z max ], where z min and z max are the minimal and maximal reachable depths values respectively. Each numerical reconstruction is defined as

P zi {H} = F -1 F{H}(f x , f y )e j2πzi √ λ -2 -f 2 x -f 2 y , (1)
where F is the Fourier transform, f x and f y are the spatial frequencies along the X and Y axis, λ is the acquisition wavelength, and z i is the sampled reconstruction depth, given by

z i = z max -z min N i + z min , (2)
where N is the number of sampled distances.

Depth From Focus (DFF) method

The focus level of each pixel in the reconstruction volume is evaluated locally using a patch R m,n,i of size s × s centered around each pixel (m, n), defined as

R m,n,i (u, v) = |P zi {H}(m + u -s/2, n + v -s/2)| .
(3) Then, the reconstruction distances at which the focus curve reaches its maximum d max m,n and minimum d min m,n values are extracted, such that

d max m,n (F M, s) = arg max i∈[1,N ] {F M (R m,n,i )} , (4)
d min m,n (F M, s) = arg min i∈[1,N ] {F M (R m,n,i )} . (5)
According to [START_REF] Madali | Automatic depth map retrieval from digital holograms using a depth-from-focus approach[END_REF], if the optimal focus measure is employed, and the reconstruction interval is close to the boundary of the scene, the sharpness measurement for highly textured patches will achieve its highest value at the depth of focus. Conversely, for uniform patches, the sharpness measurement will reach its lowest value at the depth of focus. However, if the optimal conditions are not met, the focus curve is likely to have multiple extrema, with the optimal focus depth located at the second or third extremum value. The objective of this study is to enhance the fusion of the textured and smooth regions obtained from the estimated depth maps d max and d min , respectively. This will be achieved by considering not only the information provided by each pixel focus curve but also the local neighborhood information at various patch resolutions. Assuming a uniform distribution of both regions throughout the depth map, each depth map can be perceived as a degraded version of the ground truth depth map. The degradation can be visually assessed using the in-focus map linked with each depth map, where the well-estimated areas appear sharp and the remaining areas are contaminated by speckle noise. All possible predictions produced using different pairs of focus measurements and patch size can be united into a set given by:

D = F M ∈Φ s∈S d min (F M, s), d max (F M, s) (6)
where Φ and S are all the possible focus measures and patch sizes.

Self-supervised learning

Given the set D, the task at hand is to determine the depth map d that has the minimum average deviation, according to the loss function L, by minimizing arg min

d E y∈D (L(d, y)). (7)
A methodology similar to the one proposed in [START_REF] Ulyanov | Deep image prior[END_REF] is used, which reformulates image restoration as an optimization problem. Given a degraded image ẋ and a clean image x, the conventional image denoising problem can be written as

x = arg min E( ẋ; x) + R(x), (8)
where E( ẋ; x) is a fidelity term and R(x) is a regularization term. The optimal value of x is iteratively searched in the image space starting from a random initial point until an optimal value is reached according to predefined criteria. An alternative approach outlined in [START_REF] Ulyanov | Deep image prior[END_REF] involves constructing a parameterized function G with parameters θ and optimizing them in the parameter space using gradient descent until convergence.

More formally

x = arg min E(G θ (z); ẋ) + R(G θ (z)), (9)
where G θ is a convolution network and z is random noise. The authors claim that due to the surjective nature of the function G, the two problems are equivalent. Experimental results indicate that using a suitable network architecture G initialized randomly and optimized with gradient descent, the network can converge to a naturally-looking local optimum or, at least, pass near one before reaching the optimal solution ẋ = G θ (z). To prevent the network from overfitting to the input, the authors employed an early stopping technique by manually setting a limit on the number of iterations.

