Thomas Alcaire
email: thomas.alcaire@st.com

Delphine Le Cunff
email: delphine.le-cunff@st.com

Victor Gredy
email: victor.gredy@st.com

Jean-Hervé Tortai
email: jean-herve.tortai@ltmlab.fr

Spectroscopic Ellipsometry Imaging for Process Deviation Detection via Machine Learning Approach

Keywords: Spectroscopic ellipsometry, model-less, process control, machine learning

Spectroscopic ellipsometry is a very sensitive metrology technique to accurately measure the thickness and the refractive index of the different layers present on specific dedicated metrology targets. In parallel, optical defectivity techniques are widely implemented in production lines to inspect a large number of dies and catch physical and patterning defects during the process flow. It becomes then of interest to explore a new approach overlapping metrology and defectivity by using the sensitivity of metrology tools on a full wafer scale. In our case, spectroscopic ellipsometry's optical response was collected directly on the dies to capture specific deviations such as film properties and thickness variation. This is an innovative strategy that requires a model-less approach, combining an automatic ellipsometry mapping generation and a smart classification via a machine learning algorithm. In this paper, we will present such approach on two industrial use cases and explain how an image classification algorithm can be implemented to automatically detect the process drift on the latter.

I. INTRODUCTION

In the standard approach, Spectroscopic Ellipsometry (SE) is used to measure the thicknesses and the optical indices of multistack layers deposited on a substrate. Since many years now, spectroscopic ellipsometry is also routinely used for Optical Critical Dimension (OCD) control through modeling of periodical patterned structures. Both applications require a perfect knowledge of the layers and/or of the structure constituting the metrology targets in order to develop a realistic mathematical model. By simulating the spectroscopic ellipsometry response and after a reverse problem-solving operation, one can then extract the parameters of interest (thickness, n and k, CD, Side Wall Angle…). This process can be very time consuming and request heavy engineering effort (pre-characterization, cross comparative…). Both approaches also require the design of specific blanket or periodical targets on sacrificial area of the product wafer.

In recent years, in order to accelerate the learning curve of new technological developments, model-less strategy have been introduced in the metrology field. In this context, one will rely on the strength of machine learning for data treatment rather than on an accurate underlying model. Machine Learning is a subset of artificial intelligence which enables computer systems to learn from data without being explicitly programmed. This new alternative for data treatment has already been explored for semi-conductor process control in several previous studies such as for example: predictive metrology [START_REF] Timoney | Implementation of machine learning for high-volume manufacturing metrology challenges (Conference Presentation)[END_REF][2] or detection of voids in copper lines [START_REF] Kong | Machine learning and hybrid metrology using scatterometry and LE-XRF to detect voids in copper lines[END_REF].

In this paper, we explored the benefit of spectroscopic ellipsometry mapping measurement as well as data treatment for full wafer process control. Due to the complexity of the optical response from the underlying design structure of the product wafer, we implemented a model less approach to detect local process deviation. Two use cases of process anomalies are reported in this paper: full wafer area or full chip area inhomogeneity. For the latter, the SE raw data can be processed through machine learning to automate the deviation detection protocol.

II. EXPERIMENTAL SETUP AND PROTOCOL

In this section, we describe the spectroscopic ellipsometer equipments that were used for this study as well as the protocol for SE mapping acquisition and image generation. Industrial SE metrology equipments are completely inappropriate for scanning a full wafer surface due to their small spot size. We then used two laboratory equipments with a larger footprint on the surface of the wafer. The two systems are phase-modulated ellipsometers supplied by HORIBA. Due to the high modulation frequency of the phase (50 kHz), both ellipsometers allow fast measurements of ellipsometry angles at a single wavelength (< 200 ms per point). It means that if one wavelength is chosen, the full wafer scanning by ellipsometry will be sufficiently fast.

A. Spectroscopic ellipsometers

A basic schematic of the phase-modulated ellipsometer principle is presented in Fig. 1 and the exact configuration can be found in [START_REF] Gaudiello | Electrical test prediction using hybrid metrology and machine learning[END_REF]. For phase-modulated ellipsometer, the raw data gathered are the ellipsometry parameters Is and Ic, which values are then bound between -1 and 1. These are linked to the usual ellipsometric angles Δ and Ψ through equations ( 1) and (2).

2

(1)

2 (2)
The first equipment is an UVISEL 1 designed for 200mm wafers and is therefore limited to acquisitions up to 100mm radius with an incident angles that can vary from 55° to 85°. The measuring spot size is of 0.14x0.08mm. The light source wavelength covers 190 to 850nm. It is then well suitable for localized mapping of specific region of interest. The second is an inline UVISEL 2 (IMPACT platform), which covers a large broadband spectrum (150nm -2µm), which is able to collect responses at any coordinates on the surface of a 300mm wafer, with a fixed incident angle of 70°. This equipment was then used for 300mm full wafer scanning considering its large measuring spot size of 8x5mm.

B. Acquisition Protocol and Image generation

A specific protocol was implemented for data acquisition which is summarized in Fig. 2. As a preliminary step to ellipsometry mapping acquisition, a routine is performed to select the optimum angle and wavelenght to get maximum sensitivity.

1) Angle of incidence selection

For the case of UVISEL 1 equipment, for which the angle of incidence can be varied with a goniometer, we performed spectroscopic scans of the region of interest at different angles in order to select the optimum angle providing the largest variation in amplitude for Is or Ic (Fig. 3). Indeed, ellipsometers measure the ratio of the amplitude reflection coefficients (rp/rs). Those coefficients strongly depend on the angle of incidence. At Brewster angle, the difference between the parallel coefficient (rp) and the perpendicular coefficient (rs) is maximal, and the sensitivity for the measurement is optimal. Thus, this condition could correspond to an optimal configuration for our acquisition. Unfortunately, considering the complexity of the dies, the Brewster angle can hardly be theoretically modeled, and we have to rely on experimental testing for the optimized incidence angle selection.

2) Wavelength selection

In order to get the maximal sensitivity to any changes in optical responses, a spectroscopic scan was performed in the area of interest/a typical die at the optimal angle previously chosen. The obtained spectrum allows us to choose the single wavelength that will be used to rapidly perform the Is and Ic acquisition. This wavelength is set by finding the intercepts of the Is and Ic curves of the spectroscopic scan. Among those intercepts' positions, the one that presents the maximal local slope and the maximum gain for the detector signal is chosen. Any slight variations in the material properties will then result in large variations of the ellipsometry signal on both Is and Ic components (Fig. 4). The purpose of selecting a single wavelength is to drastically reduce the acquisition time. Indeed, a single wavelength acquisition takes a couple of seconds per point to be performed, when a full spectroscopic scan would take minutes per point. Considering that we used mapping grids composed of thousands of points, a spectroscopic acquisition is therefore unrealistic. By choosing a single wavelength, we preserve the sensitivity of spectroscopic ellipsometry, while keeping acquisition times reasonable. Fig. 4 -Example of Is and Ic spectroscopic response at the optimal angle and optimal wavelength selection, which is 410nm here.

3) Image generation

The selected conditions for wavelength and potentially angle are then applied for ellipsometric (Is and Ic) acquisitions on a dense mapping grid of the area of interest. This was implemented using a custom-written Python script. The ellipsometry images are then generated by assigning each value of Is and Ic to their corresponding coordinate, and in turn obtain two ellipsometry mappings of the region of interest. The final cartographies are obtained after performing a cubic interpolation between the measured points. Examples of Ic and Is cartographies can be found in Fig. 8.

