Yosri Haddad

Jean-Charles Beugnot

Samuel Margueron

Gil Fanjoux

Distributed spectral measurement of nonlinear Raman cascade process along an optical nanofiber

Keywords: Raman cascade process, Rayleigh scattering, optical nanofiber, spectrometer

We present in this work the longitudinal characterization of the spectral evolution of a Raman cascading process along an optical nanofiber with a millimetric length and sub-micrometric size. The measurements are performed by analyzing with a confocal micro-spectrometer the light radiated by Rayleigh scattering out the nanofiber. This observation is performed with a high spatial resolution given by a confocal microscope, and a high spectral resolution given by the spectrometer.

INTRODUCTION

Simultaneous spatial and spectral measurements of the longitudinal evolution of the guided light all along an optical waveguide operating in the nominal regime is of great interest in the field of optical metrology. Indeed, this information is first useful to validate or improve the numerical and theoretical predictions. In addition, this information is crucial for waveguide engineering because of the direct access of the local losses, defects and modal behavior for example. Specifically in nonlinear regime, their operation under such conditions, as well as their fabrication methods, induce complex losses and nonlinear behaviors that remain difficult to understand and to integrate in a complete approach. Therefore, an experimental investigation of distributed measurements along waveguides is crucial for the development of micro-and nano-photonics waveguides. Conjugated spectral and spatial resolutions would allow the analysis of the evolution of spectral components generated along waveguides by nonlinear effects such as stimulated Raman scattering, four wave mixing, second harmonic generation, etc. These distributed measurements will complement the current characterization methods of optical waveguides focused either on the output or input side of the waveguide in order to analyze the transmitted or backscattered field in power, polarization, or spectrally. They will also complement other distributed measurement along the waveguide based on reflectometry or cutback techniques for example, which provide, however, non-spectrally resolved measurements or destructive method.

In this context, a spectroscopic technique is developed in this work in order to characterize spatially and spectrally the evolution of the guided light along an optical waveguide in nominal operation. This experimental technique is based on the detection of the right-angle radiated field by Rayleigh scattering (RS) process outside the waveguide by using a confocal micro-spectrometer. This system allows the observation of the longitudinal evolution of the radiated field along the waveguide optical axis. The method presents the advantage of being non-destructive, non-invasive, and to present simultaneously a micrometric spatial resolution and a high spectral resolution. Moreover, this technique can be used in linear and nonlinear regime, with continuous or pulsed laser sources, at different pump laser wavelengths, and with different geometry of waveguides.

To confirm the large flexibility of our experimental technique, the present work concerns the spectral characterization of a Raman cascading process along a sub-micrometric diameter tapered fiber (called nanofiber thereafter), homogeneous along several centimeters, and pumped in the visible wavelength range and in the sub-nanosecond regime. Optical nanofiber (ONF) present a large interest in a wide number of applications due to their specific properties, i.e., the high intensity of the confined field, and a strong evanescent field in interaction with the surrounding medium. Then, applications in both linear and nonlinear regimes go from atom trapping in the evanescent field [START_REF] Le Kien | Spontaneous emission of a cesium atom near a nanofiber: Efficient coupling of light to guided modes[END_REF][START_REF] Morrissey | Spectroscopy, manipulation and trapping of neutral atoms, molecules, and other particles using optical nanofibers: a review[END_REF] to supercontinuum generation [START_REF] Birks | Supercontinuum generation in tapered fibers[END_REF][START_REF] Ying | Supercontinuum generation based on nanofiber[END_REF], efficient evanescent Raman scattering [START_REF] Shan | Stimulated raman scattering in the evanescent field of liquid immersed tapered nanofibers[END_REF], evanescent Kerr effect [START_REF] Fanjoux | Demonstration of the evanescent kerr effect in optical nanofibers[END_REF] and provides advances in quantum information technologies [START_REF] O'shea | Fiber-optical switch controlled by a single atom[END_REF][START_REF] Petersen | Chiral nanophotonic waveguide interface based on spin-orbit interaction of light[END_REF]. Therefore, optical characterization along an ONF is highly useful for different input power and wavelength, which is not possible by any current characterization systems. The generation of the Raman cascading process in ONF is clearly visible with an exponential increase of the Stokes and anti-Stokes spectral components as excepted. Furthermore, the longitudinal evolution of the intensities of the first three Raman Stokes orders is in good agreement with *gil.fanjoux@univ-fcomte.fr; phone +33 3 81 66 64 26; https://www.femto-st.fr/ the theory, showing that our technique can provide quantitative data without any numerical post-processing. It is noteworthy that this method can be extended to others waveguides whatever its geometry which represents a strong interest for deepen optical characterization of photonics waveguides, or for other optical regimes characterized by spectral evolution of the field propagating along an optical waveguide in nominal operation.

EXPERIMENTAL SETUP

The experimental setup is illustrated in Fig. 1(a). It consists of a confocal Raman micro-spectrometer (Monovista, S-&-I GmbH) centered on the Rayleigh scattering (RS) detection (obtained by removing the Notch filter of the spectrometer). The system is equipped with a high-precision motorized stage for longitudinal and transverse displacement of the waveguide under study and a microscope objective translation , all with a step size and reproducibility better than <100 nm. This system allows the generation of an accurate recordable 3D spatial trajectory point-by-point all along the waveguide under test. Therefore, after having recorded the trajectory corresponding to the waveguide, the system allows us to measure a 1 -RS trace along the waveguide axis or 1 --RS trace perpendicular to the waveguide, or a 2 ---maps in the or planes containing the waveguide or in the plane perpendicular to the waveguide. The RS of the guided light in the optical waveguide medium, which exits to the outside, is collected in the right angle to the propagation direction (Fig. 1(a)) with a x10 microscope objective and recorded through the spectrometer. The monochromator is equipped with three diffraction gratings (300, 1500 and 2400 lines/mm), and a back-illuminated cooled CCD detector (-85 C°) leading to a low detection threshold and a high signal-to-noise ratio (SNR). Alternatively, an imaging system composed of a CCD camera can image the waveguide for positioning the waveguide at the optical axis of the RS detection system, and also for rough fiber diameter measurements. Even if the method can be used for any type of optical waveguide, this work focuses on ONF. Note that ONF study in this work represents a challenge because this waveguide is not perfectly straight and the 3D trajectory have to be done accurately to follow very precisely the transitions and ONF regions. The ONF studied in our work corresponds to a standard single-mode fiber at telecom wavelength (SMF-G652D) obtained by the heat-and-pull method [START_REF] Tong | Subwavelength and nanometer diameter optical fibers[END_REF][START_REF] Tong | Subwavelengthdiameter silica wires for low-loss optical wave guiding[END_REF] up to obtain a fiber portion with a diameter of 690 nm accurately measured by Brillouin scattering method and homogeneous up to two centimeters in length [START_REF] Godet | Brillouin spectroscopy of optical microfibers and nanofibers[END_REF]. The optical losses were measured just after manufacturing between -1.5 and -4 dB at 532 nm of wavelength, strongly depending on the injection of the laser beam in this multi-mode fiber. Therefore, this high level of losses is certainly due to the modal filtering by the taper part of the waveguide, which is not adiabatic for the all injected modes. Just after manufacturing, the ONF is placed inside closed transparent plexiglass boxes in order to protect it from external air vibrations and to reduce any kind of contamination by dust for example [START_REF] Fujiwara | Optical transmittance degradation in tapered fibers[END_REF].

Measurements were performed following two steps. First, we have evaluated the 3D-XYZ spatial trajectory of the whole taper in the linear regime in order to precisely locate the position of the ONF in the taper fiber. Then we have investigated the longitudinal evolution of the Raman cascade process in the nonlinear regime. For the linear regime, we use a power stabilized cw laser diode with a wavelength of 653 nm. The input laser power is typically between 2 and 3 mW in order to limit the thermal effects or to avoid the ONF breaking. The laser power is simultaneously monitored in the transmitted power during the measurements (see Fig. 1(a)) without significant variation of the output power. The 3D-XYZ spatial trajectory, which follows the waveguide, is made in two steps. We first roughly locate the optical taper with the CCD imaging system for the , and coordinates, and thereafter we optimize the scattering signal for each position by adjusting precisely the and positions with the minimal entrance slit width in order to have the highest spatial resolution, typically 5 µm. Then, we record several positions (X,Y,Z), point by point, along the waveguide with a step size of about 10 mm. The second step corresponds to the interpolation between these points in order to obtain a trajectory with the desired number of points. Furthermore, the exposure time is in general case equal to about 0.5 s per point. Once we identify the exact position of the ONF part of the tapered fiber, we pump the fiber with the pulsed laser source in order to investigate the longitudinal spectral evolution of the light mode in the nonlinear regime of propagation. For that purpose, the fiber is pumped by a laser source corresponds to a doubled-frequency microchip Nd:YAG laser, with a repetition rate of 21 kHz and 500 ps of pulse duration. The laser beam is first injected in a small core fiber (SM600), connected to the tapered fiber. This system allows the control of the injection in the SMF fiber core preferentially in the fundamental mode of the nonstretched part of the fiber taper. The spectrum at the fiber output is measured by an OSA (optical spectrum analyzer) every 7 seconds simultaneously with the Rayleigh scattered signal measurements in order to be sure that the Raman cascading process is stable during all the 1D-scan-X over a duration of approximately one hour. Concerning the Raman spectrometer, three Notch filters were used in order to only remove the pump Rayleigh component. The entrance slit was equal to 1 mm, and 50 points were performed, i.e., one point every 400 m.

In terms of data acquisition and treatment, the RS data were recorded in the same experimental conditions. It is noteworthy that no normalization or correction of intensity point by point has been applied for the different 1D-X scans and the raw data are directly given in this paper.

