Christine Webster
email: webster@sfr.fr

Anne Sedes
email: anne.sedes@univ-paris8.fr

Empty Room, exploring the plasticity of electroacoustic music spatialization in VR with Ambisonic 3D and Binaural techniques

Empty Room, exploring the plasticity of electroacoustic music spatialization in VR with Ambisonic 3D and Binaural techniques

INTRODUCTION

The Empty Room project is an electroacoustic composition spatialized in 3D in VR created by Christine Webster. The project is at the core part of Christine Webster's actual work in progress thesis : Electroacoustic Composition experimented in Digital 3D space. Exploration and study of emerging compositional techniques in VR. Directed by Anne Sedes, CICM, Paris 8 and co-directed by François Garnier Spatial Media Ensadlab. An overall presentation of the Empty Room project can be found on HAL archives [START_REF] Webster | Empty Room, how to compose and spatialize electroacoustic music in VR in ambisonic and binaural[END_REF]. In the present paper we will focus on Empty Room's two first spatialization experiments. The Alpha version spatialization using Unity's version 5 native 3D sound properties and the Beta version spatialization using a third part Ambisonic 3D layer in Unity provided by French company 3D Sound Labs, inspired from the code of HOA Library developped at CICM [START_REF]CICM HOA Library -High Order Ambisonics Library[END_REF].

CONTEXT AND MOTIVATION

Empty Room was created during our collaboration with EnsadLab Spatial Media [START_REF]Spatial Media EnsadLab[END_REF] in Paris (2013/2016). Our former experiments associating Electroacoustic Music with Virtual Reality were done on Second Life [START_REF][END_REF] since 2007. The distribution of the first development kit for Oculus Rift DK1 in 2013 transformed the whole VR ecosystem possibilities and opened a worldwide new era of researches and innovations. To produce Empty Room we abandoned Second Life for Unity 3D [START_REF]Unity 3D[END_REF] we considered far more efficient and flexible regarding our future investigations.

OUR HYPOTHESIS

We consider the Virtual 3D scene as a new compositional space to explore for experimental electronic music. We want to detach from the linear process commonly in use in Digital Audio Workstation as well as we want to detach from the Audio Middleware used with game engines, both optimized for Audio Visual and MultiMedia Post-production techniques for cinema and video games. Starting from this perspective how do we spatialize our composition? How do the sounds behave each other in the composition? How will the embedded 3D tools influence the creative process? How will this sound space be sensed and how will the listening be transformed? The purpose of our research is to confront the practice of electroacoustic music with a virtual 3 dimensional geometrical and graphic space we consider as a genuine place for 3D sound composition, fixation and diffusion.

EMPTY ROOM'S VR AUDITORY ECOSYSTEM

Picture 1

To start our experiments we assembled an operating environment we can compare to an ecosystem we call our VR Auditory Ecosystem. Picture 1. This system is made up with digital components accessible trough Unity 3D (multi media files, GUI, 2d screen, code), one human in immersion and the necessary external controllers, like HMD, tactile control, and tracking system. The VRAE is a sensible environment divided into 3 groups of components witch are synchronized in real time with interdependencies and capacity of interaction and control :

The VR Auditory Space Perception group. This is where we are confronted with the notion of Embodiment, (Lelièvre 2012) Bilocation and Instanciation (Amato 2008) of the Listener/Composer in VR immersion. There, the human senses navigate between two qualities of space, Reality and Virtuality, with the following distribution : the human body is bilocated in sync in the space of reality and in the virtual space, while hearing and vision disconnect naturally from reality and focus to sense only the virtual world. This kind of partial dissociation/association of the senses is very specific to human immersion in VR. The VR Auditory Space group. Here we handle all the aspects concerning auditory spatial awareness in our Unity 3D scene. Notion such as 3D sound for Virtual Reality (Durant R. Begault 2000), VAS, Virtual Auditory Space (Nicol 2010), Ambisonic in multi media (Daniel 2002), HOA Library [START_REF] Sedes | The HOA Library, review and prospects[END_REF], Hrtfs and Binaural rendering trough headphones (Blauert 1993) take part of the process. The VR Auditory Space Control group. Here we use two types of controls, the control applied in the digital domain trough coding like physical properties assigned to a 3D object and the external hardware controllers, witch are electronic devices working in relationship with the digital domain like headphones, stereoscopic Hmd, tactile controller and tracking sytems. The VR Auditory Ecosystem provides all the necessary components to a complete human immersion in a VR Auditory Space.

EMPTY ROOM'S AUDIO WORKFLOW IN UNITY3D

Picture 2

Here we describe Empty Room's audio workflow. Picture 2. For both Alpha and Beta version we used Unity 3D version 5 providing enhanced audio features and compatibility with Oculus Rift and HTC Vive stereoscopic headsets. The audio materials were recorded from a modular synthesizer and mixed in a DAW. We made a stereo premix in Pro Tools for the Alpha version and a Bipan-Transpan mix in Reaper for the additional experiments included in the Beta version. The audio files were individually bounced from the DAW's in 48khz 24bits wav format then imported and compressed in Unity in high quality Vorbis format. Then we deployed each sounds in the 3D scene attaching the audio clips to virtual sound source objects, Unity's AudioSource object. These sound objects can manage 2D sound or 3D sound properties and can be linked to a physical 3D object. In this case the AudioSource object can take advantage of the physical behaviors assigned to the 3D Object such as trajectories, bouncing, hovering or collide effects. In an audio 3D context the audio source object is in constant relationship with the AudioListener object attached to the main virtual camera. The distance, angle and movement between these two objects create the illusion of an auditory space perceived in a synthetic 3 dimensional plane, our VR Auditory Space. In the end of the process the spatial composition can be distributed as a standalone app.