In the present work, we used the assembled encoderdecoder proposed in [START_REF] Ulyanov | Deep image prior[END_REF] for the inpainting task [START_REF] Newell | Stacked hourglass networks for human pose estimation[END_REF]. It has a depth of 5 layers, 128 output channels at each stage and does not use skip connections. We used the ℓ 2 and Sobel gradient for the fidelity and regularization terms, respectively. Overall, the training process can be formalized as [START_REF] Ehret | Modelblind video denoising via frame-to-frame training[END_REF] where ẋ is a random sample from D, x is the predicted value, and ∇ x and ∇ y are the image gradients along the x and y directions respectively.

x = arg min 1 | L | 2 [ i,j ∥ ẋ -x∥ 2 + (∇ x ∥ ẋ -x∥ 2 + ∇ y ∥ ẋ -x∥ 2 )],

Experiments

To evaluate the performance of the proposed methodology, a collection of 500 holograms, divided into five categories (Piano, Table, Wood, Dices, Cars), were obtained using a layer-based method [START_REF] Gilles | Hybrid approach for fast occlusion processing in computer-generated hologram calculation[END_REF], with a resolution of 1024 × 1024, a pixel pitch of 6µm, and 100 different acquisition angles for each scene.

Each hologram is then reconstructed at 256 different distances within the range of [z min , z max ], with z min and z max set to 0.0049cm and 1.23cm, respectively. Four of the best-performing focus operators from [START_REF] Madali | Automatic depth map retrieval from digital holograms using a depth-from-focus approach[END_REF], namely the Variance of Laplacian [START_REF] Pech-Pacheco | Diatom autofocusing in brightfield microscopy: a comparative study[END_REF] (LAPV), the Ratio (WAVR) and Variance (WAVV) of wavelet coefficients [START_REF] Yang | Wavelet-based autofocusing and unsupervised segmentation of microscopic images[END_REF], and the Normalized Graylevel variance [START_REF] Santos | Evaluation of autofocus functions in molecular cytogenetic analysis[END_REF] (GLVN), are used with variable patch-sizes in the set {5, 9, 13, 17, 21, 33}.

For each hologram, the network is trained using the depth maps D for 20000 iterations with an exponential learning rate decay starting from 0.1 with a factor of 0.8 every 100 iterations.

Results

Table 1 gives the obtained ℓ 1 norm for both the autoswitch method [START_REF] Madali | Automatic depth map retrieval from digital holograms using a depth-from-focus approach[END_REF] and the proposed method.

Piano 1: The ℓ1 norm evaluates the number of reconstruction planes between the predicted and the ground truth depth. Green for the first, blue for the second, and red for the third best-performing method.

The obtained results show that for the DFF method with auto-switch, the GLVN and WAVR focus measures generally perform the best across all scenes, with the highest ℓ 1 norm for the Table scene due to multiple scene objects occlusions. The performance of the focus measures can be attributed to three factors. First, the size of the reconstruction interval and the number of sampled reconstruction distances: when the reconstruction interval and depth sampling rate increase, the focus curves become noisier, which complicates the optimal focus plane estimation. Second, estimating the depth map using a single patch size may not be adaptable as different regions may require larger or smaller patches to have a proper focus curve with single extremum values. Third, the performance also depends on the nature of the used operators. For example, gradient-based operators will have higher performance in textured regions and poor performance in smooth regions.

Fusing the depth maps leads to improved performance across all scenes. The difference in performance between the auto-switch approach and the proposed approach varies depending on the initial depth estimation. For instance, the improvement in Dices and Woods class, where initial estimations are close to the ground truth, is relatively small compared to other scenes where the performance is twice as good as the auto-switch. Despite the network improved handling of the occlusions in the Table scene, the performance remains the weakest. Indeed, while the estimated depth map presents clear boundaries between occluded objects, our proposed method fails to accurately predict the depth in occluded regions, as depicted in Figure 1.

The final results of the network are determined by the chosen regularization term and the used output space D. When no regularization is applied, the network is prone to converge to an over-smoothed solution.

To remedy this, the Sobel loss is employed to increase the output depth map sharpness. The total variation method was also considered, however, it hinders the convergence speed and gives poor results.

Discussion

In addition, the images in the set D should have a consistent degradation compared to the ground truth depth map and each region of the depth map should be accurately estimated in at least half of the images utilized. If a region is poorly estimated in several images, the network tends to average those estimates instead of propagating the well-estimated values from the other regions. In order to achieve accurate depth map estimation for each region, it may be necessary to reduce the number of images used to a minimum number where each region is accurately estimated in at least half of them. Another potential solution is to directly train the network using the depth maps produced using the auto-switch method. For example, in Figure 2, the network was trained only with the depth map generated by the WAVR operator with a patch size of 33×33. This resulted in a smoother depth map, but it lacked spatial coherence. In addition, the network struggles to properly address discontinuous areas. By supervising the network with more depth maps, the results are significantly improved and the predicted depth map is close to the ground depth.