EXPERIMENTAL RESULTS

1.1 Linear regime Figure 2(a) shows the 1D-X scattering trace obtained for the tapered fiber with the cw laser source at = 653 nm of wavelength with our technique (red curve), and compared to the trace obtained with an optical backscattered reflectometer (OBR) working at = 1566 for the same fiber (OBR4600 form Luna -blue curve). The fiber is thus in single-mode regime for the OBR trace and multi-mode regime for the RS trace. The scattering profiles are similar with a general point of view, which proves that our system is relevant. These scattering profiles are indeed due to the characteristics of the different scattering sources, corresponding to the fiber core at the beginning of the transition where light is still guided into the core of the fiber, the silica cladding when the core is no longer guiding in the tapered region, and the ONF surface for the small diameter when the surface scattering takes over the cladding scattering [START_REF] Haddad | Microscopic imaging along tapered optical fibers by right-angle Rayleigh light scattering in linear and nonlinear regime[END_REF][START_REF] Lai | Fiber taper characterization by optical backscattering reflectometry[END_REF]. As the ONF has a homogenous diameter along about 20 mm length, the scattering level remains constant as observed in the 1D-X trace. The two successive decreases of the scattering level at the beginning of the transition 1 and at the end of the transition 2 (arrow A), originate, respectively, from the expansion of different optical modes from the fiber core region into the cladding region, and from the back coupling of these modes from the cladding region into the core at the same fiber diameters at the level of the transition 2. These two characteristics are not visible in the OBR trace (blue curve) because the fiber is injected preferentially in the fundamental mode and therefore the fiber is single-mode. On the other hand, for low diameters in the transition 1, the RS signal shows a large increase of the scattering level (arrow B), contrary to the RS signal in transition 2, and contrary to the OBR trace. This scattering feature should come from the scattering of different radiation modes, which are no more supported by the transition 1 when reaching the cut-off diameter. To confirm that assumption, Fig. 2(b) represents a 2D map of the scattered intensity along the transition 1, which highlights the scattering from the surface of the fiber along the input transition level. We clearly observe the scattering process at the surface of the transition 1 by the radiation HOMs, which are no more guided by the clad of the fiber. This means that the transition is not adiabatic for all the optical modes initially propagating inside the core region of the unstretched part of the SMF-G652D fiber. The fact that this scattering in not visible in transition 2 confirms this observation.

Therefore, from this 1D-X trace in the linear regime (Fig. 2(a)), it is clear that the ONF region is exactly located between, X = 50 mm and X = 70 mm. Now, by using this information, we will present in the following paragrapth the results obtained for the Raman cascading process evolution along this targeted region of the tapered fiber in the nonlinear regime by using the sub-picosecond laser source.

Nonlinear regime: Raman cascading process along the nanofiber

In the nonlinear regime of the light propagation, all the difficulty corresponds to the detection of a weak scattering signal. Indeed, the system have to detect the Raman spectral components, which have lower intensities than the pump source, scattered by the Rayleigh scattering process originated from the refractive index fluctuations within the fiber. Second, as the laser source is pulsed in order to reach the nonlinear regime of the light propagation, the number of scattered photons decreases once again as opposed to the linear regime where the light source is a cw source and the scattering level is high. Therefore, the intensity of the scattered signal outside the fiber is extremely weak in this case. However, to overcome these limitations, the detection was made possible by the low level of detection of the cooled CCD detector of the spectrometer, and by a long exposure time of the spectrometer detector, typically 60 s per point. In addition, the large width of the entrance slit of the spectrometer (1 mm) used for this measurements increases the number of the detected Rayleigh photons. Moreover, the 1D-scan-X has been performed only for the ONF part (between X=50-70 mm), i.e., with the homogeneous diameter, where the strength of the scattering process is maximal and constant due to the large difference in the refractive index of air and dielectric [START_REF] Haddad | Microscopic imaging along tapered optical fibers by right-angle Rayleigh light scattering in linear and nonlinear regime[END_REF]. This ensures a scattered recorded signal with a high intensity and a stable SNR in order to observe the real dynamics of the Raman amplifications of the all generated spectral components along the ONF region. The spectrum is clearly stable over one hour without significant variations. Figure 3(b) shows that the first (blue curve) and the last (red curve) spectrum are almost similar. In addition, the spectrum in dashed green curve represents the spectrum at the output of the SM600 fiber. As the core diameter of the standard fiber SMF-G652D is greater than for the SM600, then the field spectrum should not strongly evolve up to the beginning of the first transition of the ONF, because of the lower intensity than in the SM600 fiber. , 0.35 -1 for the second Raman Stokes order, and 0.53 -1 for the third Raman Stokes order. The theory predicts that the slope is multiplied by a factor of two between the first and the second Raman Stokes order, and by a factor of three between the first and the third Raman Stokes orders. The experimental measurements thus show that the evolution of the spectra are in good quantitative agreement with the theory [START_REF] Boyd | Nonlinear Optics[END_REF].

CONCLUSION

The experimental results prove that the experimental technique based on right-angle Rayleigh scattering measurements is a powerful tool to characterize the conjugated spatial and spectral evolution of the field propagating along a photonic waveguide. In linear regime, this technique allows the identification of optical losses, mode behavior along a variable diameter fiber, mode beating and local defects with high reproducibility. In nonlinear regime, we have observed for the first time to our knowledge the Raman cascading process all along an optical nanofiber with quantitative evolution of the different spectral components generated during the propagation. The perspective of the technique are numerous. The technique could be extended to study other waveguides with complex geometry such as ridge, spiral, or optofluidic guides for instance. The intensity and spatial sensitivity of Rayleigh scattering can also be used to detect local scattering process induced in a Bragg grating fiber, local doping or local temperature evolution. Additionally, other nonlinear processes along a waveguide such as second harmonic generation along ridge waveguides, generation of the hyper-Rayleigh scattering process, or the development of the Raman scattering process into a supercontinuum generation along nonlinear waveguides can be studied.

Figure 1 .

 1 Figure 1. Experimental setup: (a) Schematic experimental setup (b) geometrical data of the ONF.

Figure 2 .

 2 Figure 2. (a) 1D-X Rayleigh scattering trace obtained fiber with the cw laser source at a wavelength of = 653 nm (red curve) and OBR trace obtained at = 1566 (blue curve). The black arrows: (A) show two successive decreases of the scattering level at the level of the transition 1 and 2, and (B) shows the blurred scattering level along the first transition region due to the filtering process by the transition of the lossed modes. Blue curve corresponds to a symmetrical scattering profile and is only intended to show a trend. (b) 2D scan for the input transition of the fiber. White dashed line corresponds to the fiber shape.

Figure 3 .

 3 Figure 3. (a) Evolution of the spectrum at the output of the fiber recorded by an OSA. (b) The first (blue curve) and the last (red curve) OSA spectrum. The green spectrum corresponds to the spectrum at the output of the SM600 fiber.

Figure 3 (

 3 Figure 3(a) shows the evolution of the output spectrum of the whole fiber recorded by an optical spectrum analyzer (OSA).The spectrum is clearly stable over one hour without significant variations. Figure3(b)shows that the first (blue curve) and the last (red curve) spectrum are almost similar. In addition, the spectrum in dashed green curve represents the spectrum at the output of the SM600 fiber. As the core diameter of the standard fiber SMF-G652D is greater than for the SM600,

Figure 4 (

 4 Figure 4(a) shows the evolution of the spectrum along the homogeneous part of the ONF region. It is the first time, to our knowledge, that the spectral longitudinal evolution in nonlinear regime along a ONF is observed. The Raman cascading process is clearly visible with the generation of several Raman Stokes and anti-Stokes orders. Even if the ONF length is short (about 2 cm), the Raman amplification is efficient due to the large reduction of the effective area of the guided modes which provides a strong confinement of the light modes in the ONF. Hence, this leads to increase the light-matter interactions, giving rise to high-efficiency nonlinear frequency conversions of the different Raman Stokes and anti-Stokes components. Furthermore, Fig. 4(b) compares the first and the last spectrum, and shows the dynamics of the measurement (about 40 dB) allowing the clear observation of the spectral evolution.

Figure 4 .

 4 Figure 4. (a) Evolution of the scattered Raman cascade spectrum along the ONF region. (b) The input (blue curve) and the output (red curve) spectrum for the ONF. (c) Evolution of the natural logarithm of peaks of the different spectral components in the Stokes region with the propagation distance X. The dashed lines have been used to estimate slopes.

Figure 4 (

 4 Figure 4(c) represents the evolution of the natural logarithm of the different spectral Stokes components peaks along the ONF distance X. The exponential Raman amplification, characterized by a linear evolution, is clearly visible for all the Raman Stokes orders, from their generation up to the depletion of their respective pump. Assuming that the detector response is almost flat for the spectral range concerned between the pump and the 3rd order Raman Stokes (532 nm up to 573 nm), the linear slope of the intensity evolution for the first, second and third Raman Stokes orders can be roughly estimated. The first Raman Stokes order presents a slope of about 0.18 -1

ACKNOWLEDGMENT

This work has been supported by the EIPHI Graduate School (contract ANR-17-EURE-0002) and by the Conseil Régional de Bourgogne -Franche-Comté (DECAP and CIPPIC project). Authors thank Thibaut Sylvestre for relevant and helpful discussions.