III. RESULTS

This section describes the results obtained conducting this acquisition protocol on two use cases of process variation: respectively "Intrawafer nitride passivation layers properties variation" and "Color resist layers intradie thickness variation".

A. Intrawafer nitride passivation layers properties variation

For the first use case, we tested the sensitivity of model-less ellipsometry approach to detect intra wafer signature resulting from difference in stress of nitride layers [START_REF] Kong | Machine learning and hybrid metrology using scatterometry and LE-XRF to detect voids in copper lines[END_REF] (both on blanket and on patterned wafer). These nitride layers used for the study are dielectric films acting as isolation between metallic Al contact pads of patterned wafer. For some cases, changes in mechanical properties can lead to yield issues (Fig. 5a). The analysis was then performed on the full wafer surface. The mapping grid used and presented in Fig. 5b is composed of 6000 measurement points evenly spread on the 300mm wafer surface. Is and Ic acquisitions were performed with UVISEL 2 equipment. A duration of 3.5 hours was necessary for each full wafer acquisition.

1) Blanket wafer analysis

Blanket nitride layers on silicon can easily be characterized by industrial in-line ellipsometry as already demonstrated in [START_REF] Brouzet | Full Wafer Stress Metrology for Dielectric Film Characterization: Use Case[END_REF]. For this specific use case, a strong inhomogeneity was observed linked to changes in the refractive index of the layer (Fig. 6). As a first test, we tested our model-less spectroscopic ellipsometry approach to characterize the same wafer. Following the protocol described in Fig. 2, a spectroscopic scan was performed to select the optimal wavelength for the further analysis. As it can be seen in Fig. 7, multiple wavelengths were eligible but in order to maximize luminous intensity from our source, a 500nm wavelength was retained. The Is and Ic response collected on UVISEL 2 using the mapping presented in Fig. 5b are presented in Fig. 8. The intra wafer signature cartographies obtained through model less spectroscopic ellipsometry is extremely similar to the one obtained with the in-line measurements except that they benefit from a larger amount of data. This proves the sensitivity of this approach to detect variation in the deposited film properties, without the need to develop a theoretical model of the stack.

2) Product wafer analysis

By opposition to blanket wafers, product wafers are composed of an intricate and complex stack of layers, for which an accurate model would be very arduous to produce. This, combined with the lack of metrology target structures available for some cases at the extreme edge, would make this kind of intrawafer variation hardly detectable by classical spectroscopic ellipsometry measurements.

As for blanket layers, the protocol starts with a spectroscopic scan (Fig. 9) to select the wavelength to be used for the further full wafer acquisition. The wavelength was set to 338nm to gather the raw ellipsometric data, and the resulting cartographies are displayed in Fig. 10a, where a clear intrawafer signature is observed, compatible with the one observed on the blanket wafer. The results were compared to the characterization results obtained with a full wafer interferometry technique (PWG system from KLA) which measures notably local stress and nanotopography. The PWG stress cartography is shown in Fig. 10b and displays an extremely similar signature with respect to the model-less ellipsometry ones. In conclusion, we proved the sensitivity of our model-less approach to detect changes in material properties at the wafer scale, first on a blanket wafer, but more interestingly, on a product wafer.

B. Color resist layers intra die thickness variation

For the second use case presented here, our intent was to detect intra wafer thickness variation induced by color resist deposition. The color resists are used as optical filters to perform RGB (Red, Green, Blue) pixel arrays for imagers devices. Due to complex topography effect on product wafers, the spin-coating deposition can lead to inhomogeneity in the resist thickness causing a radial striations signature. As we can see on the optical inspection image (Fig. 11a), those striations are more pronounced at the edge of the wafer than at its center. Consequently, we focused the study on dies at the edge of the wafer, and the acquisition grid was set up as illustrated in Fig. 11b. We measured 5 dies (1500 points in each die of 48mm²) for each acquisition, one at center as a reference, and 4 on the edge of the wafer. For such use case, where a full wafer acquisition is not required, we could use the UVISEL 1 and benefit from the angle optimization step presented earlier (in II.B.). We observed an enhanced sensitivity for a 55° angle of incidence (Fig. 12). It is supposed that this angle corresponds to the effective Brewster angle of the material studied (optical refractive indices in the 1.6 range), which explains the increased ellipsometric response. For this incident angle, the optimal wavelength was determined to be 315 nm (Fig. 13). This wavelength selection is even more critical here considering the relatively small amplitude variation observed in the collected spectrum. Following the acquisition protocol, we performed the multiple dies acquisitions and the cartographies obtained are show in Fig. 14. The model-less spectroscopic ellipsometry results were compared to PWG nanotopography mappings. We observe that no striations are detected at the center of the wafer both on PWG and ellipsometry cartographies. On the contrary, the 4 other dies exhibit striations that both PWG and ellipsometry could detect.

In conclusion of this use case, we were then able to detect this process deviation related to variation in thickness with a model-less spectroscopic ellipsometry approach on a product wafer, in the same way as the PWG technique. It is to be noted that PWG is limited to topography below 150nm and subsequently can suffer from reconstruction errors.

IV. MACHINE LEARNING TREATMENT

To go further in the data treatment of the ellipsometry images acquired for color resist as detailed in Section III.B, we investigate the implementation of a machine learning algorithm to automate the striation detection. We set up an artificial intelligence algorithm capable of classifying images in two categories: Striated and Flawless dies.

First of all, a specific pre-treatment of the images is needed for a machine learning algorithm to work properly. The pretreatment extracts important features from the images and was performed using descriptors which are image processing tools each one being specific to the detection of one feature (e.g. line or edge detection). Their output is often a vector composed of numerical values, which can now be treated by a supervised learning algorithm. Those features are selected so that they allow the algorithm to effectively distinguish a class from another. Because the discriminatory feature in this use case is the presence of striations, we chose the canny edge detector [START_REF] Canny | A Computational Approach to Edge Detection[END_REF] as descriptor. By definition, supervised learnings training data is required to be labeled with its corresponding class in order for the algorithm to learn. Once trained, it is able to automatically classify new cartographies, and even to use them as further training. A wide variety of supervised learning algorithms are available, but Logistic Regression gave us the best results.

As it can be seen in SE images of Fig. 14, the effects at the edge of the die could greatly influence the efficiency of the prediction algorithm. To avoid this potential issue, only the center of the die was then kept for this part (Fig. 15a). Also, being limited by the amount of experimental data at our disposal, we used synthetic mappings. They are generated to mimic the experimental ones, in order to have sufficient data to properly train the machine learning algorithm. Our dataset was then composed of a training set of 200 synthetic mappings (100 striated and 100 flawless) and a test set of 50 experimental mappings (40 striated and 10 flawless). An example of each cartography is shown in Fig. 15b. The synthetic cartographies features are then extracted by the descriptor and are used to train the machine learning algorithm. The machine learning predictive efficiency is finally tested on the experimental cartographies. For this first test of implementation, we actually obtained a very good prediction accuracy of 96 % (48/50) on our experimental ellipsometry mappings. A visualization of the dies identified (striated or flawless) by machine learning is shown in Fig. 16. Our model-less spectroscopic ellipsometry approach combined with a machine learning algorithm is then able to successfully automatically detects this process deviation. In this paper, we showed that a model-less strategy based on ellipsometry raw signal treatment is capable of detecting process deviations on full product wafer surface or on full dice of product wafer. Besides, the coupling of this technique with a machine learning algorithm, built as an image classifier, can allow an automatic detection of process deviation with a very high accuracy. Such combined approach can be applied to other sources of optical techniques more traditionally used for wafer physical defect inspection.