EMPTY ROOM ALPHA VERSION

The intention of Christine Webster, composer of Empty Room is to validate that full human body immersion in VR can be compatible with the sensing of a complex VR Auditory Space. If the VRAS is sensed correctly from the composer's cognitive perspective as well as from the Audience one, we can extend classical 2D and 3D VAS to the VR domain and imagine a new kind of compositional environment. To do so we imagined a musical composition producing different qualities of space to be sensed operating in relation with the graphical 3D space. We defined a musical structure divided in three sequences witch form a 10' composition. The composition and visual environment are linked : Seq 1 -4' -Large and aerial -environment color Blue. Seq 2 -4' -Closed, shifted perspective, lost of spatial references -environment color Red. Seq 3 -2' -generative panic room, confinement -environment color Grey-White. She released the Alpha version of Empty Room in September 2015 and developed it on Unity 3D Version 5 for Mac OSX. This version is playable only in seated mode with a gamepad used for the displacements, with Oculus Rift DK2 VR headset providing a stereoscopic view and a supra auricular headphone for a binaural hearing.

Unity's 3D sound features

Unity version 5 provided new sound features among them an integrated mixer working in conjunction with their 3D positional audio pan. This type of pan is in use in the gaming industry since the 1990s and can be more or less sophisticated. In Unity the Spatial Blend parameter 3D position of the AudioSource object takes the source sound and regulates the gains of the left and right ears contributions based on the distance and angle between the AudioListener and the AudioSource object. Providing simple directional cues for the player on the horizontal plane with an Azimuth angle of 180°. No elevation or rear perception is possible. Additional 3D settings permit to adjust the roll off curve of the Volume with a minimum and maximum attenuation distance. The sounds radiate at 360° in the distance zone. The 3D position pan creates a dynamic immersive stereo soundfield from were we can manipulate the spatial field depth to obtain different qualities of perspective. We bypassed additional integrated features like Pitch and Doppler effects or Unity's Reverb Zone and audio FX. But we used the mixer and the mixer groups.

Picture 3 -Spatialization maps of Empty Room's Seq 1, 2 and 3 (beta)

The spatialization map

Webster has created a 2D spatialization map originally designed to perform with 64 discrete virtual channels plus 10 mobile channels. She had to down sized it to 56 channels to preserve a feeling of spatial clarity. Picture 3 shows how we organized the channel distribution for each of the 3 Sequences. Seq 1 -the sound is distributed trough 4 groups : Group 0 plays without directivity (ambient air sound). Group 1 plays the sound perceived from the central platform (blue square). Group 2 plays the sounds perceived beyond the platform limits (blue circle) Group 3 plays 3 mobile channels translating in the x, y and z axe, attached to a black monolith (black dots). To stay close to the spatial dramaturgy of Seq 1 described as large and aerial, the spatialization is designed as a structure made of 5 quadraphonic layers stretching from the center to the external limits. The three central layers form a quadraphonic-octaphonic structure projecting dotted sound events responding each other in a question-answer mode. Combined with the two external quadraphonic layers (distant in red, very distant in green) we obtain a homogeneous sound field crossed by 8 perspective lines providing different depth of field. Seq 2 -Closed, shifted perspective, lost of spatial references -we have a rumble ambient sound in Group 0, in Group 1 we played with two pair of shifting stereo (yellow and blue) and four pairs of mono clusters (green and violet). Group 2 is disabled and Group 3 got a new sound element : the hectic cube, travelling trough the platform in random mode at fast pace. Seq 3 -In the Alpha version this sequence got temporary sounds. The final spatialization was set in the Beta version.

EMPTY ROOM BETA VERSION

Integration of the VR Audio Kit plugin

Unity allows replacing the standard 3D panner by more advanced ones with an open interface, the Audio Spatializer SDK, which is an extension on top of the Native Audio Plugin SDK. From this interface, French company 3D Sound Labs created a VR Audio Kit providing an High Order Ambisonic 3D [START_REF]D Sound Labs VR Audio Kit[END_REF] spatializer for Unity. The spatializer can handle simultaneously 32 audio sources in a perceptive 3D scene, with a spatial resolution going from order 1 to 6, in the horizontal and vertical plane, delivering a full spherical Auditory 3D perception. The output is binauralized with a default Hrtf characteristic [8]. The Alpha version using Unity's native 3D panner runned from end 2015 until june 2016. We developped the Beta version with 3D Sound Labs VR Audio Kit for Oculus Rift since September 2016 and achieved a full Ambisonic 3D version of Empty Room in August 2017. The HTC Vive version started in October 2017.