Adrià Grabulosa
email: adria.grabulosa@femto-st.fr

Xavier Porte

Johnny Moughames

Daniel Brunner

adiabatic 1-to-N couplers

Keywords: (3+1)D flash-printing, two-photon polymerization (TPP), one-photon polymerization (OPP), singlemode low-loss waveguides, broadband adiabatic coupling

come

INTRODUCTION

Recently, we have established the (3+1)D flash-TPP as a novel manufacturing methodology that is based on direct-laser writing (DLW) associated with two-photon polymerization (TPP) combined with one-photon polymerization (OPP) for the fabrication of polymer-cladded single-mode 3D optical waveguides 1. Typically, the fabrication of photonic circuits with large volumes, i.e. mm 3 , using the classical 3D DLW-TPP fabrication results in unfeasible fabrication times 2, 3. To accelerate this process, the (3+1)D flash-TPP lithography, first demonstrated with the commercially available IP-S photoresist 4, combines three concepts dedicated to fabricating independently the three essential parts of a photonic circuit: (i) the waveguide cores, (ii) the waveguide cladding and (iii) mechanical support ensuring overall stability. We constrain the high-resolution and slow TPPtrajectories of the waveguide cores, while use fast single-step UV blanket exposure to develop the remaining unstructured cladding volume in one flash. Waveguide cores require µm feature sizes 5, hence a fine resolution in the (x, y)-plane, i.e. a small hatching distance h, and accurate TPP writing power are crucial to ensure low propagation losses. However, areas acting as mechanical support just need to be mechanically stable and therefore are printed with maximal hatching distances in order to reduce fabrication times. At the same time, we can maximize the vertical distance between consecutive slices, or slicing distance s, which does not significantly affect the core-cladding interface roughness and equally accelerates the printing process. In photonic waveguides, light confinement is caused by total internal reflection at the core-cladding interfaces, being the refractive index of the core (n core ) larger than the one of the cladding (n cladding ) 6. To achieve sufficient refractive index contrast ∆n = n core -n cladding for optical guiding, we expose in a single-shot the entire circuit via blanket ultraviolet (UV) irradiation with a controlled exposure dosage D 7, 8. The working principle of flash-TPP is schematically depicted in Fig. 1(a-c). Figure 1(a) depicts the DLW-TPP printing procedure, where the microscope objective is directly immersed into the IP-S photoresist. We use the Nanoscribe GmbH (Photonic Professional GT) system for the TPP-fabrication of waveguide cores and mechanical supports, which is equipped with a femtosecond (fs) laser operating at 780 nm. Then, the unexposed photoresist is removed in a two-step development process using propylene-glycol-methyl-ether-acetate (PGMEA) as a developer for 20 minutes, followed by treatment in isopropyl alcohol (2-propanol) for 3-5 minutes. Finally, we transfer the developed circuit to a UV chamber (Rolence Enterprise Inc., LQ-Box model, 405 nm wavelength, 150 mW/cm 2 average light intensity) as shown in Fig. 1(b), by which we control the OPP dosage D determined by the duration of the UV exposure.

The structure in Fig. 1(c) shows a scanning electron microscopy (SEM) micrograph that illustrates the overall (3+1)D flash-TPP concept. The 3D photonic circuit is first fabricated in a single 3D DLW-TPP lithography (red region) step, which is followed by OPP via irradiation with the UV light source (blue regions) for several seconds. flash-TPP has the advantage of printing 3D microstructures using a dip-in configuration, which allows the fabrication of millimeter-tall yet µm-diameter waveguides as depicted in Fig. 1(d). Overall, we achieve low 0.26 dB injection losses and propagation losses of 1.36 dB/mm at λ = 660 nm, while printing vertical single-mode waveguides of 6 mm length that reach refractive index contrast ∆n ≈ 5 × 10 -3 . The fabrication time is reduced by ≈ 90% compared to only TPP fabrication and the the waveguide's optical performance remains stable during ∼600 hours of continuous operation with 0.25 mW injected light 1.

(3+1)D-PRINTED 1-TO-N BROADBAND ADIABATIC COUPLERS

Here, we use (3+1)D flash-TPP lithographic configuration for the fabrication of low-loss single-mode broadband adiabatic couplers with one input and up to 4 outputs. Adiabatic coupling can achieve highly efficient and broadband single-mode coupling using tapered waveguides 9, in which the optical power transfer between a tapered input and inversely tapered output waveguides is achieved through evanescent coupling. Our structures simultaneously split input light to multiple adiabatic tapered output ports. However, the majority of demonstrations and studies before consider the 2D case of 1 to 1 waveguide adiabatic tapered coupling 10, 11. Our flash-TPP fabricated adiabatic couplers have ultra-low global losses of ∼ 0.2 dB, while the losses remain < 2 dB for a remarkable range of ∆λ ∼ 500 nm.

For the adiabatic couplers here presented, we tapered the waveguide cores comparing two different strategies as shown in Fig. 2 (a), where the left (right) panel shows conical (truncated rod) geometries, respectively. In both, the waveguide cross-section continuously changes along the propagation direction from an input diameter d through a taper-length l t , which is intrinsically linked to the beating length z b 6. In the conical geometry, the waveguide cross-section shrinks equally in both, (x, y) directions. However, for the truncated rod geometry, one direction is fixed to d while the other linearly shrinks along the taper positions within l t . This unique spatiallyinvariant aspect of the truncated rod design offers coupling parallel to the splitting direction and increases the interface-area for effective signal transfer. We inversely tapered input and output waveguides with equal taperrate and therefore identical symmetry in order to match their effective modal index. We added a straight section l = 30 µm to all output ports in order to minimize the cross-talk outside the tapered regions, and we separate neighboring waveguides by a gap g between the input and output waveguides. We perform ultra-rapid fabrication by constructing the entire 3D photonic chip combining one-(OPP) and two-photon polymerization (TPP) with the (3+1)D flash-TPP lithography, saving up to ≈ 90 % of fabrication time 1. The waveguide cores are embedded in a 3D cuboid and printed in a single-step via DLW-TPP using the commercial 3D direct-laser writing Nanoscribe GmbH (Photonics Professional GT) system. The liquid negativetone IP-S photoresist is polymerized via TPP with a femtosecond (fs) laser operating at 780 nm that is tightly focused into the immersion medium thought a high-numerical aperture (NA = 0.8) 25X objective lens forming writing voxels within the focal point. The fs-laser beam is rapidly moved by galvanometer mirrors in the lateral directions and then translated along the vertical positions for a precise layer-by-layer additive manufacturing. The waveguides cores are printed with an optimal laser power (LP = 15 mW) and high-resolution (h = 0.4 µm) to ensure smooth surfaces and low propagation losses, while mechanical supports, i.e. the cuboid-surface, are printed with larger hatching distance (h = 0.8 µm). The slicing distance is maximized to s = 1 µm. After fully-TPP of the IP-S photoresist (n = 1.510), the unexposed photoresist is removed following a standard two-step development process. Finally, the entire 3D photonic circuit is treated via OPP with a UV exposure dose of D = 3000 mJ/cm 2 (see Fig. 1(a-c)). This additional photopolymerization step provides auxiliary matrix stability for complex structures and sufficient refractive index constrats ∆n ≈ 5 × 10 -3 between core-cladding waveguides and hence optical confinement for single.mode propagation 1.

We evaluate the global losses (injection, coupling and propagation) for gaps g ∈ {0.4, 0.8, 1.2} µm between input and output 1 to 2 adiabatic splitters. Noteworthy, we present here the extensive results for the truncated rod geometry only as we found to have on average ∼ 3 times less global losses. The waveguide cores are linearly tapered from an input-taper diameter d = 3.3 µm. We tune the taper-length l t from 100 to 500 µm and find adiabatic conditions for a taper-length l t > 200 µm, as shown in the output intensity profiles of Fig. 2(b). It is clear that for l t = 200 µm (left) the outputs profiles for g = 0.4 µm (top) and g = 0.8 µm (middle) are not single-mode, and for g = 1.2 µm (bottom) are not completely separated. However, we obtain full splitting of optical single-modes for l t = 300 µm (right) for all g. Consequently, the adiabatic criterion of the 1 to 2 couplers is fulfilled for a taper-length l t > 200 µm, which agrees with the theoretical prediction for adiabaticity l t > z b = λ/∆n ≈ 132 µm, where λ = 660 nm is the wavelength in the vacuum. Finally, Fig. 2(c) shows the global losses of the 1 to 2 adiabatic couplers with truncated rod geometry for different gaps g. Global losses are minimized to ∼ 0.2 dB and the output intensities only differ by ∼ 3.4 % between the two output ports for l t = 500 µm and g = 0.4 µm. The experimentally measured high optical power transfer certifies the adiabatic signature of our splitters, which is a hallmark of evanescent transition and further demonstrates the high-quality of our adiabatic couplers with truncated rod geometry. Generally, adiabatic transfer offers wavelength-independent splitting of optical signals. We inject light ranging from λ = 520 nm to λ = 980 nm to our 1 to 2 adiabatic couplers with l t = 500 µm, and Fig. 3(a) shows the output intensity profiles of the couplers across this entire range of wavelengths. For λ ≥ 660 nm, the lower confinement of the fundamental mode leads to a signal spreading towards the cladding, and therefore a larger gap g = 0.8 µm is required for its complete adiabatic transfer. However, for λ = 520 nm, higher-order propagation modes are excited and launched into the waveguides. Finally, Fig. 3(b) depicts global losses that remain below ∼ 2dB for the 1 to 2 adiabatic couplers over the ∆λ ∼ 500 nm range here evaluated.

In a last investigation, we fabricate 1 to 3 and 4 adiabatic couplers, where the cross-sections of the waveguides are triangular and quadrangular pyramids for the 1 to 3 and 1 to 4 adiabatic couplers, respectively. Figure 3(c) shows the single-mode splitting outputs for 1 to 3 (top) and 1 to 4 (bottom) adiabatic couplers with l t = 500 µm and g = 0.4 µm. Total losses are only ∼ 0.7 dB and the intensity difference between output ports is ∼ 4.6 % for 1 to 3 and ∼ 6.1 % for 1 to 4 adiabatic couplers, respectively. Overall, we use the novel (3+1)D flash-TPP lithography configuration for the manufacturing of single-mode low-loss broadband 3D adiabatic couplers with one input and up to 4 output ports. After comparing different tapering geometries for adiabatic transfer, the global losses are reduced to ∼ 0.2 dB and the adiabatic broadband functionality with losses < 2 dB over ∆λ ∼ 500 nm is demonstrated. The scalability of output ports here addressed can only be achieved by using the three spatial dimensions, hence unlocking the third dimension results crucial for such implementation 12, 13. This demonstrates our flash-TPP printing technology as a powerful tool for complex and CMOS compatible 3D integrated photonic circuits.