Fig. 1 -

 1 Fig. 1 -Schematic of the phase-modulated ellipsometer working principle.

Fig. 2 -

 2 Fig.2 -Overview of the ellipsometry imaging protocol.

Fig 3 -

 3 Fig 3 -Example of Ic variable angle spectroscopic response and optimal angle of incidence selection, which is 60° here.

Fig. 5 -

 5 Fig.5 -a) Yield wafer map showing a diametrical intra-wafer signature and b) dense mapping grid defined for 300 mm wafer surface scanning.

Fig. 6 -

 6 Fig.6 -In-line spectroscopic ellipsometry measurement map on the blanket wafer for refractive index (37 points).

Fig. 7 -

 7 Fig.7 -Ic and Is spectroscopic scan on the blanket wafer for the optimal wavelength selection.

Fig. 8 -

 8 Fig.8 -Model-less spectroscopic ellipsometry cartographies (6000 points) of the shifted blanket wafer.

Fig. 9 -

 9 Fig.9-Is and Ic spectroscopic scan on the patterned wafer for the optimal wavelength selection, which is 338 nm here.

Fig. 10

 10 Fig.10 -a) Model-less spectroscopy ellipsometry cartographies (6000 points) and b) PWG stress cartography of a patterned wafer.

Fig. 11

 11 Fig.11 -a) Optical inspection showing the presence of striations on the edge of the wafer, and their absence in the center and b) schematic of the mapping grid used for multiple die ellipsometry imaging.

Fig. 12 -

 12 Fig.12 -Spectroscopic scan (only Is is displayed for visibility) done at 55° and 70° showing a stronger ellipsometric response for the smaller angle of incidence.

Fig. 13 -

 13 Fig.13-Spectroscopic scan done at an angle of incidence of 55° for the optimal wavelength selection.

Fig. 14 -

 14 Fig.14 -Comparison between PWG nanotopography and model-less spectroscopic ellipsometry cartographies (1500 points) on the same die at the center and edge of wafer.

Fig. 15

 15 Fig.15 -a) Reduced region studied to remove uncertainty due to the positioning and b) examples of each of the cartographies used during the machine learning treatment: Flawless and Striated cartographies obtained through model-less spectroscopy ellipsometry, and Flawless and Striated cartographies generated artificially.

Fig. 16 -

 16 Fig.16 -Visualization of the prediction from the machine learning algorithm on the experimental ellipsometry cartographies

ACKNOWLEDGMENTS This work has received funding from the Electronic Component Systems for European Leadership Joint Undertaking under grant agreement No 826589 (project MADEin4). This Joint Undertaking receives support from the European Union Horizon 2020 research and innovation program and, Austria, Belgium, France, Germany, Hungary, Ireland, Italy, Netherlands, Romania, Sweden. This work has also been supported by the French ANR program Investissements d'Avenir EQUIPEX, contract number ANR-10-EQPX-33


Jianhao Zhang
email: jianhao.zhang@c2n.upsaclay.frcarlos.ramos@c2n.upsaclay.fr

Omar Ortiz

Xavier Le Roux

Eric Cassan

Laurent Vivien

Delphine Marris-Morini

N D Lanzillotti-Kimura

Carlos Alonso-Ramos

Norberto Daniel Lanzillotti-Kimura

Subwavelength engineering for Brillouin gain optimization in silicon optomechanical waveguides

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Brillouin scattering is a third-order optical nonlinear process that results from an optomechanical coupling [1]. This effect has been widely explored in the context of laser physics, microwavefrequency phonon excitation, and precision spectroscopy, among others [2]. Brillouin scattering process entails the excitation of mechanical waves, with frequencies ranging between a few tens of MHz and hundreds of GHz, which are frequently referred to as acoustic phonons [3]. The availability of materials co-confining phonons and photons in the same device has enabled the demonstration of Brillouin scattering on chip, showing exceeding strengths over Kerr and Raman interactions [4][5][6]. There have been remarkable achievements in the control of on-chip Brillouin optomechanics using chalcogenides [4], gallium arsenide [7][8][9], and silicon [10,11]. Optical phonons with frequencies up to 10 GHz [5] in nanometric-scale waveguides have wavelengths close to one micron, enable the strong interaction with photons in the near-infrared [11][12][13][14][15], remarkably large Brillouin gain [16] and Brillouin lasing [17] in silicon. While Brillouin scattering in optical fibers is mainly governed by the electrostriction effect in the bulk material, Brillouin scattering in nanophotonic silicon waveguides comprises a complex interplay between the bulk electrostriction and the geometry-dependent radiation pressure at the waveguide boundaries [10,11,15,18]. Different strategies have been proposed to optimize Brillouin gain in silicon waveguides [13,17,19,20]. Pedestal waveguides provide strong photon-phonon overlap [13] with a good Brillouin gain of up to 3000 (W • m) -1 . However, minimizing phonon leakage towards the silica under-cladding requires a silica pedestal with a narrow width, which may be challenging to fabricate. Silicon membrane rib waveguides can achieve remarkably low photonic and phononic losses by implementing silicon air gaps that partially isolate the waveguide core and the lateral silicon slab [16,17]. Still, the photonic mode is mainly confined in the waveguide core while the phononic mode expands all over the slab. This results in a comparatively small photon-phonon overlap that limits the intra-mode Brillouin gain to 1000 (W • m) -1 [16]. Theoretical calculations predicted a remarkably large Brillouin gain of 8000 (W • m) -1 in phoxonic crystals, exploiting bandgap confinement [20] of photons and phonons, simultaneously. Nevertheless, the performance of these phoxonic crystals may be hampered by the narrow bandwidth and high optical propagation loss, typically linked to bandgap confinement [21]. On the other hand, low optical loss and ultrawideband operation have been demonstrated in silicon subwavelength grating waveguides exploiting index-contrast confinement [22][23][24][25]. Silicon optomechanical waveguides using index-contrast confinement for photons and bandgap confinement for phonons have been recently proposed [19]. Such optomechanical waveguides could provide a Brillouin gain of 1750 (W • m) -1 . Yet, the performance of the bandgap-confined phononic modes might be hampered by the intrinsically limited bandwidth of this approach and by loss arising from tight sensitivity to fabrication imperfections.

Here, we propose a new waveguide geometry that exploits subwavelength index-contrast engineering to confine optical modes and subwavelength mechanical softening to minimize the loss of the acoustic modes. By leveraging the nanostructuration, the proposed subwavelength-lattice waveguide allows to enhance simultaneously the photon-phonon interaction and increase the quality factor of the mechanical mode. Our calculations show that the proposed waveguide may provide Brillouin gain as high as 10000 (W • m) -1 .