BiPan-Transpan audio scene transposition

The Beta journey started with a primary test we made with Jean-Christophe Messonnier, sound engineer at the CNSMDP [8] in Paris. Messonnier works on recording and spatialization of complete audio scenes with the Binaural/Transaural system developed by Jean-Marc Lyzwa CNSMDP and Alexis Baskind [START_REF] Baskind | Binaural and transaural spatialization techniques in multichannel 5.1 production[END_REF]. We transposed an audio scene previously recorded and spatialized by Messonnier with the Transaural Bi-pan panner, into our VR Auditory Space using 3D Sound Labs VR Audio Kit. The scene consisted of 16 monophonics stems recorded from a Jazz trio, including the direct sounds, the early reflections, the later reflections and the high reverberations. The placement of the audio stems on the spatialization map Picture 4, creates a very realistic, transparent and homogeneous 3D sound scene trough witch the virtual camera linked to the binaural audio output can move freely in all dimensions, 6 degrees of freedom, between each recorded layer, without audio clicks or directivity loss. Most important of all, in this situation and from the listener's perspective, we entirely get rid off the initial sweet spot constraint. In addition to this primary and successful test we decided to add a second recorded sound scene made by Messonnier in our Unity 3D scene. This time with got two pianos, a cello and one harp. We included this scene alongside to the first one. Thus we got two complete different sound scenes, each one with it's own personal reverb and reflection specificities we could position in the 3D space, walking in subjective view from one to another, freely. With 16 stems per audio scene we got a total of 32 stems playing at the same time in the 3D space. These primary tests authorize us to go further and to adapt this technique to achieve Empty Room's Seq 3.

Picture 4 -The Jazz Trio BiPan-TranPan spatial configuration

Working together with Messonnier on Empty Room's Seq 3 we had to adapt his recording and spatializing process to Empty Rooms dramaturgy. Originally Seq 3 takes place in a sort of containment chamber or panic room. Because we need a little bit of reflections to work with the Transaural Bi-pan panner we choose to record a session doing close-up miking on 5 Tibetan percussions instruments, a gong and 4 bowls. For each instrument recording we made a close-up mike in order to take the direct sound and all around we positioned 6 omnidirectional mikes to obtain a 360° primary reflections sound field. From this session Messonnier build a new sound scene with the Transaural Bi-pan panner placing the 5 elements together with their primary reflections, adding digital reverb to glue the whole scene together and to work on the height reverb perception at +45° elevation. On Picture 3 at the far right we can see the spatial positioning of Seq 3 : the direct sounds are symbolized in red, the primary refection in grey and the reverb in elevation in blue.

Adapting the spatialization map from 2D to 3D

After Seq 3's finalization we adapt in the same way Empty Room's Seq 1 and 2. Going from a 2D pan positioning 3D sounds in a virtual scene to an Ambisonic 3D soundfield changed several aspects of the sound perception in our VR Auditory Space : now some elements of the spatialization map will be sensed at different heights. Picture 5. This gives more space between them, more clarity and unfolds the space in all 6 dof directions. As a result the amplitude of the sounds are perceived at a lower gain rate. So we had to adjust completely the distance between each element to obtain a well balanced and natural impression of spatial presence for each AudioSource. 3D Sound labs VR Audio Kit force us to bypass Unity's mixer, because all sound sources are mixed trough the ambisonic interface and could not be treated individually outside from it.

The VR Auditory Space Perception issues

Once we got a stable Beta version for OSX and Oculus Rift DK2 we adapt it to be used with the HTC Vive. Upgrading to HTC Vive gives us a better stereoscopic visual resolution and the possibility to walk very naturally in standing position trough our virtual platform in HTC's room scale mode. Displacement could be made physically or with the hand controllers. From this point we have to work now on three important aspects regarding the human immersion in our VR Auditory Scene : -We have to extend the interaction field from a simple spatial interaction sensed trough the listener's displacement to direct touch interactions with the virtual sound field. The question for us is : what do we touch, how do we touch and what for ? Considering that these gestures have to stay in harmony with the compositional purpose.

-Because the body is now full engaged in the virtual scene we have to think about how we want to represent this body trough the representation of an avatar. Do we render a full body shape ? Part of the body, or only the hand ? Can the virtual hand become a multi functional tool ? -How can we organize the VR Auditory Ecosystem as a dynamic 3D workspace for the composer ? Therefore we need to develop a dedicated immersive 3D user interface displaying the virtual spatialization map and all usefull elements the composer needs to workout in space.

All these questions are still in processing and will be worked out in the near future.

RESULTS

Empty Room's Alpha and Beta experimentations shows that we can completely integrate the classical principles of the Virtual Auditory Space displayed trough loudspeakers configurations and transpose them into a VR3D scene in projection perspective. These two layers of virtuality merge together and form a new environment we called a VR Auditory Space. The VRAS is accessible for the composer and the audience in complete VR immersion with a Stereoscopic Hmd and Binaural audio displayed trough headphones. VRAS, Listener and Controllers form our VR Auditory Ecosystem, the basis of our research field.

Picture 5 -Sound positioning in our VR Auditory Space : Left, Unity's 2D distance pan in the horizontal plane without rear perception or elevation. Middle, Ambisonic 2D perception at 360° without elevation. Right, Ambisonic 3D perception at 360° in the horizontal, frontal and median plane.