Figure 1 .

 1 Figure 1. flash-TPP for 3D integrated photonics. (a) Illustration of the dip-in process in which the IP-S photoresist is polymerized via DLW-TPP using a fs-laser to fabricate the waveguide cores. (b) A UV light source polymerizes the unexposed regions of the structure, i.e. waveguide cladding. (c) SEM micrograph of a 3D-printed cross-section cutting through a cuboid integrating 16 waveguides with diameters ranging from d ∈ {0.8 : 0.4 : 6.8} µm. The waveguide cores are printed in high-resolution for maximizing optical performance, while structures only serving mechanical stability are printed with lower spatial resolution. Red colour represents the TPP-printed regions, blue colour represents regions that are polymerized via OPP. (d) Photography of a structure integrating a 6 mm long waveguide size-scaled to a match.

Figure 2 .

 2 Figure 2. Design and global losses of 1 to 2 adiabatic couplers. (a) Design of the conical (left) and truncated rod geometries of (3+1)D flash-TPP printed 1 to 2 adiabatic splitters. (b) Output intensity profiles of the 1 to 2 adiabatic couplers with truncated rod geometry for taper-length lt = 200 µm (left) and lt = 300 µm (right) for gaps g ∈ {0.4, 0.8, 1.2} µm (top to bottom). (c) Global losses of 1 to 2 couplers with truncated rod geometry for gaps g and taper-lengths lt ∈ {100 : 100 : 500} µm

Figure 3 .

 3 Figure 3. Broadband functionality and 1 to 3 and 4 adiabatic couplers. (a) Global losses versus injection light wavelength λ of the 1 to 2 adiabatic couplers with lt = 500 µm and g = 0.8 µm. (b) Output intensity profiles from (a) over ∆λ ∼ 500 nm. (c) Output intensity profiles of the 1 to 3 (top) and 1 to 4 (bottom) adiabatic couplers with lt = 500 µm and g = 0.4 µm.


Na A Xiao

Chen Xie
email: xie_chen@tju.edu.cn

François Courvoisier

Minglie Hu

Caustics of the Axially Symmetric Vortex Beams: Analysis and Engineering

published or not. The documents may come

Introduction

Vortex beams are structured light fields with doughnut-shaped spatial profiles and phase singularities. Their distinctive feature is the presence of the helical phase fronts described as exp (ilθ), where l is the so-called topological charge and θ is the azimuthal angle [START_REF] Allen | Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes[END_REF]. Among the vortex beams, the spiral vortex beams are shown to be strong in terms of the structural stability [START_REF] Abramochkin | Spiral light beams[END_REF][START_REF] Volyar | Structural stability of spiral vortex beams to sector perturbations[END_REF]. Due to these notable properties, vortex beams have attracted widespread interests and play critical roles in various applications, like particle micromanipulation [START_REF] Yang | Optical trapping with structured light: a review[END_REF][START_REF] Marzo | Acoustic virtual vortices with tunable orbital angular momentum for trapping of Mie particles[END_REF], microscopy [START_REF] Yan | Q-plate enabled spectrally diverse orbital-angular-momentum conversion for stimulated emission depletion microscopy[END_REF], material processing [START_REF] Hnatovsky | Materials processing with a tightly focused femtosecond laser vortex pulse[END_REF][START_REF] Sharma | Selective edge enhancement using anisotropic vortex filter[END_REF][START_REF] Xie | Tubular filamentation for laser material processing[END_REF], and optical communication [START_REF] Wang | High-volume optical vortex multiplexing and de-multiplexing for freespace optical communication[END_REF][START_REF] Yan | High-capacity millimetre-wave communications with orbital angular momentum multiplexing[END_REF]. By superimposing the helical phase on the axially symmetric phases of different host beams like the Gaussian beam, the Bessel beam and the abruptly autofocusing beam, a series of axially symmetric vortex beams (ASVBs) are investigated [START_REF] Indebetouw | Optical Vortices and Their Propagation[END_REF][START_REF] Orlov | Propagation of Bessel beams carrying optical vortices[END_REF][START_REF] Dai | Propagation dynamics of an optical vortex imposed on an Airy beam[END_REF][START_REF] Jiang | Propagation dynamics of abruptly autofocusing Airy beams with optical vortices[END_REF]. In this way, vortex beams can be adapted with diverse intensity tubes in focusing, diverging and non-diffracting shapes.

Two major schemes are basically implemented to interpret and engineer the light fields.

The integral-based scheme is the mostly-used method, involving with the diffraction integral using the stationary phase method [START_REF] Born | Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light[END_REF][START_REF] Chremmos | Fourier-space generation of abruptly autofocusing beams and optical bottle beams[END_REF][START_REF] Zannotti | Shaping caustics into propagation-invariant light[END_REF][START_REF] Khonina | Sudden autofocusing of superlinear chirp beams[END_REF]. In some cases, the use of this method induces "singular points" whose intensity tends to infinity [START_REF] Soifer | Caustics of vortex optical beams[END_REF]. Various asymptotic methods have also been discussed to analyze the field around the irremovable singular points [START_REF] Soifer | Caustics of vortex optical beams[END_REF][START_REF] James | Geometrical theory of diffraction for electromagnetic waves[END_REF][START_REF] Weinberg | Asymptotic Methods in Equations of Mathematical Physics[END_REF][START_REF] Kharitonov | Caustics of the vortex beams generated by vortex lenses and vortex axicons[END_REF]. As the approximation is indispensable to simplify the diffraction integral, error estimates as well as complex algebra are required. In parallel, the differentiation-based scheme is also developed in the realm of the geometric optics. In this scheme, vortex light fields can be decomposed into families of rays emerging from any cross-section along propagation [START_REF] Berry | Exact and geometrical optics energy trajectories in twisted beams[END_REF][START_REF] Xie | Light trajectory in Bessel-Gauss vortex beams[END_REF]. And the envelope of the ensemble of rays constitutes the framework of the caustics [START_REF] Berry | Exact and geometrical optics energy trajectories in twisted beams[END_REF][START_REF] Karpov | Spontaneously formed autofocusing caustics in a confined self-defocusing medium[END_REF]. In this way, a relatively simple and efficient method without further approximation is available to interpret the caustics of ASVBs and to predict the focusing properties in real space [START_REF] Xiao | Caustic Interpretation of the Abruptly Autofocusing Vortex beams[END_REF]. However, some clarifications of the compositions of caustics in real space have not been discussed. In addition, for applications like microscopy, laser ablation and fabrication of two-photon polymerization, tailoring the caustics have also been an increasing interest in recent years [START_REF] Dorrah | Frozen waves following arbitrary spiral and snake-like trajectories in air[END_REF][START_REF] Chremmos | Bessel-like optical beams with arbitrary trajectories[END_REF][START_REF] Zhao | Observation of self-accelerating Bessel-like optical beams along arbitrary trajectories[END_REF][START_REF] Froehly | Arbitrary accelerating micron-scale caustic beams in two and three dimensions[END_REF][START_REF] Goutsoulas | Tunable self-similar Bessel-like beams of arbitrary order[END_REF]. With a comprehensive understanding of the caustics, engineering the caustics of the ASVBs is also an issue of fundamental importance in applications.

In this paper, we show a developed approach to classify the caustics of different ASVBs. In Section 2, we firstly introduce the theoretical results as a set of concise expressions developed for reproducing the caustics of the ASVBs. Comparing with the existing results, these expressions are obtained without any mathematical approximation. Then, the third Section compares our theories with the numerical simulations and the experimental results for several exemplary beams, illustrating the geometries of the vortex caustics. In particular, finer features can be well outlined in more intuitive physical images as shown in the discussion of the Bessel vortex beam and the abruptly autofocusing vortex beam, which are absent in previous studies. Furthermore, we also developed a method to engineer vortex beams with tailored novel tubular caustics based on our results. Our theoretical results are in excellent agreement with both numerical simulations and experimental results.

The caustics of the ASVBs

Analytical results

From a given axially symmetric host beam with the phase profile 𝜙 host (r), the vortex version can be synthesized by combining the spiral phase 𝜙 vortex (θ) =lθ with 𝜙 host (r) in any crosssection along propagation [START_REF] Xiao | Caustic Interpretation of the Abruptly Autofocusing Vortex beams[END_REF]:

        host vortex host , = + r r r l          (1)
In geometrical optics, the corresponding radial components of the k vector can be defined as

k⋅sin γ(r) ≡ -∂ 𝜙 (r, θ)/ ∂r = -∂ 𝜙 host /∂r,
where k = 2π/λ is the wave number depending on the wavelength λ [START_REF] Xiao | Caustic Interpretation of the Abruptly Autofocusing Vortex beams[END_REF]. Note that this angle is only related to the host beam. With the parameters N and V defined as:

2 2 2 2 ( ) sin + ( ) ( ) 1 N r l kr V r N      (2)
we found that the rays emerging from the ring of a specific radius r in the initial transverse plane z = 0 lie on the hyperboloid [Fig. 1(a)] specified as:

  2 2 2 2 2 2 2 2 2 2 2 1 | | / w w z z x y R L l k V l R L RV z r N N N k            , , (3)
where R, L and z w are all dependent on r for a given topological charge l. Moreover, the whole light field can be represented in geometrical optics as the superposition of ray families lying on different hyperboloids [Fig. 1(b)], leading to their envelope defined as the caustic of the synthesized vortex beam from the host field. Here we postulate the existence of the caustic, and we found that the z positions of the constituent points on the caustic (or the characteristic points in [START_REF] Xiao | Caustic Interpretation of the Abruptly Autofocusing Vortex beams[END_REF]) are simply the solutions to the equation:

      2 2 2 2 + =0 = , , 4
w w w A z z B z z C z R R A LVN B C R L R B AC              (4)
where the prime denotes a derivative with r. Since the Eq. ( 4) is quadratic, there are two sets of constituent points (ρ(z 1 ), z 1 ) and (ρ(z 2 ), z 2 ) for each given parameter r:

1 2 2 2 w w B z z A B z z A         (5)
Note that the host phase 𝜙 host (r) is represented by the specific distribution of sin γ(r) through differentiation. And the introduction of the parameters N and V in Eq. ( 2) can further facilitate the understanding on the geometrical image of the vortex beams. One can easily calculate the complex caustic by substituting (N, V) of a specific host phase into Eqs. ( 3) - [START_REF] Marzo | Acoustic virtual vortices with tunable orbital angular momentum for trapping of Mie particles[END_REF]. Since no integral is involved in this procedure, our differentiation-based method is very friendly to implement without further approximations. (the blue and red half tubes), with their intersections with y=0 plane (blue and red solid curves) projected into the bottom plane. The intersections between other hyperboloids and y=0 plane are also superimposed (gray solid curves). The whole beam propagates in the z direction.