As illustrated in Fig. 1 (a), the proposed membrane waveguide consists of a central strip and a lattice of lateral arms that anchor the waveguide core to the infinite silicon slab. The phonon confinement in our structure is achieved by a large impedance mismatch between the air and the silicon core, instead of a phononic bandgap, while the arm lattice with subwavelength period introduces a small mechanical dissipation. The core has a width of 𝑤 𝑤𝑔 = 500 nm and a thickness of 𝑡 𝑤𝑔 = 220 nm, while the arms have a width of 𝑤 𝑎 = 2.5μm. The acoustic leakage is reduced by exploiting the height contrast at the core-nanoarm interface and nanorm-slab interface, where the spatial mismatch of acoustic mode causes a potential well (difference in frequency) for a shared wavevector. To maximize this height contrast, we choose a nanoarm, thickness of 𝑡 𝑎 = 50 nm. We refer "width" to dimension along x while "length" to that along z. This width (2.5 µm) is chosen to ensure a low optical loss, while avoiding acoustic resonance in nanoarms. An increase of this width causes slight variation in Brillouin gain and is not detailed here. We choose a period of 𝛬 = 240 nm, well below the acoustic wavelength (650 nm in silicon, at 9 GHz) and optical wavelength (640 nm in silicon, i.e., 1550 nm in vacuum). b) and (c), respectively. The optical and mechanical mode profiles are collected at the I plane with 𝐿 𝑎 = 50 nm. Low optical loss down to 2 dB/cm using subwavelength gratings is achievable, reported in earlier works [22][23][24][25] and is not detailed here. The mechanical band diagram of the proposed waveguide is obtained using a Finite-Element-Method (FEM) model, considering the 1 st degenerate Brillouin zone, with the wavevector along the direction of light propagation (z). The waveguide supports more than 30 mechanical modes with frequencies varying between a few MHz and 15 GHz. As an illustration, Fig. 2 To assess the performance of the proposed optomechanical waveguide we use the Brillouin gain, 𝐺 𝑚 . [10][11][12]15] From coupledmode theory it follows that the evolution of the power in the Stokes wave is governed by

𝒅𝑷 𝒔 𝒅𝒛 = -𝑮 𝒎 𝑷 𝒑 𝑷 𝒔 + 𝜶 𝒔 𝑷 𝒔 (1)
being 𝑃 𝑠 and 𝑃 𝑝 the power in the pump and Stokes waves, respectively, and 𝛼 𝑠 the propagation loss of the Stokes waves. Assuming that the optical mode and mechanical mode (induced by the optical force) co-propagate along the waveguide, the Brillouin gain coefficient, with units of (𝑊 -1 • 𝑚 -1 ), can be calculated via [10,15,26,27]: with 𝑆(𝛺) the Lorentzian spectrum of the Brillouin gain and 𝑆(𝛺 𝑚 ) = 1 is obtained with eigen acoustic frequency 𝛺 𝑚 . In Eq.

( Here, we consider the thermo-elastic loss (𝑄 𝑇𝐸 ), the thermoviscous loss from surrounding air (𝑄 𝑇𝑉 ), the viscoelastic loss (𝑄 𝑉𝐸 ) and the loss from the acoustic leakage to the lateral silicon slab (𝑄 𝐿 ). The quality factor is defined via 𝑄 𝑖 = 𝛺 𝑚 (𝛾 𝑖 ) ⁄ where 𝛾 𝑖=𝑇𝐸,𝑇𝑉,𝑉𝐸,𝐿 /2 is the decay rate of the mechanical mode. The mechanical quality factor changes nonlinearly with the length of the arms. For silicon nanostructures, the thermoelastic effect and the thermo-viscous effect typically yield quality factors of 𝑄 𝑇𝐸 ~60000 and 𝑄 𝑇𝑉 ~3000, respectively [28]. The viscoelastic loss of suspended silicon membrane is negligible due to the strain release through the liberation [29,30]. At atmospheric pressure, the maximum achievable mechanical quality factor is limited to 𝑄 𝑚 ~3000 by the thermo-viscous loss induced by the surrounding air. However, to achieve this optimum 𝑄 𝑚 , it is necessary to minimize mechanical leakage towards the lateral silicon slab. The 𝑄 𝐿 is mainly governed by the geometries of the waveguide and the nanoarms anchoring it to the lateral silicon slab. Note that we include 8-µm-wide slab sections at the two sides of the structure to void any reflection artefact that may result in an overestimation on the quality factor.

The phonon confinement within the waveguide core is mainly governed by the filling factor in the arms, described as ( 𝑤 𝑎 • 𝑡 𝑎 )/( 𝛬 • 𝑡 𝑤𝑔 ). The lower the filling factor, the higher the confinement and the lower the leakage towards the lateral silicon slab. Maximum period length is fixed by subwavelength condition, while minimum value is determined by achievable feature size. Thus, we set 𝑡 𝑎 = 50 nm, 𝑡 𝑤𝑔 = 220 nm, 𝑤 𝑤𝑔 = 500 nm and 𝛬 = 240 nm, 𝑤 𝑎 = 2.5 μm, as a compromise between high phonon confinement and achievable minimum feature size. In Fig. (3) we show the mechanical quality factor and the Brillouin gain, calculated as a function of the arm length (𝐿 𝑎 ). A remarkable 𝑄 𝑚 of up to 1200 can be obtained for an arm length of 25 nm. Here 𝑫 is the electric displacement field and 𝛿𝜀 = 𝜀 1 -𝜀 2 and 𝛿𝜀 -1 = 𝜀 1 -1 -𝜀 1 -2 describe the permittivity difference between core 𝜀 1 and surrounding 𝜀 2 . The 𝑻 𝑝𝑒 and 𝑻 𝑠 represent photoelastic tensor and strain tensor induced by the mechanical waves. In Equ.

(2), 𝑑𝐴 and 𝑑𝑉 indicates a surface integral and volume integral, respectively. As shown in Fig. 3, the Brillouin gain follows a similar trend as quality factor, with a sharp increase for arm lengths below 30 nm. For an arm length of 20 nm, the Brillouin gain reaches 10000 (𝑊 • 𝑚) -1 . For an arm length of 50 nm, compatible with electron-beam lithography, the Brillouin gain is 2000 (𝑊 • 𝑚) -1 . This gain value compares favorably with previously proposed silicon membranes with periodic silicon arms [19], while obviating the need for transversal Bragg phononic mirrors. Increasing the air gap in the nanoarms reduces Brillouin linewidth reaching 34 MHz (𝑄 𝑚 = 260) for a duty cycle of 20%.

The distribution of modulus of these two force densities 𝑓 𝑛𝑑,𝑖 = 𝑓 𝑑,𝑖

√𝑃𝑎𝑝𝑃𝑎𝑠•𝑢𝑚

where 𝑖 = 𝑝𝑒, 𝑚𝑏, are shown in the inset of Fig. 3 for an arm length of 𝐿 𝑎 = 50 nm. Both density distributions are taken at the cross-section I in Fig. 1. The 𝑓 𝑛𝑑,𝑝𝑒 presents a distribution similar to that of the optical mode shown in Fig. 1, since it is directly related to the vectorial product of the strain profile and the electric field 𝐸 𝑥 . Conversely, the 𝑓 𝑛𝑑,𝑚𝑏 exists at waveguide boundaries, associating with the tangential electric field components (𝐸 𝑧 in this case).

Taking the linear integral of 𝑓 𝑛𝑑,𝑝𝑒 along the 𝑥 direction, we find that it has values with same order of magnitude but opposite signs to 𝑓 𝑛𝑑,𝑚𝑏 . These two forces are driven along different direction, partially cancelling each other, reducing the overall Brillouin gain. This result is in agreement with previous studies on the interactions between electrostriction and radiation pressure in silicon waveguides [11,15]. The Brillouin gain could be further increased by achieving constructive interaction between the photoelastic and the moving-boundary effects, e.g. by geometry optimization or interaction between different optical modes [15,17].