Although the network yields meaningful results, it has limitations. Firstly, the choice of neural network architecture influences the final results. The selected network should have a long path from input to output while avoiding too many parameters that cause fast overfitting. Secondly, finding the optimal number of iterations can be challenging, since stopping too soon leads to poor depth maps, while stopping too late degrades the results as shown in Figure 3. Lastly, the optimization step takes considerable time, around 10-15 minutes in our experiments, which may hinder online use.

Conclusion

In this work, we present a learning-based approach to fuse information obtained by applying the DFF method on a holographic reconstruction volume using various focus measures with varying patch sizes.

The experimental results showed that combining different focus measures can significantly improve the obtained results when selecting the appropriate hyperparameters, such as network architecture, number of iterations, optimization, and a regularization term.

Figure 1 :

 1 Figure 1: An illustration of the estimates produced on the Table scene, where occluded areas in the ground truth depth map in (a) are misestimated on the predicted depth map in (b).

Figure 2 :

 2 Figure 2: In (b) the obtained depth map when the network is supervised using only one depth map computed using the WAVR operator in (a), (c) the depth map when trained using the proposed methodology, and (d) the ground truth depth map.

Figure 3 :

 3 Figure 3: The intermediate results generated by the network after 1,000, 4,000, and 20,000 iterations in (a), (b), and (c), respectively, are compared to the ground truth depth map in (d). During the early iteration stages, the network converges towards a relevant solution but lacks the intricate details present in the ground truth depth map. Nevertheless, as the iteration count increases, the network gradually integrates the missing details into its predicted depth map.

Table Woods

 Woods

					Cars Dices
	LAPV 22.4	37.17 7.65	9.47 21.4
	WAVV 24.46	39.4	7.12	9.53 19.01
	WAVR 10.24	22.64 4.89	6.1	9.99
	GLVN 15.63	26.3	4.2	7.24 5.54
	Ours	4.41	8.54	2.53	3.82 3.02
	Table

* This work has been achieved within the Research and Technology Institute b<>com, dedicated to digital technologies. It has been funded by the French government through the National Research Agency (ANR) Investment referenced ANR-A0-AIRT-07. Authors can be reached at {nabil.madali, antonin.gilles, patrick.gioia, luce.morin}@b-com.com.


Ismaël Charlet
email: ismael.charlet@cea.fr

Bruno Reig #
email: 2bruno.reig@cea.fr

Corentin Mercier $#

Julien Delprato #

Vincent Puyal #

Clémence Héllion #

Marjolaine Allain #

Stéphane Monfray $

Alain Fleury $

Frédéric Gianesello $

Philippe Cathelin

Jean-François Robillard

Emmanuel Dubois

Jose Lugo-Alvarez

RF Performance of Large Germanium Telluride Switches for Power Application

Keywords: PCM, GeTe, phase-change RF switches, sub-6G

This paper reports the integration and characterization of germanium telluride (GeTe) phase-change material (PCM)-based RF switches in series configuration with a 89 µm wide PCM element to address sub-6G power handling requirements. Switching conditions using indirect heating, as well as small-and large-signal measurements are performed in this work. Devices handle power up to 37 dBm and 29 dBm respectively in ON-and OFF-state at 915 MHz. These results estimate for the first time the performances of such wide GeTe switches compatible with cellular applications and investigate technological improvement guidelines.

I. INTRODUCTION

The development of multiband cellphones pushes the requirements of the radio-frequency (RF) devices. In particular, RF switches set in the Frond-End Module play a key role since they route the signal to the different RF blocks and/or allow RF reconfigurability. Over the last decade, RF switches based on micromechanical system (MEMS) have been used taking benefit from good RF performance [START_REF] Pacheco | Design of low actuation voltage RF MEMS switch[END_REF]- [START_REF] Bovadilla | Optimization of RF MEMS phase shifter for microwaves applications[END_REF]. Alternatively, RFSOI switches based on PIN diodes or MOS transistors have been an industrial and mature technology for the last 10 years [START_REF] Foissey | 85 fs RonxCoff and CP1dB@28GHz>25dBm Innovative PIN Diode Integrated in 55 nm BiCMOS Technology mmW 5G and 6G Front End Module[END_REF]- [6]. In this context, phase-change material (PCM) switches represent an appealing recent solution to address sub-6G and millimetre-wave applications, thanks to their low insertion loss and small footprint. They rely on the large resistance ratio between the amorphous and crystalline states of chalcogenide compounds. More precisely, germanium telluride (GeTe) is widely studied for these applications thanks to its low ON-state resistivity and large ON/OFF resistivity ratio.