Discussions on the topology of the caustics

In this work, we focus on the caustics in real space of an optical setup (z >0) since the beam is usually generated in such ways. In general, the constituent points of the global caustics of the ASVBs can be rather complex: either a single set of {z 1 } (or{z 2 }) or both sets of {z 1 } and {z 2 } can get involved in constituting the caustic in real space (as will be shown in Section 3.2). After some algebra, we find that the expression below can well distinguish the above two general cases in terms of z w :

1 2 2 1 1 2 sgn( ) Both sets: , 0, 0 2 sgn( ) Single set: , 0 (or 0 ) 2 w w B R z z z A B R z z z z z A                 (6)
where sgn(R′) is the sign function. Besides, the relative locations of z 1 and z 2 for a specific host beam are determined by R'. More details of the cases are summarized in Table 1.

Table 1. Constituent points in real space (z > 0)

No. of solution sets

Sign of R' We use the same setup as in [START_REF] Xiao | Caustic Interpretation of the Abruptly Autofocusing Vortex beams[END_REF] to demonstrate our analytical results in section 2. As shown in Fig. 2, the Gaussian beam centered at 1065nm from a home-made ultrafast fiber laser is firstly injected into a beam expander. The output collimated beam with an expanded width of ~8mm is then incident upon a phase-only spatial light modulator (Holoeye SLM-Pluto, 1920×1080 pixels). The exposed power is controlled by adjusting the half-wave plate before a polarizer. With the phases of the synthesized vortex beams in Table 2 encoded on the SLM together with an additional grating phase, ASVBs can be generated right after SLM in the first order. Then a telescope is used to shrink the ASVBs down to the micron-scale. The telescope is built with a lens (focal length = 1m) and a microscope objective (50×, NA = 0.8). An iris is installed close to MO 1 to block the undesired orders, retaining the first order of the diffraction lights as shown in the inset of Fig. 2. An imaging system made of another identical microscope objective (MO 2 ) followed by a lens (focal length = 0.5m) is used to capture the intensity profiles at each z position along beam propagation. The 2D side views of the intensity profiles are then extracted from the stacked data. Careful alignments only introduce weak perturbation in the setup. This guarantees that the vortex beams appear with negligible deviation from the ideal caustics.

Two sets: {z 1 } and {z 2 } A single set: {z 1 } (or{z 2 }) + 0<z 2 < z 1 z 2 <0<z 1 - 0< z 1 < z 2 z 1 <0<z 2

Numerical and experimental demonstration of the caustics of different ASBVs

Setup

Experimental demonstrations

In our previous study on vortex beams hosted by the abruptly autofocusing beams [START_REF] Xiao | Caustic Interpretation of the Abruptly Autofocusing Vortex beams[END_REF], it is proved that the caustics of their central tube can be well described by the set of constituent points determined by Eq. ( 3)-( 5). To further demonstrate the universality of our formulae for the whole family of axially symmetric vortex beams (ASVBs), vortex beams with four additional host beams, such as Gaussian beams, Bessel beams, Bessel-like beams [START_REF] Litvin | Digital generation of shape-invariant Bessel-like beams[END_REF] and parabolic toroidal lens beams [START_REF] Khonina | Caustics of non-paraxial perfect optical vortices generated by toroidal vortex lenses[END_REF] are generated in this work. Table 2 lists their phases and the corresponding sin γ(r).

    , = sin 0 r l r      2 2 ( / ) 2 / l k z l k            Bessel vortex beam     , = sin sin constant r k r l r          2 2 2 2 2 2 sin ( / ) / ( / ) sin l k z l k l k                 Bessel-like vortex beam [33]       1 1 , = sin n m n m r k ar br l r nar mbr            Parabolic vortex toroidal lens [34]         2 0 0 , = 2 2 sin r k r r r f l r r r f         
According to section 2, the caustics of the ASVBs with a given topological charge l are only related to the corresponding host phases. Without loss of generality, the topological charge l is chosen to be l = 5 for this section. Based on the angular spectrum method [START_REF] Goodman | Introduction to Fourier Optics 2nd Ed[END_REF], we also numerically simulate different ASBVs with parameters corresponding to the setup in Fig. 2.

The simulated and experimental results are shown in Fig. 3, where the measured 2D intensity profiles are extracted from the 3D data acquired with our setup. For Gaussian vortex beams, the phase of the host Gaussian beam as well as the corresponding total phase of the ASVB are shown in Fig. 3(a1) -(a2). The flat phase of the host Gaussian beam results in z w = 0 and B = 0 for any r, which gives z 2 = -z 1 with z 1 > 0. According to Table 1, the caustic in real space is determined only by the single set of z 1 , shown by the blue dash-dotted line in Fig. 3(a3) -(a5).

In the second example, the phases of the host Bessel beam with sin γ(r) ≡ 8.87×10 -4 and the corresponding vortex beam are shown in Fig. 3(b1) -(b2). As Bessel vortex beam determines that R′(r) > 0 and z w < (B+Δ) /2A, the sign of z 1 and z 2 can be solved as z 1 > 0 and z 2 < 0. In this way, the caustic in real space is also determined by the single set of z 1 as shown in Fig. 3(b3) -(b5) with the blue dash-dotted line. Besides, the set of (ρ(z 1 ), z 1 ) can be well approximated by the set of (R(r), z w (r)) for Bessel vortex beams. Comparing with the cylindrical caustic expected for Bessel vortex beams [START_REF] Kharitonov | Caustics of the vortex beams generated by vortex lenses and vortex axicons[END_REF], our solution in Table 2 can well outline the finer features of the transitionally-expanding tube caustic as shown in [START_REF] Xie | Light trajectory in Bessel-Gauss vortex beams[END_REF]. The cylindrical caustic is the special case where the approximation l/(kr) ≪ sin γ is applied in the expression R(r) in Eq. (3). We stress here that the analytical results in this paper also address the puzzling correspondence between the ensemble of the hyperboloidal waists and the central tube caustics of Bessel vortex beams in our previous study [START_REF] Xie | Light trajectory in Bessel-Gauss vortex beams[END_REF].

A family of perfect optical vortices (POVs) are recently designed by integrating the spiral phase into the phase of the parabolic toroidal lens [START_REF] Khonina | Caustics of non-paraxial perfect optical vortices generated by toroidal vortex lenses[END_REF]. This POV with a large range of topological charge l has a quasi-static focal ring radius r 0 at a focal length f. The set of equations determining the caustics of these beams are too complex in [START_REF] Khonina | Caustics of non-paraxial perfect optical vortices generated by toroidal vortex lenses[END_REF], resulting in analytical estimates in two special cases. Since our analytical results are deduced for any axially symmetric vortex beam without any further estimate, we also generate one of such beams to demonstrate the validity of our method. The corresponding beam parameters after the telescope in Table 2 are selected as f = 30μm and r 0 = 7μm. After some algebra, we found the caustic of this specific POV can be decomposed into multiple sections. When a given r is small enough that z w (r) < [B+Δ⋅sgn(R′)] /2A and R′ > 0, the single set of z 1 (r) define the most front section of the caustic [the blue dash-dotted line in Fig. 3 Besides, when the vortex order is increased, the section {(ρ(z 1 ), z 1 (r))} before the focus remains quasi-static and the section {(ρ(z 2 ), z 2 (r))} after the focus is shorter. In short, the complex global caustics can be well analyzed with our method. Our analytical results are in excellent agreement with both numerical simulation and experimental results. In addition, our results can well reproduce the deviation of the propagating behavior of the synthesized vortex beam from that of the corresponding host beam with specific parameters.

The topological charge l is also chosen to be l = 5. Here, we firstly selected the Bessel-like beam in [START_REF] Litvin | Digital generation of shape-invariant Bessel-like beams[END_REF], i.e. a family of shape invariant beams, as the host beam. For specific Bessel-like beams with a linearly ramped central lobe along propagation, the parameters are selected as n = 2, m = 1 in Table 2. As for the Bessel-like beams featuring the linearly diverging central lobe, the parameters are chosen as a = -0.17 m -1 , b = 0.0018. With R′ > 0 and z w < (B+Δ) /2A, z 2 is negative. Therefore, the global caustic is also defined by the single set of z 1 , shown as the blue dash-dot lines in Fig. 4

Engineering the caustics of the ASVBs in the paraxial optics

Previous sections show how to calculate the caustics of the ASVBs from the their phases 𝜙(r,θ).

With the help of the caustic expressions, we also demonstrate that the propagation behavior of the ASVBs can deviate significantly from that of the corresponding host beams due to the spiral phase. Therefore, a challenge arises here that the host beams cannot always well predict the tube shape of their vortex "brother" beams.