From our spatialization maps, Picture 3, we show how we can stack and handle for the same musical composition multiples layer of virtual transducers configurations and raise them from 2D to 3D without affecting the general purpose of the sound work. And how we could introduce a non linear auditory experience of the composition, dissolve the sweet spot constraints and mold our spatialization close to any spatial dramaturgy. Following the same principle we show how the listener can sense the VR Auditory Space gradually either with a perception in the horizontal plane in 2D at 180° or 360° or from within a full Ambisonic 3D sphere. Picture 5. Our experiments state that Ambisonic 3D with a high spatial resolution is the best candidate for our compositional experiments event if we are limited to 32 AudioSources. We also observed that during the Ambisonic 3D tests and demos we used the same default Hrtfs for the binaural output without any major hearing problems reported. Concerning the use of reverberations, we bypassed Unity's default reverb zone during the Alpha version and designed our sounds with more or less reverb mixed directly on our audio files. The spectral content and the position in the spatialization map accentuate or not the illusion of distance in the field depth. At first this was motivated to preserve the esthetical approach generated by our modular synthesis sound system, at second doing so we spared a lot of precious CPU for the project. In contrast Messonnier's workflow with the BiPan-TranPan audio scene shows another acoustical approach, how we can spatialize the reverb like an autonomous sound object in the 3D scene using natural recorded reverb and artificial ones in the same scene in conjunction with the direct sounds. During our experimentations the Ambisonic 3D sound field displaying a high spatial resolution operating in a VR context freed from external loudspeaker constraints acts like a virtual universal host. We managed mono, stereo, quadriphonic, octophonic, and object oriented audio in the same project. We scratched the surface of our VR Auditory Ecosystem and the potential is promising, there is lot more to investigate for us in the future. We have to optimize the immersion in VR, to define specific 3D tools for the composer and want to involve more dynamically the listener in the compositional process.

CONCLUSIONS

Our VR Auditory Ecosystem is a dynamic environment running on the basis of a projection perspective plane in Unity 3D, translating and rotating with six degrees of freedom (6dof), displaying a VR Auditory Space in High Order Ambisonic 3D with Binaural headphone output. This 3 dimensional sensory scene is accessible in VR immersion with a stereoscopic Hmd, a headphone and tactile controllers. The Ambisonic3D, Binaural VR Auditory Space represents a new dedicated area for the composer from where he can operate innovative compositional gestures : The composer can integrate, handle and display numerous audio dissemination formats, such as mono, stereo, multichannel stems, multiphonic, quadriphonic or octophonic within the same VR scene. The transposition of complex audio scenes originally linked to a sweet spot turn to an open spatial structure and can be perceived from the composer or the audience from any spatial locations in the VR scene.

The manipulation of virtual transducers in the VR scene transforms the spatialization process. It becomes more adaptative and flexible and allows the consideration of a renewed spatial esthetical expression taking full advantage of the topological characteristics from a VR scene rather than from the classical loudspeaker configurations. All these aspects opens the way to a new harmonization in our VR Auditory Space being at the same time a creation/diffusion platform, suited for complex sound spatialization restitution or a custom made auditory space freed from classical post-production techniques.

ACKNOWLEDGEMENTS

EnsadLab Spatial Media, Ircam, SCAM Brouillon d'un rêve Pierre Shaeffer, La Francogrid, LE CUBE Centre d'art numérique d'Issy-les-Moulineaux, 3D Sound Labs, CNSMDP Conservatoire National de Musique et de Danse de Paris, Institut Français.


Hadrien Pujol

Éric Bavu

Alexandre Garcia

Source localization in reverberant rooms using Deep Learning and microphone arrays

Keywords: Localization, Deep-Learning, Dataset, Reverberant

Sound sources localization (SSL) is a subject of active research in the field of multi-channel signal processing since many years, and could benefit from the emergence of data-driven approaches. In the present paper, we present our recent developments on the use of a deep neural network, fed with raw multichannel audio in order to achieve sound source localization in reverberating and noisy environments. This paradigm allows to avoid the simplifying assumptions that most traditional localization methods incorporate using source models and propagating models. However, for an efficient training process, supervised machine learning algorithms rely on large-sized and precisely labelled datasets. There is therefore a critical need to generate a large number of audio data recorded by microphone arrays in various environments. When the dataset is built either with numerical simulations or with experimental 3D soundfield synthesis, the physical validity is also critical. We therefore present an efficient tensor GPU-based computation of synthetic room impulse responses using fractional delays for image source models, and analyze the localization performances of the proposed neural network fed with this dataset, which allows a significant improvement in terms of SSL accuracy over the traditional MUSIC and SRP-PHAT methods.