We have as well studied the impact of the silicon thickness on the Brillouin gain and mechanical quality factor. In this case, we fixed the thickness of the nanoarms to 𝑡 𝑎 = 50 nm and the length to 𝐿 𝑎 = 50 nm (compatible with electron-beam lithography). As observed in Fig. 4 (a), both the gain and quality factor increase with a thicker Si layer, up to a thickness of 300 nm. For a silicon thickness of 290 nm, the Brillouin gain reaches 3000 (W • m) -1 . When the silicon thickness is larger than 300 nm, the maximum displacement shifts from the waveguide core to the nanoarms, substantially increasing the mechanical quality factor but is simultaneously responsible for a reduced photon-phonon overlap. It is a direct indication that, the figure of merit 𝐺/𝑄 drops sharply for waveguide thicknesses above 300 nm (inset). To illustrate this behavior, we plot the displacement profile of the mechanical mode for a silicon thickness of 𝑡 𝑤𝑔 = 280 nm, approximately maximizing the gain, and of 𝑡 𝑤𝑔 = 310 nm, in Figs. In summary, we have proposed and analyzed a new strategy to optimize Brillouin scattering in silicon membrane waveguides. The proposed waveguide geometry confines mechanical and optical modes by subwavelength softening and subwavelength index engineering, respectively, and allows to simultaneously enlarge the photon-phonon interaction and improves the quality factor of the mechanical mode. Our calculations show that the proposed optomechanical waveguide could provide a Brillouin gain exceeding 3000 (W • m) -1 for a minimum feature size of 50 nm, compatible with electron-beam lithography, and up to 10000 (W • m) -1 for minimum feature size of 20 nm. These results showcase the great potential of subwavelength engineering to shape mechanical and optical modes with silicon. We believe these results will expedite the use of subwavelength-lattice waveguides to engineer photon-phonon interaction in silicon.

Fig. 1 .

 1 Fig. 1. (a) Schematic of the proposed optomechanical silicon waveguide based on a subwavelength lattice. Optimized dimensions: 𝑡 𝑎 = 50nm, 𝑡 𝑤𝑔 = 220 nm, 𝑤 𝑤𝑔 = 500 nm and 𝛬 = 240 nm, 𝑤 𝑎 = 2.5 μm. (b) Normalized intensity of the electric field collected at plane I. (c) Normalized displacement profile of the mechanical mode we are interested in.As an example, the normalized intensity profile of the subwavelength optical mode at 195 THz (effective index is 2.4) and the displacement profile of the mechanical mode at 9.1 GHz are shown in Figs1 (b) and (c), respectively. The optical and mechanical mode profiles are collected at the I plane with 𝐿 𝑎 = 50 nm. Low optical loss down to 2 dB/cm using subwavelength gratings is achievable, reported in earlier works[22][23][24][25] and is not detailed here. The mechanical band diagram of the proposed waveguide is obtained using a Finite-Element-Method (FEM) model, considering the 1 st degenerate Brillouin zone, with the wavevector along the direction of light propagation (z). The waveguide supports more than 30 mechanical modes with frequencies varying between a few MHz and 15 GHz. As an illustration, Fig.2(a) shows the band structure of 8 modes of different kinds. The normalized displacement distributions of these mechanical modes (excited by

  (a) shows the band structure of 8 modes of different kinds. The normalized displacement distributions of these mechanical modes (excited by the same mechanical power) are plotted in Fig. 2 (b). We can distinguish modes with flexural displacement concentrated in the nanoarms (Fig. 2 b I), or in entire structure (Fig. 2 b II, III) and modes with hybrid flexural-torsional displacement (Fig. 2 b V-VIII). The optoacoustic overlap is evaluated using 𝜂 = 𝑉 ∫ 𝒖 • 𝑬 𝑝 * • 𝑬 𝑠 𝑑𝑉 𝑉 / [∫ |𝑢|𝑑𝑉 𝑉 • ∫ 𝑬 𝑝 * • 𝑬 𝑠 𝑑𝑉 𝑉 ], where 𝒖 is the vectorial distribution of the displacement (all the vectorial quantity is labelled in bold) over the entire waveguide. The overlap values are indicated in the figures. A few of these modes present a considerable absolute displacement. However, only the mode with mainly transverse displacement along x in the core and nanoarms (mode IV, shown in Fig. 2(b)) allows strong optomechanical coupling with the TE optical mode. The angular frequency of this mechanical mode is 𝛺 𝑚 ~2π • 9GHz. According to the energy conservation condition, the angular frequencies of the optical pump (𝜔 𝑝 ) and Stokes (𝜔 𝑠 ) are related by 𝛺 𝑚 = 𝜔 𝑝 -𝜔 𝑠 . Concurrently, the wavevector of the mechanical mode matching the momentum conservation condition can be calculated as 𝑞 = 𝑛 𝑔 (𝜔 𝑝 -𝜔 𝑠 ) 𝑐 ⁄ . The fundamental optical Bloch mode of the optomechanical waveguide has a group index of 𝑛 𝑔 ~4.6 near 1550 nm wavelength. This yields 𝑞 < 1000 (2π • m -1 ) [16] and limits the interacting mode at the band edge.

Fig. 2 .

 2 Fig. 2. (a) Band diagram of the mechanical modes of the proposed optomechanical waveguide with 𝑤 𝑎 = 50 nm. The mode indicated with a blue curve is used later for the Brillouin gain evaluation. (b) Normalized displacement profile of some selected modes, including flexural modes (I-III), longitudinally strained mode (IV), and flexuraltorsional hybrid mode (V-VIII).

  ,𝒎𝒃 (𝒛)𝒅𝑨 + ∫ 𝒇 𝒅,𝒑𝒆 (𝒛)𝒅𝑽| 𝟐

  ), 𝑃 𝑎,𝑖=𝑝,𝑠 = (∫ 𝑬 𝑖 × 𝑯 𝑖 * • 𝑧d𝐴)/2 is the time-average energy flux carried by the unnormalized optical pump (i=p) and Stokes (i=s) modes while 𝑢 𝑚 gives the maximum value of the displacement. The effective linear mass density of the mechanical mode is 𝑚 ̃𝑒𝑓𝑓 = 𝟏 𝜦 ∫ 𝜌 0 |𝑢/𝑢 𝑚 | 2 𝑑𝑉. The quality factor of the mechanical mode, 𝑄 𝑚 , is related to the full width at half maximum (FWHM) of the gain spectrum, 𝛾 𝑚 , through 𝑄 𝑚 = 𝛺 𝑚 /𝛾 𝑚 . It is limited by different loss mechanisms, described through their associated quality factors as:

Fig. 3 .