In fact, the inverse problem of tailoring the caustic by an engineered host phase is more interesting. The target profile ρ(z) = c(z) can be engineered by solving the host phase 𝜙 host (r) from z caustic (r) =c -1 (z) based on Eq. ( 4) and [START_REF] Marzo | Acoustic virtual vortices with tunable orbital angular momentum for trapping of Mie particles[END_REF]. However, this usually involves a mathematical challenge. Here, we clarify a fundamental limit and show the possibility of tailoring the tube shape based on our results with two preconditions. Since travelling waves are generated in numerous applications, this leads to a fundamental requirement that the topological charge l should not be very large within the radial contents of the beam, i.e. |l|≤ kr| cos γ(r)|. In practice, paraxial optics are adopted in most cases, requiring that tan γ(r)≃sin γ(r). If we further focus on the ASVBs with the azimuthal components of k vector much smaller than the corresponding radial components, we found r ≫ |-l /𝜙'(r,θ)| and sin 2 γ(r) + [l/(kr)] 2 ≃ sin 2 γ(r). With these preconditions implemented in Eqs. ( 4) and ( 5), the tubular caustic can be well approximated as:

    tan sin host host z r r kr r l r              (7)
The 𝜙 host (r) can be solved by substituting the target caustic profiles ρ(z) = c(z) into Eq. ( 7) with the help of the computer. To demonstrate the validity of our method, several exemplary ASVBs with tailored tubular profiles after the telescope can be generated in our setup. The target profiles and the specific parameters of these beams are listed in Table 3. All the measured caustics present excellent agreements with the corresponding tailored profiles as shown in Fig. 5.

Table 3. Target profiles ρ(z) = c(z) of different shapes

Types of profiles

The target profiles and parameters after the telescope Quartic    

         Logarithmic     2 log 0.2, 1μm, 0.5μm z a z b c a b b        Parabolic     2 0 2 1 0 9 10 m , 50μm, 2.5μm z a z z b a z b           Exponential     0 4 1 0 exp 1, 20μm, 5 10 m , 1.5μm z a b z z c a z b c              

Conclusion

In this paper, we demonstrate that the set of analytical equations developed in our previous study on the caustics of the abruptly autofocusing vortex beams [START_REF] Xiao | Caustic Interpretation of the Abruptly Autofocusing Vortex beams[END_REF] have a wider universality to well reproduce the caustics of the axially symmetric vortex beams. Based on a couple of vortex beams synthesized from different host beams, the universality is proved by the excellent agreements of our theory with the numerical and experimental results. Features of the vortex caustics can also be well addressed with our theoretical methods, including the components of the global caustics and the deviation of the vortex caustics from the host caustics. Besides, we have also shown that it is possible to pre-engineer the vortex caustics based on our theory in the paraxial regime where the polarization components can be decoupled. Interesting opportunities also arise for extending our work to the nonparaxial regime. We expect that these results will promote the development of numerous applications, such as material processing, microscopy, particle micromanipulation and synthesis of novel electro-magnetic wavepackets [START_REF] Wong | The complex charge paradigm: A new approach for designing electromagnetic wavepackets[END_REF][START_REF] Wong | Propagation-invariant space-time caustics of light[END_REF]. Also, the structural stability of the different solutions in the nonlinear regime can represent interesting future work.

Fig. 1 .

 1 Fig. 1. Schematics of the axially symmetric vortex beams (ASVBs) in geometrical optics: (a) a single hyperboloid formed by rays emerging from a specific ring in the transverse plane z=0; (b) Two different hyperboloids formed by the corresponding families of rays from the plane z=0

Fig. 2 .

 2 Fig. 2. Schematic of the experimental setup. HWP: half-wave plate; P: polarizer; BE: beam expander; SLM: spatial light modulator; L: lenses; MO1 and MO2: microscope objectives.

  (c3)-(c5)]. As r grows until z w (r) > [B+Δ⋅sgn(R′)] /2A, both the solution sets in Eq. (5) are involved in forming the caustic: {(ρ(z 1 ), z 1 (r))} define a tiny part of the caustic very close to the original focal plane of the host beam [dark blue lines in Fig. 3(c3)-(c5)]; {(ρ(z 2 ), z 2 (r))} determine the long opening tube behind, as shown by the green dash-dotted line in Fig. 3(c3)-(c5). The intermediate caustic section [orange dots] in the range of [z 1 (r max ), z 2 (r max )] between the short dark blue line and the green line is composed by part of the hyperboloid formed by the rays emerging from the edge of the effective aperture.

Fig. 3 .

 3 Fig. 3. The results of different ASVBs with topological charge l = 5. (1)-(2): The phases of the host beams and the corresponding ASVBs. (3)-(5): The theoretical, numerical and experimental results for (a) Gaussian vortex beam, (b) Bessel vortex beam and (c) perfect optical vortex beam generated by parabolic vortex toroidal lens. Light blue dash-dot lines together with dark blue short lines represent the caustic defined by (ρ(z 1 ), z 1 ). Green dash-dot lines represent the caustic defined by (ρ(z 2 ), z 2 ). Orange dot lines represent part of the caustic formed by part of the hyperboloidal surface between (ρ(z 1 ), z 1 ) and (ρ(z 2 ), z 2 ) occurred in the perfect optical vortex.

  (a). For comparison, the central lobe profile of the Bessel-like beam without the spiral phase is superimposed as the purple line in Fig.4(a3). It is evident that the spiral phase further accelerates the divergence along propagation. In the second Bessel-like beam [shown in Fig. 4(b)], we select the parameters as a = 0.056 m -1 and b = 0.0006 without changing n and m. Instead, this Bessel-like host beam has a linearly-tapered central lobe along propagation, presented by the purple line in Fig. 4(b3). Interestingly, the corresponding ASVB has an extremum on the ring caustic [shown by the single red circle in Fig. 4(b3)] determined by R′(r extrema ) = 0. This behavior is totally different from the monotonous tapering of central lobe in the host beam along propagation. In fact, this deviation was also observed in our early work on abruptly autofocusing vortex (AAFV) beams hosted by the polynomial phase [27]. For such AAFV beams with specific parameters (for instance, n = 4, a = 0.3×10 -4 m -3 and r 0 = 1.08 mm as in [27]), the sets of z 1 and z 2 are both involved in constituting the global caustics. Besides the significant deviation of the vortex caustics from the host polynomial trajectories [shown by the purple line in Fig. 4(c3)], the introduction of the spiral phase into the host beam also brings the interesting feature: once the local maximum in the waist distribution R(r) exists, an extremum occurs accordingly on the global caustic despite of the original trajectory of the host beam as shown in Fig. 4. In short, the analyses based on our method can exactly reproduce the fine features of the caustic shape. This further demonstrates the validity of our analytical results, serving as a powerful tool for analyzing the vortex beams.

Fig. 4 .

 4 Fig. 4. Propagating behavior deviation of the ASVBs (l = 5) from their host beams. Bessel-like vortex beam with parameters: (a) a = -0.17 m -1 , b = 0.0018 and (b) a = 0.056 m -1 , b = 0.0006; (c) Abruptly autofocusing vortex beams with: n = 4, a = 0.3×10 -4 m -3 and r 0 = 1.08 mm. Purple lines in (a3) and (b3) show the central lobe size of the host Bessel-like beam without the spiral phase, and the purple line in (c3) represents the host polynomial trajectory. Blue dash-dot lines represent the caustic defined by (ρ(z 1 ), z 1 ) and green dash-dot lines represent the caustic defined by (ρ(z 2 ), z 2 ). Orange dot lines in (c1) -(c3) represent part of the caustic formed by part of the hyperboloidal surface between (ρ(z 1 ), z 1 ) and (ρ(z 2 ), z 2 ) like the perfect optical vortex. Insets of (b3) and (c3) show zoomed extrema points.

Fig. 5 .

 5 Fig. 5. Synthesized vortex beams of different vortex orders with: (a) quartic, (b) logarithmic, (c) parabolic and (d) exponential tubular profiles and tailored parameters listed in Table 3. The blue dash dotted lines s represent the pre-engineered target caustics. The caustic profiles extracted from the simulation and experiments shown in the third column are all defined as in [27]. In addition, two exemplary ASVBs with l = 6 are pre-engineered and shown in Fig. 6, presenting one beam carrying an opening central tube profile ρ(z) = a + bz with a = 2μm, b = 0.058 [Fig. 6(a)] and the other with a tapered central tube profile ρ(z) = a -bz with a = 5μm, b = 0.058 [Fig. 6(b)]. By inserting Eq. (7) into the above target tube profiles, the engineered phase can be readily obtained as 𝜙 host (r)= -(|l|/a)r ± r 2 bk/(2a). The blue solid lines represent the predetermined caustics ρ(z). Fig. 6(a3) and Fig. 6(b3) also present the simulated and measured tube profiles. The discrepancies from the target tapered profile in Fig. 6(b) mainly originate from the limited active aperture on the SLM (8.64mm×8.64mm). This is proved by simulating the same beam with SLMs of different areas, where the larger active area (12mm×12mm) allows a better agreement with the predefined geometry [Fig. 6(b5)]. Despite that such a large SLM is unavailable in our experiment, this distortion can be corrected in tailored caustics with a smaller initial ring and a less steep taper (a = 4μm, b = 0.038) as shown in Fig. 6(c).

Fig. 6 .

 6 Fig. 6. Numerical simulations and experimental results of several particular examples with the tunable hollow core radius. (a) ρ = a + bz with a = 2μm, b = 0.058; (b) ρ = a -bz with a = 5μm, b = 0.058 and (c) ρ = a -bz with a = 4μm, b = 0.038. SLMs with a smaller (7mm×7mm) and a larger (12mm×12mm) area are adopted in the simulation in (b4) and (b5), respectively. The blue dash dotted lines represent the pre-engineered target caustics. The caustic profiles extracted from the simulation and experiments shown in the third column are all defined as in [27].

Table 2 . Phases of four vortex beams synthesized from specific host beams

 2

	Specific ASVBs	Applied phase and their sin γ(r): caustics of two simplest cases are given in the parentheses
	Gaussian vortex beam

Funding. National Natural Science Foundation of China (61605142, 61827821); Tianjin Research Program of Application Foundation and Advanced Technology of China (17JCJQJC43500); Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences (Open Fund of the State Key Laboratory of High Field Laser Physics); European Research Council (682032-PULSAR); Agence Nationale de la Recherche (ANR-15-IDEX-0003, ANR-17-EURE-0002).