INTRODUCTION

Sound sources localization (SSL) in reverberant or noisy environments is a challenging task, with wide applications ranges, such as sensitive areas acoustic surveillance, speaker localization for videoconferencing tasks, acoustic diagnosis in industrial engineering, or acoustic survey systems of animal species. In order to solve this inverse problem using acoustic measurements, many classes of algorithms have been developed in the last decades using microphone array signal processing, propagation models, and signal models. However, when the background noise is high and the propagating medium differs from the model, these acoustic localization methods are either less efficient or require significant computation time. In the last years, Deep Learning methods have been demonstrated to be efficient for a large number of tasks, especially for computer vision and machine hearing. More specifically, for acoustic based applications, Deep Learning approaches have been successfully applied for automatic speech recognition (ASR) or speaker recognition and allow to achieve state of the art performances. More recently, the use of Deep Learning methods and data-based paradigms have been explored by the scientific community for sound sources localization tasks. In order to reach this goal, several promising approaches have been proposed. Some are them are based on pre-processed features calculated from the measured multichannel signals, either in the frequency domain [START_REF] Adavanne | Sound Event Localization and Detection of Overlapping Sources Using Convolutional Recurrent Neural Networks[END_REF]- [START_REF] Vera-Diaz | Towards end-to-end acoustic localization using deep learning: From audio signals to source position coordinates[END_REF] or in the ambisonic domain [START_REF] Perotin | CRNN-based multiple DoA estimation using acoustic intensity features for Ambisonics recordings[END_REF]. In this paper, we propose to perform a SSL task using raw multichannel audio signals. The main intent is to perform the localization inference in real-time, without having to rely on hand-crafted features or signal based hypothesis. We therefore propose a neural network architecture that aims at performing a joint feature learning process, in order to solve the SSL inverse problem from raw multichannel audio, in a similar way to what we recently proposed for sound recognition tasks using raw audio [START_REF] Bavu | TimeScaleNet : a Multiresolution Approach for Raw Audio Recognition using Learnable Biquadratic IIR Filters and Residual Networks of Depthwise-Separable One-Dimensional Atrous Convolutions[END_REF]. For data-based methods, the learning efficiency strongly relies on the dataset used for the training process. For a SSL task, several approaches can be exploited in order to build the multichannel audio dataset. These ap-proaches can be either based on realistic numerical simulations of the direct problem, on extensive experimental measurements, or on hybrid methods which use real-life (or simulated) room impulse responses (RIRs) in order to perform a convolution with real-life recordings [START_REF] Adavanne | A Multi-room Reverberant Dataset for Sound Event Localization and Detection[END_REF]. For simulation-based or hybrid-based approaches, the dataset has to be realistic enough in order to match real recording conditions. In addition, a large variability in the dataset should be available during the learning process, so that features can be efficiently extracted from the raw multichannel data without overfitting to a specific configuration. We therefore developed an efficient tensor GPU-based computation of synthetic RIRs, using fractional delays for image source models. Using this dataset for a compact microphone array, the proposed neural network architecture allows to perform a 2D DOA inference with a significant improvement in terms of SSL accuracy of 1 o to 25 o in anechoic and reverberant environments and strong noisy conditions over the MUSIC and SRP-PHAT methods, for speech signals.

METHODS AND EVALUATION 2.1 Multichannel reverberant audio Dataset

In the context of the development of this Deep Learning based approach for SSL using raw audio data, we have built several datasets using the MiniDSP UMA-8 compact microphone array (see https://www.minidsp.com/ products/usb-audio-interface/uma-8-microphone-array) geometry, either using simulated data or measured data. This geometry matches those of the microphone arrays used in personal assistants such as Amazon Alexa, Google Home, or Apple HomePod. The UMA-8 array is composed of 7 digital MEMS microphones, the first one being at the center of the array, and the 6 others being evenly distributed on a 8 cm diameter circle. For both the dataset development and the neural network architecture, the proposed approach has been tailored in order to acomodate any microphone array configurations. However, in this paper, for sake of clarity and concision, we chose to illustrate the results using this particular array, both for model-based and data-based approaches. This approach is used for more than 20 × 10 9 image source-receiver configurations in order to compute the 48000 × 7 RIRs used to build the dataset corresponding to a typical classroom geometry, with a T R of 0.5 s.

The multichannel reverberant dataset relies on the computation of realistic RIRs using the image method [START_REF] Allen | Image method for efficiently simulating small-room acoustics[END_REF], for 48000 source positions in the considered room, and for every microphone position of the compact measurement array. As a consequence, the dataset relies on the computation of 336000 RIRs for this particular microphone array. For each of these RIRs, the whole number of image sources positions and corresponding attenuations that contribute to the acoustic field for a time interval of [0; T R ] are computed using the Pyroomacoustics library [START_REF] Scheibler | Pyroomacoustics: A python package for audio room simulation and array processing algorithms[END_REF]. For a typical classroom size of 7x10x3.70 m with a T R of 0.5 s, the whole number of image sources therefore represents more than 80000 image sources for each RIR.

For such a large number of RIRs and image sources, the existing frameworks did not allow to perform the RIR computation efficiently and accurately enough. This is the reason why we developed for this specific application a fast parallel batch RIR computation performed on the GPU using the Tensorflow APIs [START_REF] Abadi | TensorFlow: Large-scale machine learning on heterogeneous systems, Software available from tensorflow[END_REF]. This implementation is achieved using sparse tensors computations and an efficient fractional delay filters implementation (see Fig. 1). For a compact microphone such as the UMA-8, there is a critical need to keep the precision of the time delays corresponding to the distance between each of the image sources involved in the RIR computation and the microphones. In order to implement the RIR with non integer sample delays, we therefore used a 7 th order Lagrange interpolation [START_REF] Smith | Introduction to Digital Filters with Audio Applications[END_REF], that allows to perform a better accuracy that standard truncated sinc interpolations, with a much lower computation cost. The integer part of the sample delay is represented as a sparse array, and the 7 coefficients of each fractional delay filters along with the individual dampings corresponding to the cumulative effect of each wall of the room and the distance between the image source and the microphones are then used (see Fig. 1) to compute the RIRs using every source image contributions (see Fig. 2). Using this large number of RIRs, the complete dataset is built using batch convolutions with real-life recordings.

Proposed neural network architecture

The proposed neural network architecture is directly fed with raw multichannel audio data measured by the microphone array (see Fig. 3), without any further preprocessing, in order to allow a joint feature learning along with the SSL task. We also specifically developed this architecture in order to allow real-time SSL inference after convergence of the training process. The convolutional neural cells, which are widely used in deep neural network architectures, can be thought as a strict equivalent to learnable finite impulse response (FIR) filtering in standard digital signal processing (DSP). Conventional beamforming methods can be thought as a filter and sum approach [START_REF] Brandstein | Microphone arrays: signal processing techniques and applications[END_REF]. The global neural architecture depicted on Fig. 3 has been therefore developed in a similar way, with specific operations allowing the neural network to be expressive enough to achieve the SSL task in reverberant and noisy environments using raw multichannel audio data.