 3 Fig. 3. The quality factor (black) of the mechanical mode and the overall Brillouin gain (red) as a function of the length of silicon arms 𝐿 𝑎𝑟𝑚𝑠 . Upper panel (inset) is the modulus of normalized force density regarding to the photoelastic effect 𝑓 𝑛𝑑,𝑝𝑒 , while the lower one is the boundary motion-induced force density along the interface. Both density distribution is taken at the cross-section I that shown in Fig. 1. The Brillouin gain is calculated via Equ. (2), with force density associated with the moving boundary effect [10,18] 𝒇 𝑑,𝑚𝑏 = 𝒖 * (𝛿𝜀𝑬 𝑝,‖ * • 𝑬 𝑠,‖ -𝛿𝜀 -1 𝑫 𝑝,ꓕ * • 𝑫 𝑠,ꓕ )/4 and force density produced by the photoelastic effect 𝒇 𝑑,𝑝𝑒 = -𝜀 0 𝜀 1 2 𝑻 𝑝𝑒 • 𝑻 𝑠 • 𝑬 𝑝 * • 𝑬 𝑠 /4 [10].Here 𝑫 is the electric displacement field and 𝛿𝜀 = 𝜀 1 -𝜀 2 and 𝛿𝜀 -1 = 𝜀 1 -1 -𝜀 1 -2 describe the permittivity difference between core 𝜀 1 and surrounding 𝜀 2 . The 𝑻 𝑝𝑒 and 𝑻 𝑠 represent photoelastic tensor and strain tensor induced by the mechanical waves. In Equ.(2), 𝑑𝐴 and 𝑑𝑉 indicates a surface integral and volume integral, respectively. As shown in Fig.3, the Brillouin gain follows a similar trend as quality factor, with a sharp increase for arm lengths below 30 nm. For an arm length of 20 nm, the Brillouin gain reaches 10000 (𝑊 • 𝑚) -1 . For an arm length of 50 nm, compatible with electron-beam lithography, the Brillouin gain is 2000 (𝑊 • 𝑚) -1 . This gain value compares favorably with previously proposed silicon membranes with periodic silicon arms[19], while obviating the need for transversal Bragg phononic mirrors. Increasing the air gap in the nanoarms reduces Brillouin linewidth reaching 34 MHz (𝑄 𝑚 = 260) for a duty cycle of 20%.The distribution of modulus of these two force densities 𝑓 𝑛𝑑,𝑖 =

  4 (b) and 4(c), respectively.

Fig. 4 .

 4 Fig. 4. (a) The quality factor (black) of the mechanical mode and the overall Brillouin gain (red) as a function of the thickness of silicon rib 𝑡 𝑤𝑔 . The dimension of the nanoarms 𝑡 𝑎 and 𝐿 𝑎 are both fixed at 50 nm. Inset is gain over the mechanical 𝑄 with varying 𝑡 𝑤𝑔 . (b) and (c) are the normalized displacement profile of the mechanical mode with 𝑡 𝑤𝑔 = 280 nm and 𝑡 𝑤𝑔 = 310 nm, respectively.

Funding. Agence Nationale de la Recherche (ANR) (MIRSPEC ANR-17-CE09-0041, BRIGHT ANR-18-CE24-0023-01); European Research Council (ERC) (No. 715939, Nanophennec) Disclosures. The authors declare no conflicts of interest.


Coherent phonon generation by optical pump-probe experiments has enabled the study of acoustic properties at the nanoscale in planar heterostructures, plasmonic resonators, micropillars and nanowires. Focalizing both pump and probe on the same spot of the sample is a critical part of pumpprobe experiments. This is particularly relevant in the case of small objects. The main practical challenges for the actual implementation of this technique are: stability of the spatio-temporal overlap, reproducibility of the focalization and optical mode matching conditions. In this work, we solve these three challenges for the case of planar and micropillar optophononic cavities. We integrate the studied samples to single mode fibers lifting the need for focusing optics to excite and detect coherent acoustic phonons. The resulting excellent reflectivity contrast of at least 66% achieved in our samples allows us to observe stable coherent phonon signals over at least a full day and signals at extremely low excitation powers of 1µW. The monolithic sample structure is transportable and could provide a means to perform reproducible plug-and-play experiments.

Fiber-integrated microcavities for efficient generation of coherent acoustic phonons

O. Ortiz 1* , F. Pastier 2* , A. Rodriguez 1 , Priya 1 , A. Lemaitre 1 , C. Gomez-Carbonell 1 , I. Sagnes 1 , A. Harouri 1 , P. Senellart 1 , V. Giesz 2 , M. Esmann 1~, N.D. Lanzillotti-Kimura 1+

Coherent phonon generation by optical pump-probe experiments [START_REF] Thomsen | Surface generation and detection of phonons by picosecond light pulses[END_REF][START_REF] Thomsen | Coherent Phonon Generation and Detection by Picosecond Light Pulses[END_REF][START_REF] Ruello | Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action[END_REF] has enabled the study of acoustic properties at the nanoscale in planar heterostructures [START_REF] Fainstein | Strong Optical-Mechanical Coupling in a Vertical GaAs/AlAs Microcavity for Subterahertz Phonons and Near-Infrared Light[END_REF][START_REF] Beardsley | Coherent Terahertz Sound Amplification and Spectral Line Narrowing in a Stark Ladder Superlattice[END_REF][START_REF] Czerniuk | Impact of nanomechanical resonances on lasing from electrically pumped quantum dot micropillars[END_REF][START_REF] Jäger | Resonant driving of magnetization precession in a ferromagnetic layer by coherent monochromatic phonons[END_REF][START_REF] Brüggemann | Laser mode feeding by shaking quantum dots in a planar microcavity[END_REF][START_REF] Matsuda | Acoustic phonon generation and detection in GaAs∕Al0.3Ga0.7As quantum wells with picosecond laser pulses[END_REF][START_REF] Sun | Coherent Acoustic Phonon Oscillations in Semiconductor Multiple Quantum Wells with Piezoelectric Fields[END_REF][START_REF] Bartels | Coherent Zone-Folded Longitudinal Acoustic Phonons in Semiconductor Superlattices: Excitation and Detection[END_REF][START_REF] Arregui | Coherent generation and detection of acoustic phonons in topological nanocavities[END_REF], plasmonic resonators [START_REF] Picca | Tailored Hypersound Generation in Single Plasmonic Nanoantennas[END_REF][START_REF] O'brien | Ultrafast acousto-plasmonic control and sensing in complex nanostructures[END_REF][START_REF] Medeghini | Controlling the Quality Factor of a Single Acoustic Nanoresonator by Tuning its Morphology[END_REF][START_REF] Xu | All-optical in-depth detection of the acoustic wave emitted by a single gold nanorod[END_REF][START_REF] Kelf | Ultrafast Vibrations of Gold Nanorings[END_REF], cells [START_REF] Viel | Picosecond ultrasounds as elasticity probes in neuron-like cells models[END_REF], micropillars [START_REF] Czerniuk | Impact of nanomechanical resonances on lasing from electrically pumped quantum dot micropillars[END_REF][START_REF] Anguiano | Micropillar Resonators for Optomechanics in the Extremely High 19-95-GHz Frequency Range[END_REF][START_REF] Anguiano | Scaling rules in optomechanical semiconductor micropillars[END_REF][START_REF] Lagoin | High spectral resolution of GaAs/AlAs phononic cavities by subharmonic resonant pump-probe excitation[END_REF] and nanowires [START_REF] Belliard | Vibrational response of free standing single copper nanowire through transient reflectivity microscopy[END_REF].

All these experiments rely on the optical mode matching between the incident pump and probe laser fields and the optical modes of the structure under study. The efficient generation of coherent acoustic phonons relies on an efficient coupling of the pump field into the system, while the sensitive detection of phonons requires an efficient coupling of the probe to the optical mode undergoing a phononinduced modulation. The main practical challenges for the actual implementation of this technique are therefore: 1) stability 2) reproducibility and 3) high power densities limiting the range of compatible samples [START_REF] Bartels | Ultrafast time-domain spectroscopy based on high-speed asynchronous optical sampling[END_REF].

The implementation of these experiments, also known as time domain Brillouin scattering [START_REF] Gusev | Advances in applications of time-domain Brillouin scattering for nanoscale imaging[END_REF], usually requires a long mechanical delay line. In terms of stability, this implies issues due to changes in beam pointing, spot size and laser power over the scan of the delay line. The diversity of possible experimental configurations makes identical measurements in different laboratories virtually impossible. Even more, successive realignments of a single setup might result in slightly different excitation and collection mode matching. These shortcomings have so far been a roadblock in establishing pump-probe as a quantitative spectroscopy tool for nanoacoustics.