Electrical switching of the PCM is achieved by applying short thermal pulses, for instance with Joule effect using microheaters [START_REF] El-Hinnawy | A four-terminal, inline, chalcogenide phasechange RF switch using an independent resistive heater for thermal actuation[END_REF]- [START_REF] Borodulin | Recent advances in fabrication and characterization of GeTe-based phase-change RF switches and MMICs[END_REF]. Switches are set in OFF-state when the PCM is heated above the melting temperature and cooled quickly to quench in the amorphous state. On the opposite, the ON-state is obtained with longer thermal pulse slightly above the crystallization temperature, allowing atoms to set in the preferred crystalline state. So far, small PCM switches has been reported, with lengths and widths about 1 µm and 30 µm respectively [START_REF] El-Hinnawy | A four-terminal, inline, chalcogenide phasechange RF switch using an independent resistive heater for thermal actuation[END_REF]- [START_REF] Singh | Characterization of phase change material germanium telluride for RF switches[END_REF]. Nevertheless, such small dimensions with reported power handling below 32 dBm will not be sufficient to handle cellular applications requiring 37 dBm.

In this work, we present the performances of GeTe based 89µm-wide and 4 µm-long RF switches. In section II, we present the switch design and fabrication. In section III, we present electrical characterization results. In section IV, we discuss the switch limitations and technological improvement guidelines. To the best of our knowledge, this is the first measurements of PCM switches with dimensions compatible with cellular applications for sub-6GHz 4G/5G FEM architectures.

II. PCM-BASED SWITCH DEVICES AND FABRICATION

Using formulas presented in [START_REF] Leon | RF power-handling performance for direct actuation of germanium telluride switches[END_REF] and [START_REF] Wainstein | Compact modeling and electrothermal measurements of indirectly heated phase-change RF switches[END_REF], we plotted different abacus which show that high-power cellular applications require large PCM switches. Indeed, for 37 dBm handle power and minimum FoM of 35 fs, the length must be greater than 3.5 µm and the width has to be at least 85 µm. Thus, we came up to the conclusion that a length of 4 µm was mandatory to handle 37 dBm in the OFF-state. On the other hand, to keep a reasonable (and thus insertion losses) a width of 89 µm has been chosen.

CPW lines are used to access the PCM element; while the material heating is achieved using a tungsten micro-heater located above the PCM (indirect heating).

The fabrication process is based on the integration of GeTe within the back-end-of-line (BEOL) of a CMOS compatible 200 mm flow. The process flow starts with a thermal oxidation of a high-resistivity (HR) silicon wafer. A first metallic layer made of AlCu is deposited, etched and filled with silica. Next, the 100 nm-thick 0.5-0.5 GeTe is deposited and backside contacted to the underlying metal. A 100 nm-thick PVD AlN layer is deposited over the PCM as a dielectric layer, followed by a silica deposition and planarization. Heaters above the PCM are then formed, using tungsten connected to the first metallic layer. Finally, vias and the 1 µm-thick second metallic layer are formed.

III. ELECTRICAL PERFORMANCES

A. Electrical Actuation

Indirect PCM switching using the heater is performed here. High speed pulses are applied between the heater electrodes using a 50 Ω -matched HP 8114A pulse generator and the resulting PCM resistance is measured with a source measurement unit (SMU) by applying a 10 mV signal. Switches are set in OFF-state by setting the PCM in the amorphous phase. To do so, the PCM must be heated above the melting temperature of 700 °C, followed by a short cooling time. Pulses used for amorphization are 400 ns wide, with 10 ns rise and fall times. Then, switches are reset in ON-state by applying a 1 µs long crystallization pulse. The pulse voltage has been swept to find the adequate conditions to achieve crystallization and amorphization and are illustrated in Fig. 2. Amorphization is achieved when a 12.7 V pulse is applied to the heater electrodes while crystallization is done for a 5.6 V pulse.