The proposed neural network is built using successive subnetworks that act as learnable filterbanks. Each subnetwork is implemented as a residual network of 1D depthwise separable atrous convolutional layers with exponentially increasing dilation factors, which have recently emerged as an efficient architecture for audio generation [START_REF] Van Den Oord | Wavenet: A generative model for raw audio[END_REF], denoising [START_REF] Rethage | A wavenet for speech denoising[END_REF], neural translation [START_REF] Kaiser | Depthwise separable convolutions for neural machine translation[END_REF], and raw audio recognition [START_REF] Bavu | TimeScaleNet : a Multiresolution Approach for Raw Audio Recognition using Learnable Biquadratic IIR Filters and Residual Networks of Depthwise-Separable One-Dimensional Atrous Convolutions[END_REF]. The atrous convolutions are performed independently over every input channel, and allow to achieve a large receptive field with only 6 sets of one-dimensional convolutions with kernels of size 1 × 3. This presents the considerable advantage of making a much more efficient use of the parameters available for representation learning than standard convolutions [START_REF] Kaiser | Depthwise separable convolutions for neural machine translation[END_REF].

These computed depthwise convolutions are then projected onto a new channel space for each layer using a pointwise convolution. Each atrous convolution layer is followed by a layer normalization process and a tanh activation function. After M successive filterbanks (M = 3 on Fig. 3), a pseudo-energy computation is achieved using a mean computation of the squared output channels followed by a Selu activation function [START_REF] Klambauer | Self-normalizing neural networks[END_REF]. This nonlinear activation function has been recently introduced in the literature in order to avoid standard batch normalization processes, without degrading the computational efficiency of deep neural networks. The computed pseudo-energies are then fed to a dense layer in order to compute the overall output of the neural network, which corresponds to the estimation of the source's position in cartesian coordinates, similarly to what Adavanne et al. proposed in [START_REF] Adavanne | Sound Event Localization and Detection of Overlapping Sources Using Convolutional Recurrent Neural Networks[END_REF], in order to avoid any ambiguity due to discontinuies at the angular wrap-around for DOA retrieval, which is treated as a regression task. The cost function is computed using the mean square error loss on the spherical distance between the reference and the estimated locations, in order to solve the SSL as a regression task. In order to enforce overall energy conservation between the measured input channels and the output of the successive learnable filterbanks, a second term is added to this cost function in order to minimize the L1-norm of the difference between these energies. The learning and backpropagation of errors through the neural network is optimized using the Adaptive Moment Estimation (Adam) [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF] algorithm, which performs an exponential moving average of the gradient and the squared gradient, and allows to control the decay rates of these moving averages. The models have been implemented and tested using the Tensorflow open source software library [START_REF] Abadi | TensorFlow: Large-scale machine learning on heterogeneous systems, Software available from tensorflow[END_REF].

Methods

As explained in the previous section, the dataset is computed using a large number of RIRs, which are batch convoluted with real life recordings, similarly to what has been proposed in [START_REF] Vera-Diaz | Towards end-to-end acoustic localization using deep learning: From audio signals to source position coordinates[END_REF], [START_REF] Perotin | CRNN-based multiple DoA estimation using acoustic intensity features for Ambisonics recordings[END_REF], [START_REF] Adavanne | A Multi-room Reverberant Dataset for Sound Event Localization and Detection[END_REF]. The dataset is split into training, validation and test sets, with a ratio of 80:10:10. The proposed approach has been tested both in an anechoic environment and in a reverberant environment with a typical classroom size of 7 × 10 × 3.7 m and a reverberation time of 0.5 s. In this case, the microphone array is positioned at coordinates [4, 6, 1.5] m, the origin of the Cartesian coordinate system corresponding to a corner of the room.

For each environments, the source positions are randomly drawn from a uniform distribution in a torus of mean radius R = 2 m, centered on the microphone array position, with a section dR.Rsin(dφ ) (see Fig. 4). Since we aim at achieving 2D-DOA retrieval of the angular position θ , this process allows to introduce a spatial variability in order to increase the robustness of the proposed method to unseen conditions. For each environment, the dataset is equally distributed over 6 kinds of real life signals convoluted with 8000 random positions, each signal corresponding to different kinds of frequency contents : a 1000 Hz sinusoidal tone, a car horn, anechoic recordings (see https://users.aalto.fi/~ktlokki/Sinfrec/sinfrec.html) of symphonic music [START_REF] Lokki | Recording of anechoic symphony music[END_REF], and outdoor recordings of a people's multiple talking in Danish (mostly women) (see https://odeon.dk/downloads/anechoic-recordings/). At the training phase, each multichannel recordings of the training mini batches are added to a random Gaussian noise with randomly picked signal-to-noise ratio (SNR) values in the range of [+∞, +10] dB. This process can be interpreted as a data augmentation strategy, which allows to ease the generalization capabilities of the proposed method during the learning process.