A partial solution mainly addressing stability considerations is provided by the asynchronous optical sampling (ASOPS) technique by removing the need for a mechanical delay line [START_REF] Bartels | Ultrafast time-domain spectroscopy based on high-speed asynchronous optical sampling[END_REF]. However, alignment and power requirements are still inherent of the experiment. In this work, we integrate fibered systems to eliminate any optical alignment on the sample, thus simultaneously solving the three aforementioned main problems. We glued single mode fibers to a planar optophononic microcavity and to an optophononic micropillar [START_REF] Czerniuk | Impact of nanomechanical resonances on lasing from electrically pumped quantum dot micropillars[END_REF][START_REF] Anguiano | Micropillar Resonators for Optomechanics in the Extremely High 19-95-GHz Frequency Range[END_REF][START_REF] Anguiano | Scaling rules in optomechanical semiconductor micropillars[END_REF][START_REF] Lagoin | High spectral resolution of GaAs/AlAs phononic cavities by subharmonic resonant pump-probe excitation[END_REF][START_REF] Lamberti | Optomechanical properties of GaAs/AlAs micropillar resonators operating in the 18 GHz range[END_REF][START_REF] Esmann | Brillouin scattering in hybrid optophononic Bragg micropillar resonators at 300 GHz[END_REF][START_REF] Kargar | Direct observation of confined acoustic phonon polarization branches in freestanding semiconductor nanowires[END_REF][START_REF] Garcia-Sanchez | Acoustic confinement in superlattice cavities[END_REF]. The integration with fibers might establish the missing link between high frequency acoustic phonon engineering [START_REF] Volz | Nanophononics: state of the art and perspectives[END_REF][START_REF] Ortíz | Phonon engineering with superlattices: Generalized nanomechanical potentials[END_REF][START_REF] Balandin | Nanophononics: Phonon Engineering in Nanostructures and Nanodevices[END_REF][START_REF] Beardsley | Coherent Terahertz Sound Amplification and Spectral Line Narrowing in a Stark Ladder Superlattice[END_REF][START_REF] Akimov | Ultrafast Band-Gap Shift Induced by a Strain Pulse in Semiconductor Heterostructures[END_REF] and stimulated Brillouin scattering in structured optical fibers [START_REF] Eggleton | Grating resonances in air-silica microstructured optical fibers[END_REF][START_REF] Godet | Brillouin spectroscopy of optical microfibers and nanofibers[END_REF][START_REF] Merklein | A chip-integrated coherent photonic-phononic memory[END_REF][START_REF] Stern | Tunable sharp and highly selective microwave-photonic band-pass filters based on stimulated Brillouin scattering[END_REF][START_REF] Pant | On-chip stimulated Brillouin scattering[END_REF]. Figure 1a shows a field of micropillars obtained by electron beam lithography and inductively coupled plasma etching of a planar microcavity. The planar cavity is formed by two (λ/4 Ga90Al10As, λ/4Ga5Al95As) distributed Bragg reflectors (DBRs) enclosing a λ/2 spacer of GaAs. 19 [START_REF] Bartels | Ultrafast time-domain spectroscopy based on high-speed asynchronous optical sampling[END_REF] DBR layer pairs on the air (GaAs substrate) side result in a balanced optical cavity. We grew the sample by molecular beam epitaxy with a thickness gradient across the wafer, which implies a spatially dependent optical resonance. Figure 1b presents a single circular micropillar of 2.8 m diameter. The inset shows a close-up of the central pillar section. The DBR structures and the cavity spacer are clearly visible. This micropillar presents an optical mode at 912.1 nm evidenced as a reflectivity dip (black line in Fig. 1c) whereas the planar microcavity has an optical resonance at 917.6 nm (dashed blue line in Fig. 1c). This wavelength difference is due to the spatial gradient across the wafer. For the individual micropillar, we spatially overlap the core of a single mode optical fiber (Thorlabs 780HP, core diameter 4.4 m) with its top surface [START_REF] Cadeddu | A fiber-coupled quantum-dot on a photonic tip[END_REF][START_REF] Haupt | Fiber-connectorized micropillar cavities[END_REF][START_REF] Schlehahn | A stand-alone fiber-coupled single-photon source[END_REF][START_REF] Snijders | Fiber-Coupled Cavity-QED Source of Identical Single Photons[END_REF]. To this end, we monitor the micropillar reflectivity dip through the fiber while adjusting the sample chip position on a 3D piezo motor with submicron precision. In contrast, for the planar cavity a precise lateral alignment of the fiber core is not necessary. Once spatial alignment is completed, we permanently glue the fiber to the sample chip (UV-curing glue Norland NOA 81). During the curing process of the glue the optical mode frequency undergoes only minimal shifts (<0.1%). Figure 1d presents a picture of a finished device. A ceramic ferrule (white tube) provides additional mechanical stability. The mode overlap of the fiber to the microresonators achieved during the gluing is permanent and stable. No further alignment steps are required.

Our measurement setup for the integrated samples is based on a fiber beam splitter (see schematics in Fig. 2a). To evaluate the loss budget in the system, we identify three main contributions: The fiber beam splitter itself, the fiber coupler (FC), and the fiber-to-sample coupling. We define the coupling efficiency as the ratio of collected power in arm 3 of the beam splitter to the input power injected from arm 1. Based on standard values for the beam splitter transmission (50%) and FC transmission (80%), we obtain a theoretical maximum coupling efficiency of 16% for the planar cavity at a wavelength outside the cavity mode. Note that this coupling efficiency takes into account both input and output coupling. The measured coupling efficiency is 12.7% with the main potential source of additional losses being the fiber-to-sample coupling (distance and angle). In the case of the micropillar an additional loss is the geometrical overlap between the fiber core and the micropillar top surface (40%). We obtain a theoretical maximum coupling efficiency of 2.5% for a wavelength outside the cavity mode. We measure a coupling efficiency of 0.8%, again mainly determined by additional potential misalignment of distance and angle between pillar and fiber core.

The reflectivity contrast for the optical mode is the additional parameter which determines the sensitivity to detect acoustic phonons, since the detected magnitude is usually the transient reflectivity change Δ𝑅/𝑅. In the present cases, the reflectivity contrast is 66% for the micropillar and 80% for the planar cavity.

(d) and (e) present corresponding Fourier spectra at selected low and high pump powers for the micropillar and planar cavity, respectively. Factors on the left of each plot indicate vertical magnification with respect to the top curve. Note that the lowest pump power used was 1W for the planar cavity.

To investigate the fibered devices, we have implemented the degenerated pump-probe setup sketched in Figure 2a. A Ti:Sapphire laser (Spectra Physics Tsunami) delivers pulses of 3.8 ps full width at half maximum (FWHM) with 80 MHz repetition rate and at a central wavelength of 912.08 nm for the micropillar (917.58 nm for the planar resonator). We divide the beam into cross-polarized pump and probe beams. A mechanical delay line (3.6 m maximum delay) delays the probe with respect to the pump. We use an acousto-optical crystal to modulate the pump (800 kHz modulation frequency, 50% duty cycle) enabling synchronous detection of pump-induced changes in the probe signal. Both beams recombine and enter arm 1 of a fiber beam splitter, while arm 2 connects to the sample. The same arm 2 collects reflected pump and probe signals and sends them to a photodetector through arm 3. A power meter at arm 4 of the beam splitter monitors input powers. Since pump and probe are equal in wavelength and are contained in the same spatial mode of the fiber, the synchronous detection critically relies on polarization filtering of the pump probe signals. We compensate any rotation of polarization in the fibers by means of a combination of a quarter-and half-wave plate. A linear polarizer then filters out the pump before the signal reaches the photodetector. To minimize random fluctuations of polarization rotation within the fibers, we mechanically stabilize all fibers in the system.