Data availability. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Disclosures.

The authors declare no conflicts of interest.


Rachid Riad

Hadrien Titeux

Xuan Nga Cao

Emmanuel Dupoux

Laurie Lemoine

Justine Montillot

Agnes Sliwinski

Jennifer Hamet Bagnou

Anne-Catherine Bachoud-Lévi

A comparison study on patient-psychologist voice diarization

Conversations between a clinician and a patient, in natural conditions, are valuable sources of information for medical follow-up. The automatic analysis of these dialogues could help extract new language markers and speed up the clinicians' reports. Yet, it is not clear which model is the most efficient to detect and identify the speaker turns, especially for individuals with speech disorders. Here, we proposed a split of the data that allows conducting a comparative evaluation of different diarization methods. We designed and trained end-to-end neural network architectures to directly tackle this task from the raw signal and evaluate each approach under the same metric. We also studied the effect of fine-tuning models to find the best performance. Experimental results are reported on naturalistic clinical conversations between Psychologists and Interviewees, at different stages of Huntington's disease, displaying a large panel of speech disorders. We found out that our best end-to-end model achieved 19.5% IER on the test set, compared to 23.6% achieved by the finetuning of the X-vector architecture. Finally, we observed that we could extract clinical markers directly from the automatic systems, highlighting the clinical relevance of our methods.

Introduction

During the last decades, it became easier to collect large naturalistic corpora of speech data. It is now possible to obtain new realistic measurements of * ⋆ Equal contribution. We are very thankful to the patients that participated in our study. We thank Katia Youssov, Laurent Cleret de Langavant, Marvin Lavechin, and the speech pathologists for the multiple helpful discussions and the evaluations of the patients. This work is funded in part by the Agence Nationale pour la Recherche (ANR-17-EURE-0017 Frontcog, ANR-10-IDEX-0001-02 PSL*, ANR-19-P3IA-0001 PRAIRIE 3IA Institute) and Grants from Neuratris, from Facebook AI Research (Research Gift), Google (Faculty Research Award), Microsoft Research (Azure Credits and Grant), and Amazon Web Service (AWS Research Credits).

turn-takings and linguistic behaviours [START_REF] Ash | Why study connected speech production[END_REF]. These measurements can be especially useful during clinical interviews as they augment the current clinical panel of assessments and unlock home-based assessments [START_REF] Matton | Into the wild: Transitioning from recognizing mood in clinical interactions to personal conversations for individuals with bipolar disorder[END_REF]. The remote automatic measure of symptoms of patients with Neurodegenerative diseases could greatly improve the follow-up of patients and speedup ongoing clinical trials.

Yet, this methodology relies on the heavy burden of manual annotation to reach the necessary amount needed to draw significant conclusions. It is now indispensable to have robust speech processing pipelines to extract meaningful insights from these long naturalistic datasets [START_REF] Lahiri | Learning domain invariant representations for child-adult classification from speech[END_REF]). Huntington's Disease represents a unique opportunity to design and test these speech algorithms for Neurodegenerative diseases. Indeed, individuals with the Huntington's disease can exhibit a large spectrum of speech and language symptoms [START_REF] Adam | Speech acoustic markers of early stage and prodromal huntington's disease: a marker of disease onset?[END_REF] and it is possible to follow gene carriers even before the official clinical onset of the disease [START_REF] Hinzen | A systematic linguistic profile of spontaneous narrative speech in presymptomatic and early stage huntington's disease[END_REF]. The first unavoidable computational tasks to extract speech and linguistic information from medical interviews is the diarization: (1) the detection of speaker-homogeneous portions of voice activity [START_REF] Graf | Features for voice activity detection: a comparative analysis[END_REF] and (2) the identification of speaker [START_REF] Bigot | Looking for relevant features for speaker role recognition[END_REF]. Speaker turns are clinically informative for diagnostic in Huntington's Disease [START_REF] Perez | Classification of huntington disease using acoustic and lexical features[END_REF][START_REF] Adam | Speech acoustic markers of early stage and prodromal huntington's disease: a marker of disease onset?[END_REF].

First, a number of studies are trying to solve this problem directly from the audio signal and linguistic outputs, also referred to as Speaker Role Recognition. They are taking advantage of the specificities (ex: prosody, specific vocabulary, adapted language models) of each role in the different domains: Broadcast news programs [START_REF] Bigot | Looking for relevant features for speaker role recognition[END_REF], Meetings [START_REF] Sapru | Automatic speaker role labeling in ami meetings: recognition of formal and social roles[END_REF], Medical conversations [START_REF] Flemotomos | Combined speaker clustering and role recognition in conversational speech[END_REF], Child-centered recordings [START_REF] Lavechin | An open-source voice type classifier for childcentered daylong recordings[END_REF] Figure 1: Two approaches for the diarization of conversational clinical interviews. The steps for the Speaker Enrollment Protocol are in Blue, and Green for the Speaker Role Recognition. [START_REF] Rao Koluguri | Meta-learning for robust child-adult classification from speech[END_REF].

Another approach relies on Speaker Enrollment [START_REF] Snyder | Deep neural network embeddings for text-independent speaker verification[END_REF][START_REF] Heigold | End-to-end text-dependent speaker verification[END_REF], it aims to check the identity of a given speech segment based on a enrolled speaker template. Our study differ from these studies as they are evaluating their pipelines with already segmented speakerhomogeneous speech segments. Another related approach is Personal VAD (Voice Activity Detection) model from [START_REF] Ding | Personal vad: Speaker-conditioned voice activity detection[END_REF] where they used enrolled speaker template to detect speech segments from each individual speaker.

None of these approaches have been compared under the same evaluation metric, despite prior works aiming at solving both these tasks [START_REF] García | Speaker detection in the wild: Lessons learned from jsalt[END_REF] and their high degree of similarities.

Here in this paper, we aimed to detect automatically the portions of speech and to identify the speakers in medical conversation between Psychologists and Interviewees. These interviewees are either Healthy Controls (C), gene carriers without overt manifestation of Huntington's Disease (preHD) and manifest gene carriers of Huntington's Disease (HD). We introduced a novel way to split the datasets so that we are now capable to compare two different speech processing approaches to deal with these 2 problems (Figure 1): Speaker Role Recognition and Speaker Enrollment Protocol. We showed the clinical relevance of these pipelines with the extraction speech markers that have been found predictive in Huntington's Disease.

2 Data, evaluation splits, metrics 2.1 Dataset Ninety four participants were included from two observational cohorts (NCT01412125 and NCT03119246) in this ancillary study at the Hospital Henri-Mondor Créteil, France): 72 people tested with a number of CAG repeats on the Huntingtin gene above 35 (CAG > 35), and 22 Healthy Controls (C). Mutant Huntington gene carriers were considered premanifest if they both score less than five at the Total Motor score (TMS) and their Total functional capacity (TFC) equals 13 [START_REF] Tabrizi | Biological and clinical manifestations of huntington's disease in the longitudinal track-hd study: cross-sectional analysis of baseline data[END_REF] using the Unified Huntington Disease Rating Scale (UHDRS). All participants signed an informed consent and conducted an interview with an expert psychologist. Therefore in the diarization setting, there are two roles in each interview: a Psychologist and an Interviewee. The speech data were annotated with Seshat (Titeux* et al., 2020) and Praat [START_REF] Boersma | Praat, a system for doing phonetics by computer[END_REF] softwares. The dataset is composed of K = 94 interviews I 1...K . We designed a new way to split of speech dataset to compare different diarization approaches: an end-to-end Speaker Role Recognition model and a Speaker Enrollment pipeline (See Figure 2). The dataset is split in three sets which we refer to metatrain set M train , meta-dev set M dev and meta-test set M test with the ratio of 60%, 20%, and 20%, respectively. Interview I ∈ I 1...K is composed of N I segments I = {U 0 , U 2 , . . . , U N I }. Each segment U i is pronounced by a speaker s i . We summarized the corpus statistics in Table 1.

Each interview I in the meta-dev and meta-test is split in two sets which we refer dev set X dev and test set X test . X test is always kept fixed through all experiments, and we study the influence of the size of the X dev based on T dev that filters the segments (cf Figure 2).

All the data from the meta-train set M train is used to train or fine-tune the neural network models for voice activity detection, speaker change detection, speaker role recognition, and speaker enrollment. The dev set X dev of the meta-dev set M dev and the dev set X dev of the meta-test set M test are only used for the speaker enrollment experiments, to build the template representation of each speakers. The results on the test set X test of the meta-dev set M dev are used to select all the hyper-parameters and select the best model for each experiment. The final comparison is done with the test set X test of the meta-test set M test .

Metrics

To compare final performance of each approach, we use the Identification Error Rate (IER) taking into account both the segmentation and confusion errors. IER is obtained with pyannote.metrics [START_REF] Bredin | pyannote.metrics: a toolkit for reproducible evaluation, diagnostic, and error analysis of speaker diarization systems[END_REF]:

IER = T false alarm + T missed detection + T confusion T Total
The T confusion T Total component in the IER is related to the Miss-classification Rate (MR%) used in Speaker Role Recognition study [START_REF] Flemotomos | Role specific lattice rescoring for speaker role recognition from speech recognition outputs[END_REF], which is based on Frames and not duration of the turns. We compared the different approaches as a function of the size of the enrollment T dev in Figure 3.