Evaluation

Since the output of the proposed neural network corresponds to continuous cartesian coordinates (x, y), the estimated angular position of the sources are inferred using θ = 180 π arctan 2( y x ). In order to evaluate the 2D-DOA SSL performances of the proposed algorithm and to compare it in a reproducible way to model-based algorithms (MUSIC [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF] and SRP-PHAT [START_REF] Dibiase | Robust localization in reverberant rooms[END_REF]), each method is tested using the exact same testing dataset. This testing dataset consists in 360 sources, uniformly distributed over a circle of radius R = 2 m, using speech data (unseen during the training phase). The mean DOA mismatch is evaluated over these 360 positions, for each proposed environments, and for different deterministic (SNR) values, ranging from +∞ (no noise) to 0dB in order to evaluate the robustness of each methods to noisy measurements.

RESULTS AND DISCUSSION

In this section, the SSL performances of the proposed method, denoted as "BeamLearning" are compared with those obtained using the MUSIC and SRP-PHAT methods. The performance metric used here corresponds to the mean absolute angular mismatch over the whole 360 test positions.

Ideal case of non-noisy measurements : robustness to reverberation

In order to assess the capabilities of the BeamLearning approach, the proposed deep neural network has been trained using two different datasets, corresponding to an anechoic environment, and the reverberant environment with a reverberation time of 0.5 s described in the previous sections. The neural network has been trained using the proposed data augmentation, with a random SNR value for each element of the training mini-batches. Even if the neural network already allows to obtain good results after a small number of epochs (50 epochs for the anechoic case and 250 epochs for the reverberant case), the training phase has been performed during 2600 epochs, which improved SSL performances by a few tenths of degrees, without any sign of overfitting.

Method

Anechoïc room Reverberant room (T R = 0. Table 1 shows the obtained results using MUSIC, SRP-PHAT, and BeamLearning, after convergence of the learning process. Interestingly, these results show that the BeamLearning approach performs significantly better than both model-based methods in the reverberant environment. In the simpler case of an anechoïc room however, the obtained angular mismatch obtained using the proposed approach is slightly higher than with MUSIC and SRP-PHAT, but the DOA estimation error remains very low for the three methods. Interestingly, when the neural network is trained using non-noisy data (no data augmentation), the BeamLearning approach allows to obtain a DOA mismatch of 0.28 o in an anechoïc room, which matches the performances of MUSIC and SRP-PHAT. As shown in the following section, a further exploration of the SSL performances in noisy environments shows that the data augmentation method allows BeamLearning to be more robust to noise than the model-based methods, which explains the reasons why Table 1 shows the testing results obtained using data augmentation, even if the obtained results are slightly worse for the ideal case of non-noisy measurements.

Robustness to measurement noise

These competitive results obtained for the SSL task in ideal conditions using the BeamLearning approach motivated a systematic analysis of performances of the proposed deep neural network after convergence under noisy conditions, including heavy cases, where the SNR value equals to 0 dB (the data augmentation during the training phase only included higher SNR values than +10 dB). The obtained results are presented on Fig. 5, both for the anechoïc case and the reverberant case. In both environments, the obtained results show that the BeamLearning approach allows to achieve better SSL performances under noisy conditions than the model-based algorithms, even for lower SNR values than those used in the training phase. As expected, SRP-PHAT gives the worst performances for noisy measurements, and MUSIC allows to improve the DOA estimation. However, the BeamLearning approach allows to improve the DOA estimation precision by 1 o to 10 o over the MUSIC method (resp. 10 o to 24 o over the SRP-PHAT method), for SNR values lower than 20 dB in reverberant environments. In the anechoic case, the same trend is observed, with a 2 o to 4 o improvement over the MUSIC method, and a 3 o to 10 o improvement over the SRP-PHAT method, for SNR values lower than 20 dB. In order to further investigate the influence of the data augmentation process, the neural network has also been trained in reverberant environments, with added random Gaussian noise with randomly picked SNR values in the range of [+∞, +28] dB instead of [+∞, +10] dB (see dotted red curve on Fig. 5). The obtained results show that in this case of less intensive data augmentation, the BeamLearning approach is less robust to noise for SNR values lower than 15 dB, therefore validating the proposed approach of data augmentation with a high variability in SNR values, in order to increase robustness to measurement noise for SSL assessment.

Computational efficiency

As shown in the previous sections, the proposed Deep Learning approach strongly relies on a large sized dataset, with a high variability in terms of sources positions, signals, and SNR values. Using this process, the Beam-Learning approach allows to achieve the SSL task with a higher precision and a better robustness to measurement noise than MUSIC and SRP-PHAT methods, in both reverberant and anechoïc environments, including under unseen heavy noise conditions. Using the proposed datasets, the whole learning process took approximately 190 hours of computation using a Nvidia GTX 1080Ti GPU card, for a total of 645 hours of audio waveforms processed by the proposed model. This computation time in the learning phase includes the feed forward propagation, loss function computations, back-propagation, gradients computations and variables updates using Adam. For inference at the testing phase however, the neural network model parameters are frozen, and the only computations involved correspond to the feed forward propagation of the input data to evaluate the source's angular position. In this case, the mean computation time for one DOA estimation reduces to 0.2 ms on the same GPU card, which is significantly faster than the mean computation times of 660 ms and 47 ms for the MUSIC and SRP-PHAT methods using an Intel Core i7-6900K CPU card.

Figure 1 .

 1 Figure 1. 7 th order Lagrange interpolation for the fractional delay computation, for a single source/microphone.This approach is used for more than 20 × 10 9 image source-receiver configurations in order to compute the 48000 × 7 RIRs used to build the dataset corresponding to a typical classroom geometry, with a T R of 0.5 s.