B. Small-signal Measurements

S-parameters of the RF switch have been carried out using PNAX from Keyseight. A Line-Reflect-Reflect-Match calibration is performed on a commercial Impedance Standard Substrate (ISS), using Ground-Signal-Ground (GSG) Infinity probes with a 100 µm pitch. S-parameters results are presented in Fig. 3 for the ON-state and Fig. 4 for OFF-state. A thru deembedding is applied to extract ON-state resistance ( ) using the equation:

= 2 10 ( )/ -1 . (1)
Where = 50 Ω is the reference impedance, ! the measured transmission of the thru de-embedding structure and "#$ the switch transmission. is estimated about 2.5 Ω below 6 GHz (Fig. 3), corresponding to a crystal GeTe conductivity of 1.8E5 S/m which meets expectations. Next, the OFF-state capacitance (' (( ) is extracted using the formula:

' (( = 4*+ √10 /--1 -. (2)
Where + is the frequency, = 50 Ω the reference impedance and "#$ the measured transmission of the SPST switch. Thus, the extracted OFF-capacitance below 6 GHz is 37.5 fF.

C. Large-signal Measurements

Power handling is then measured at 915 MHz fundamental frequency using the bench described in Fig. 5 and calibrated with in a thru configuration. The device studied in series configuration is exposed to a power sweep from 0 dBm to 40 dBm. Fig. 6 presents the output powers for ON-and OFFstate of the 89 µm-wide switch. A 30 dBc isolation from ONstate and OFF-state is presented. The maximal RF power before the device breakdown in the ON-state 0 1(234 is measured at 37 dBm which is in good agreement with expectation for such PCM dimensions. In the OFF-state, the maximal RF power before the device breakdown 0 1(234 (( is measured at 29 dBm. This result is lower than the target since 37 dBm are expected in the OFF-state for such dimensions. This effect will be detailed in next section as well as guidelines to improve this parameter. IV. SWITCH LIMITATIONS AND IMPROVEMENTS One well known difficult aspect of large PCM switches concerns the capability to turn "all" the material in amorphous state, since amorphization is linked to the capability to cool the material fast enough. The assumption when sizing the switch was that 2/3 of the material was amorphized, but the OFF-state power handling measurements suggest that this condition is not satisfied.

To verify this assumption we use the property that amorphization length can be accessed by fitting ' (( measurements using the following equation:

' (( = ' 1( 5 ' 67$7 5 ' 8 2 5 ' ( . ( 3
)
With ' 1( the capacitance between the RF contacts, ' 67$7 the capacitance between crystalline GeTe, ' 8 the capacitance between the crystalline GeTe and the heater and ' ( the fringe capacitance, given by: ' 1( = : . : ;4 . < 1( . = >;? @ 1( (4)

' 67$7 = : . : 67$7 . < 67$7 . = 67$7 @ 1( . A (5) ' 8 = : . : BC . < 67$7 . < 8 -@ 1( . A 2 = "D . (6)

Where < 1( is the PCM width here set to 89 µm, @ 1( the RF gap set to 4 µm, = >;? the contact metal thickness, = 67$7 the GeTe thickness, : : ;4 : BC : 67$7 respectively the vaccum, oxide, AlN and GeTe permitivities, = "D the dielectric barrier thickness, < 8 the heater width set to 3 µm and A the PCM amorphization ratio. Herein, A is an adjustable variable corresponding to the PCM amorphization ratio such as A = 0 means the PCM is fully cristalline while A = 1 corresponds to a fully amorphized PCM. Fig. 7 presents the ' (( measurements of 4 µm long PCM with different widths and theoretical ' (( from (3) for different amorphization ratios A. This shows a clear and sensitive influence of A on the theoretical ' (( . In particular, theoretical ' (( fit well with the experimental ones for A = 0.5, which is lower than the required A = 2/3 to ensure OFF-state power handling expectation. This reduced amorphization length can be explained with thermal simulations performed with COMSOL (Fig. 8). The switch stack shown in Fig. 8a is simulated with an amorphization pulse sent through the heater. Fig. 8b corresponds to the resulting temperature alongside the GeTe width. This simulation shows a strong non-uniform heat profile within the PCM since temperature reaches 1300 K at the PCM center and increases to 1630 K at the PCM edges. These heat hotspots are due to lower heat dissipation at the PCM edges. Thus, the PCM element required a longer quenching time to cool down on its edges, which ultimately degrades the amorphization process and leads to a smaller amorphization length. Besides, this non uniform temperature distribution involves a non uniform amorphization profile, which also degrades the OFF-state power handling.