Methods

Speaker Role Recognition

We adapted the approach from [START_REF] Lavechin | An open-source voice type classifier for childcentered daylong recordings[END_REF] for the Speaker Role Recognition. We trained on M train a unique model to detect each role (Psychologist,Interviewee), and selects the best epoch on M dev . This is a multi-label multiclass segmentation problem. A threshold parameter for each role is optimized on the Meta-dev set M dev for the two output units of the model. Therefore the two classes can be activated at the same time, i.e. we can also detect overlapped speech. To solve and model this task, we used SincNet filters [START_REF] Ravanelli | Speaker recognition from raw waveform with sincnet[END_REF] to obtain adapted speech features vectors from the audio signal. The Sinc-Net output is fed to a stack of 2 bi-recurrent LSTM layers with hidden size of 128, then pass to a stack of 2 feed-forward layers of size 128 before a final decision layer. We used a binary cross-entropy loss and a cyclic scheduler as training procedure. The hyper-parameters to train our model can be found here1 .

Speaker enrollment protocol

The Speaker enrollment protocol can be decomposed into four tasks: (1) Voice Activity Detection (2) Speaker Change Detection, (3) Enrollment, (4) Identification. We extended the speech processing toolkit from [START_REF] Bredin | Pyannote. audio: neural building blocks for speaker diarization[END_REF] pyannote.audio to run our experiments. Clinical laboratories can not all re-train in-domain speech processing models due to data scarcity or a lack of computational resources. Therefore, we evaluated pretrained models on open-source datasets and transfer models on our dataset to evaluate these out-of-domain performances with real clinical conversational conditions.

Voice Activity Detection

The first step is the Voice Activity Detection (VAD), i.e. obtain the speech segments in the audio signal. It can be modeled as an audio sequence labeling task. There are 2 classes (Speech or Non-Speech). The VAD labels for each interview I are the presence or not of a segment U i at time t.

The model can be used already Pretrained or Retrained on the meta-train set M train of our dataset. We choose the DIHARD dataset [START_REF] Ryant | The second dihard diarization challenge: Dataset, task, and baselines[END_REF] as a potential pretrained dataset as it contains multiple source domain data (clinical interviews among them). When trained from scratch, the training is done for 200 pyannote epochs and the model is selected on the Meta-dev M dev . The model is also composed of SincNet filters with 2 bi-recurrent LSTM layers and 2 feed-forward layers. The full specifications can be found here2 .

Speaker Change Detection

The second step is the Speaker Change Detection (SCD), i.e. obtain the moment when one of a speaker starts or stops talking. It can aslo be modeled as an audio sequence labeling task. There are 2 classes (Change or No-Change). The SCD labels for each interview I are the start or end of a segment U i at time t. We also compared Pretrained on DIHARD and Retrained models. We used the same model as for the Voice Activity Detection. The full specifications can be found here.

Based on VAD and SCD outputs, for each Interview I we obtain a set of N ′ I candidates speakerhomogeneous segments { Û1 , . . . ÛN ′ I }.

Enrollment

In the enrollment stage, we need to get a Speaker Embedding function f θ for our specific task. We combined SincNet filters and the X-vector architecture [START_REF] Snyder | Deep neural network embeddings for text-independent speaker verification[END_REF] as in [START_REF] Bredin | Pyannote. audio: neural building blocks for speaker diarization[END_REF]. For finetuning, we froze all layers and finetuned the last layer. We used the VoxCeleb2 dataset [START_REF] Nagrani | Voxceleb: a large-scale speaker identification dataset[END_REF] as a pretraining dataset as it contains a diverse distribution of speakers and recording conditions.

Then, we used the set of segments from the dev set X dev of the meta-dev and meta-test to build a template vector m j for each speaker j in the interview I. X dev contain a set of segments U enrollment speaker j from each speaker j. The start of each segment U enrollment speaker j needs to be smaller than T dev . We computed the average of the representations for each speaker j:

m j = 1 |U enrollment speaker j | U ∈U enrollment speaker j f θ (U )
(1) In principle, the more data you have to build template of each speaker, the easier it is to distinguish them. Thus, we studied the effect of the size of the enrollment based on the parameter T dev ∈ (90s, 100s, . . . , 180s) to build the template m j [START_REF] Larcher | Text-dependent speaker verification: Classifiers, databases and rsr2015[END_REF].

Identification

For the identification stage, we use the function f θ and the different representation m j of the speakers from the enrollment stage. We used the following

D( Û , m j ) = 1 2   1 - f θ ( Û ) ⊤ m j ∥f θ ( Û )∥ ∥m j ∥   (2) argmin j D( Û , m j ) : Selects Speaker j (3)
In addition, we analysed topline performance of the speaker embedding models when the Ground Truth Segmentation is provided. Finally, we computed a chance baseline based on speaker Enrollment by randomly permutating all the cosine distances. Spearman correlation is computed to compare clinical markers extracted from our best system to ground truth extractions (Figures 4 and5).

Results and discussions

Figure 3 shows results in term of IER for the different approaches. Both approaches greatly improved over chance. If we consider pipelines solving both segmentation and identification, our best performance is obtained using the Speaker Role Recognition approach with IER=19.5% while the Speaker Enrollment Protocol obtained at best IER=23.6% at T dev = 120s, with Retrained VAD/SCD models and Finetuned Speaker Embedding. Even though, the Speaker Enrollment protocol has per-speaker templates, it is not surpassing the Speaker Role Recognition approach. The topline with Ground Truth Segmentation (IER=9.1%) indicated that Speaker Enrollment could benefit greatly from a better detection of speaker-homogeneous turns.

Errors of Speaker Enrollment are accumulated through the steps and can not be recovered, while Speaker Role Recognition takes advantage of solving all steps together in an end-to-end approach.

Increasing the size of the Template Enrollment m j for each speaker with T dev lead to slight improvements to all Speaker Enrollment methods. The finetuning of the X-vector speaker embedding model with in-domain is especially crucial (ex: Based on retrained VAD/SCD the IER decreases from 28.2% to 23.6%). We ran an additional ablation experiment (Table 2) for the Speaker Role Recognition to measure the amount of data necessary. This ablation study informed us on the necessary amount of data to reach certain level of performance. Even though models are better than Chance, we found out that at least 50% of our dataset (28 Interviews) is necessary to outperform the Speaker Enrollment Protocol pipeline (IER of 20.7% vs 23.6%). The analysis of the pattern of errors showed that the most important component is the False Alarm (FA), and a tenfold increase in dataset size allows to gain 4 points of FA. Therefore, most of the errors come from the voice activity detection part of the system. One of our hypothesis is that the system is confused by too much ambient noises from the hospital environment and thus potentially trigger too much positive presence of speech.

In previous studies in Huntington's Disease [START_REF] Adam | Speech acoustic markers of early stage and prodromal huntington's disease: a marker of disease onset?[END_REF][START_REF] Perez | Classification of huntington disease using acoustic and lexical features[END_REF], the Ratio of Silence and Statistics on utterances were informative to distinguish between classes of Individuals. These speech markers can be extracted directly from the predictions of the Speaker Role Recognition outputs. We computed the Ratio of Silence and the Standard Deviation of Duration of Utterances on the test set of the Meta-test set M test . This computation was done both from the Ground Truth Segmentation and the segmentation provided by the Speaker role recognition system (Figures 4,5. We observed that the automatic system outputs behaved differently as a function of clinical marker. The Ratio of Silence was better predicted (significant spearman correlation of r = 0.579, p = 0.009) than the SD of Duration of Utterances (non significant spearman correlation of r = 0.325, p = 0.175). One potential interpretation of our results is that the difference between the ratio and the standard deviation reveals that our pipeline is great overall to obtain summary statistics of the interview, but its precision at the turntaking level is not sufficient to obtain turn statistics. Some bias of the predictive system might not hurt the IER metric but hurt the reliability of some clinical measures.

Conclusion and future work

Detection and Identification of speaker turns are fundamental problems in speech processing, especially in healthcare applications. While works studying these problems in isolation has provided valuable insights, in this work, we showed that Speaker Role Recognition was the most suitable approach for Interviewees at different stages of Huntington's Disease. For future work, we plan to investigate the use of these methods to derive robust biomarkers automatically and compare them to more classic approaches [START_REF] Riad | Vocal markers from sustained phonation in huntington's disease[END_REF][START_REF] Perez | Classification of huntington disease using acoustic and lexical features[END_REF][START_REF] Romana | Classification of manifest huntington disease using vowel distortion measures[END_REF].

Figure 2 :

 2 Figure 2: Illustration of the data split with 4 interviews. Each line I i represents an interview between the Interviewee and the Psychologist. The elevation of each row indicates 'who speaks when'. The segments can overlap.

Figure 3 :

 3 Figure 3: Identification Error Rates for the different combination of approaches on the test set X test of the meta-test set M test as a function of the size of the enrollment parameter T dev . Spk Emb., VAD,SCD stand for Speaker Embedding, Voice Activity Detection and Speaker Change Detection. Best performance of each approach is displayed at the best T dev .

Figure 4 :

 4 Figure 4: Ratio of Silence from the Ground truth segmentation and from the best Speaker role recognition pipeline.

Figure 5 :

 5 Figure 5: Standard Deviations (SD) of the Duration of Utterances of Interviewees from the Ground truth segmentation and the best Speaker role recognition system.

Table 1 :

 1 Corpus statistics. P stands for Psychologist. IT stands for Interviewee. Dur stands for Duration and reported in hour. Durations are reported in hours.

		M train	M dev	M test
	#Interviews	57	18	19
	#Segments IT	21400	7503	7788
	#Segments P	4184	1381	1517
	Dur Role IT	7.65	3.02	3.21
	Dur Role P	3.54	1.14	1.15
	Dur Overlap	1.10	0.50	0.45
	C/preHD/HD 13/11/33 4/4/10 5/3/11

Table 2 :

 2 Speaker Role Recognition Ablation study: Identification Error Rates on the test set X test of the meta-test set M test as a function of the percentage of interview in the meta-train set M train . MD stands for Missed detection, FA for False Alarm and Conf. for Confusion

	% of M train MD FA Conf. IER
	10%	8.0 14.5	3.9	26.5
	20%	7.8 12.4	3.8	24.0
	50%	7.5 10.4	2.5	20.7
	100%	7.1 10.2	2.3	19.5

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