Figures 2b andc display typical pump-probe differential reflectivity time traces measured in fibercoupled planar and micropillar resonators. Pump powers of 500 W and probe powers of 700 W were measured in arm 4 of the fiber beam splitter. At zero delay, we observe a sharp pump-induced reflectivity change followed by a slow decay of the signal. While this decay happens on a time scale of 4ns for the planar resonator the decay only takes approx. 0.5 ns for the micropillar. This difference might be attributed to faster recombination of electronic excitations at surface defects [START_REF] Anguiano | Optical cavity mode dynamics and coherent phonon generation in high-Q micropillar resonators[END_REF][START_REF] Hamoumi | Microscopic Nanomechanical Dissipation in Gallium Arsenide Resonators[END_REF]. The coherent phonon-induced oscillations superimposed with this electronic response are not evident to the naked eye. Only by Fourier transforming the time traces, we obtain the coherent acoustic phonon spectra in Figure 2d and e. We observe a main peak around 18.43 GHz (18.22 GHz) corresponding to the fundamental acoustic cavity mode of the micropillar (planar resonator) and additional peaks at 58 GHz corresponding to their third harmonic. The acquisition time for each of these measurements was just five minutes for the micropillar as well as for the planar cavity. In free space, similar experiments using a mechanical delay line usually take on the order of one hour and higher laser powers.

Our fiber-coupled samples provide inherently stable mode overlap with no further alignment involved.

The measurements thus only depend on the characteristics of the laser and the stability of the external fiber coupling. To assess this stability, we performed long-term pump-probe measurements over the course of 42h only adjusting the incident power (measured in arm 4 of the fiber beam splitter) once after 24h to compensate for drift of the laser. Figure 3a presents the amplitude of the differential reflectivity at zero delay (blue) and the spectral peak power of the mode at 18.43 GHz (red) over time.

Both are normalized to the first datapoint. The curves show variations of 15% dominated by a 24h temperature cycle in the laboratory. As long as the coupled laser characteristics remain constant, the resulting pump-probe traces are the same. This means, that by transferring the integrated sample exactly the same pump-probe measurement could be performed in any other experimental pumpprobe setup.

Figure 3b andc show intensity maps of pump-probe traces as a function of pump power for micropillar (b) and planar (c) resonators. In these experiments, we kept the incident probe power constant at 700µW. Figures 3d ande present corresponding Fourier spectra at selected low and high pump powers. Note that for the case of the planar cavity we observe phonon signals down to 1µW of pump power (100 µW for the pillar). The essential features of the pump-probe spectra remain unchanged with pump power. We measured these powers in arm 4 of the fiber beam splitter, equivalent to the power injected into the FC coupler at arm 2. Similar phonon dynamics have been reported before, yet for much higher pump powers in the range of 5mW [START_REF] Anguiano | Scaling rules in optomechanical semiconductor micropillars[END_REF].

The observation of complex optical mode dynamics is mainly related to the efficient pump injection whereas the observation of coherent phonons at extremely low pump powers indicates an efficient probe coupling. This implies that our mode matching for both pump and probe enables a new generation of pump-probe experiments where low powers are required.

The impulsive generation of coherent acoustic phonons has a broad of applications spanning from nondestructive testing to the study of quantum optomechanical systems. Experimental challenges typically include mechanical stability and reproducing equivalent experimental conditions over time and between setups. Furthermore, high power densities are typically needed limiting the range of accessible samples. We proposed a solution to these three challenges consisting of integrating the studied sample in a fibered system. We showed that gluing optophononic semiconductor cavities to a single mode fiber lifts the need for focusing optics to excite and detect coherent acoustic phonons. The mode matching achieved in our samples allowed us to observe stable signals over at least a full day and coherent phonon signals at extremely low excitation powers of 1W. The monolithic sample structure is transportable making it a potential means to perform reproducible plug-and-play pumpprobe experiments in individual microstructures.

Our results could enable a new generation of nanophononic experiments in systems such as planar heterostructures (quantum wells, multilayers, lasers, quantum dots, magnetic thin films [START_REF] Jäger | Resonant driving of magnetization precession in a ferromagnetic layer by coherent monochromatic phonons[END_REF], piezoelectric materials, 2D materials), organic materials (not resistant to high powers usually needed in coherent phonon generation experiments) single cells and metallic nanostructures (antennas, metamaterials, metasurfaces). In our case, we distinguished pump and probe signals based on polarization. However, the proposed technique would also allow two-color experiments as a straightforward extension.

Figure 1

 1 Figure 1 Integrated fibered system for pump probe experiments. (a) SEM image of a field of GaAs/AlAs micropillars presenting different shapes and sizes. (b) SEM image of an individual micropillar cavity of 2.8 m diameter. The inset shows a detail of the DBRs (green) and the central GaAs spacer (orange). (c) Optical reflectivity spectrum of the fibered micropillar after we permanently glued it to the core of a single mode fiber (solid black). The fundamental optical mode appears as a dip at 912.1 nm. The spectrum of the connected planar cavity (dashed blue) shows a fundamental optical mode at 917.6 nm (d) Final device with one individual micropillar on the chip coupled to the single mode fiber. A ceramic ferrule provides additional mechanical stability. Inset: microscope image of fiber core and cladding.

Figure 2 :

 2 Figure 2 : (a) Schematic of the pump-probe setup used to characterize the optophononic response of fiber-coupled microcavities. We replaced the standard focalization optics by a fibered system. (b,c) Differential reflectivity time traces of the micropillar and planar resonators. (d,e) Fast Fourier transform (FFT) of the experimental time traces featuring the acoustic resonance at 18.43 GHz (18.22 GHz) for the micropillar (planar resonator) and its third harmonic. Powers were 500W for the pump and 700W for the probe.

Figure 3 :

 3 Figure 3 : (a) Stability analysis of the fibered micropillar resonator over 42h; amplitude of the time trace at zero delay (blue) and spectral peak power of the acoustic cavity mode at 18.43 GHz (red). (b) Pump-probe traces as a function of pump power for the fibered micropillar cavity and (c) for the planar resonator. The results shown in the color maps are the interpolation of 19 and 25 individual measurements for the micropillar and the planar cavity, respectively. We kept the incident probe power constant at 700W. (d) and (e) present corresponding Fourier spectra at selected low and high pump powers for the micropillar and planar cavity, respectively. Factors on the left of each plot indicate vertical magnification with respect to the top curve. Note that the lowest pump power used was 1W for the planar cavity.

Acknowledgements:

The authors thank P. Voisin for enlightening discussions. The authors acknowledge funding by the European Research Council Starting Grant No. 715939, Nanophennec. This work was supported by the European Commission in the form of the H2020 FET Proactive project TOCHA (No. 824140). The authors acknowledge funding by the French RENATECH network and through a public grant overseen by the ANR as part of the "Investissements d'Avenir" program (Labex NanoSaclay Grant No. ANR-10-LABX-0035). M.E. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Project 401390650. F. P. and V. G. acknowledge support from the IAD -ANR program (Project ANR-18-ASTR-0024 LIGHT).