An achievable area of improvement relies on the heat confinement within the heater. So far, the heater has a top encapsulation made of oxide and SiN (as in Fig. 8a) which are poor thermal conductor leading to a strong heater overheat. Neverthless, adding an AlN layer between the oxide and SiN top layers can help to reduce the hotpsot temperature thanks to its good heat conduction. Thermal simulations are performed for the stack of Fig. 8a but with an extra 50 nm-thick AlN layer as illustrated in Fig. 9a. The temperature alongside the PCM is shown in Fig. 9b. Here, the presence of AlN significantly lowers hotspots to 1330 K. From this, a better amorphization uniformity and a shorter quenching time can be expected, leading to a longer and more uniform amorphization area and ultimately to a better power handling in the OFF-state. Fig. 10. Results of this work compared to state-of-the-art switches FoM and estimation for our future work with a better amorphization control. 'RFSOI [START_REF] Gianesello | Advanced 200-mm RF SOI Technology exhibiting 78 fs[END_REF] with SP9T' corresponds to a SP9T device using stacked transistors of reference [START_REF] Gianesello | Advanced 200-mm RF SOI Technology exhibiting 78 fs[END_REF]. Measured performances of this device were shared to the authors by STMicroelectronics.

All the results present here are summarized in Fig. 10 and compared to state-of-the-art switches. In particular, reaching . ' (( = 35 fs requires a better amorphization control as explained, but also a better GeTe conductivity in the ON-state [START_REF] King | Development of cap-free sputtered GeTe films for inline phase change switch based RF circuits[END_REF].

V. CONCLUSION Electrical performances of GeTe-based SPST switches with dimensions compatible with cellular applications have been studied for the first time. Switches are set OFF and ON by applying a 12.7 V pulse of 400 ns and a 5.6 V pulse of 1 µs respectively. This work shows that the 89 µm wide switches can handle power up to 37 dBm at 915 MHz in the ON-state, which meets requirements for cellular applications. Besides, it is shown that future work must focus on the amorphization control. Indeed, switches handle power up to 29 dBm in the OFF-state, limited by the amorphization ratio. To reach 37 dBm in the OFF-state, mandatory for cellular applications, several guidelines are highlighted. Numerical simulations have shown that the amorphization process is degraded by hotspots at the heater edges. Thus, OFF-state power handling can be improved by reducing these hotspots, for example by adding an extra AlN layer above the heater.

Fig. 1 .

 1 Fig. 1. (a) Optical photograph of the SPST switch (b) schematic layout of the PCM element (grey) and the heater (green).

Fig. 2 .

 2 Fig.2. Indirect switching of the SPST switch using a 400 ns long pulse for amorphization (dark) and a 1 µs long pulse for crystallization (red).

Fig. 3 .

 3 Fig. 3. Measured transmission loss of the 89 µm wide SPST in ON-state (Insertion Loss) and the thru results; with the associated %& extraction.

Fig. 4 .

 4 Fig. 4. Measured transmission loss of the SPST in OFF-state (Isolation) and the open structure results; with the associated ' %// extraction.

Fig. 5 .

 5 Fig. 5. Measurement setup for the power handling.

Fig. 6 .

 6 Fig. 6. Measured transmission as a function of the input power at 915 MHz for SPST in both ON and OFF states.

5 δ = 2 / 3 Fig. 7 .

 5237 Fig. 7. Measured OFF-state capacitance of 4 µm-long switches for different widths (red) and fit curves (black) for different amorphization ratios using (3).

Fig. 8 .

 8 Fig. 8. COMSOL heat simulation of the studied switch (a) simulated stack cut alongside the XX'-axis (b) PCM temperature distribution alongside the YY'axis.

Fig. 9 .

 9 Fig. 9. COMSOL heat simulation of the studied switch with an extra AlN layer (a) simulated stack cut alongside the XX'-axis (b) PCM temperature distribution alongside the YY'-axis.