Figure 2 .

 2 Figure 2. Example of the simulated RIR for a source in a room with a reverberation time of 0.5 s using the proposed GPU-based fast implementation.

Figure 3 .

 3 Figure 3. Global neural network architecture for the SSL task, based on successive learnable filterbanks based on residual subnetworks of dephwise separable atrous 1D convolutions with increasing dilation rates and pseudoenergy computation. The output corresponds to the estimated source position in cartesian coordinates.

  the sources at the angle θ

Figure 4 .

 4 Figure 4. Random distribution of the source positions at a given angular position θ

Figure 5 .

 5 Figure 5. Mean absolute angular mismatch obtained using the speech testing dataset (360 positions) with SNR values ranging from +∞ to 0 dB. These DOA errors are plotted for the MUSIC method (solid blue curve), the SRP-PHAT method (solid green curve), and the proposed BeamLearning approach (dotted curves), in both an anechoïc environment (a) and a reverberant environment (b). The proposed neural network has been trained using noisy measurements, with random SNR values ranging from +∞ to +10 dB (orange dash-dotted curve) and random SNR values ranging from +∞ to +28 dB (red dotted curve).

Table 1 .

 1 Mean absolute angular mismatch obtained using the speech testing dataset (360 positions) for the two model-based methods and the proposed data-based methods for a SSL task in the ideal case of non-noisy measurements. The proposed neural network has been trained using noisy measurements, with random SNR values ranging from +∞ to +10 dB. Bold font indicates best result obtained.

	5 s)

CONCLUSIONSIn this paper, we presented BeamLearning, a machine learning approach for the sound source localization task, using raw multichannel audio. The proposed neural network allows to estimate a sound source's angular position, with a higher accuracy and a better robustness to noisy measurements than the traditional model-based MUSIC and SRP-PHAT methods. The proposed data-based approach is inspired by the filter and sum approach used in traditional beamforming, and allows to estimate a source position in real time, using a neural network which mainly consists in successive learnable filterbanks based on residual subnetworks of depthwise separable atrous 1D-convolutions, followed by a pseudo-energy computation over the learnt channels. We also show that the success of this approach strongly relies on the dataset used in the training phase, which has been built using an efficient implementation on the GPU of a large number of RIR, and a batch convolution on the GPU of these computed RIRs with real life recordings. In a future work, an experimental dataset will be presented, based on computed RIRs and the use 3D sound field synthesis using higher ambisonics. Preliminary results on this experimental dataset show that similar improvements are observed, with the main advantage of allowing the neural network to build the best representation in order to compensate the individual microphone frequency responses and the diffraction effects induced by the microphone array structure, which are not taken into account using standard model-based methods.


G Lapouge

H Younes
email: hatem.younes@univ-grenoble-alpes.fr

P Poignet

S Voros

J Troccaz

Needle Segmentation in 3D Ultrasound Volumes Based on Machine Learning for Needle Steering

INTRODUCTION

To achieve needle steering and guide a flexible needle, it is necessary to localize it first. In the case of 3D ultrasound (US) in B-mode, the poor imaging quality and artefacts make it difficult to determine the needle pose. Needle localization issues specific to needle steering are reviewed in [START_REF] Rossa | Issues in closed-loop needle steering[END_REF]. In [START_REF] Lapouge | Multi-rate unscented Kalman filtering for pose and curvature estimation in 3D ultrasound-guided needle steering[END_REF], we proposed an observer of the needle tip pose. In [START_REF] Younes | Automatic needle localization in 3D ultrasound images for brachytherapy[END_REF], we detailed a machine learning approach for needle localization in 3D US volumes. In this paper, we describe the interconnection of both methods for precise curved needle localization in the context of needle steering.

MATERIALS AND METHODS

Approach

3D US compatible needle localization algorithms are reviewed in [START_REF] Zhao | Evaluation and comparison of current biopsy needle localization and tracking methods using 3D ultrasound[END_REF]. Localizing the needle in the whole US volume is computationally expensive. To manage realtime localization, we propose the interconnection of an observer that automatically provides a volume of interest (VOI), and a segmentation algorithm. Because the needle visibility in 3D US volume changes during the insertion, we propose machine learning to select adaptively needle voxels among the VOI voxels.

Needle Pose Estimation

The needle tip behavior is modeled by a unicycle kinematic model. The tip follows a circular path when it is inserted without rotation. In this work, we consider that the needle shaft follows the path of the needle tip. In this hypothesis, segmenting the needle tip over time is equivalent to segmenting the whole needle shaft. This hypothesis is validated experimentally.

The needle tip position and orientation are estimated in real time by the multi-rate unscented Kalman filter (MUKF) described in [START_REF] Lapouge | Multi-rate unscented Kalman filtering for pose and curvature estimation in 3D ultrasound-guided needle steering[END_REF]. It takes as inputs the robot control commands and the measurement coming from the robot's sensors, needle tip segmentation results in the 3D US volumes and stiffness information of the surrounding tissue. The estimated path traced by the needle tip Φ defines a tubular volume of interest, of radius set by the user, for needle segmentation.

If the needle cannot be segmented in the ultrasound volume, the estimation continues until the needle becomes visible again. The segmentation results are then considered as measurements to update the estimation.

Naïve Bayes classification

As explained in [START_REF] Younes | Automatic needle localization in 3D ultrasound images for brachytherapy[END_REF], the VOI histogram can be modeled using an additive Gaussian Mixture Model (GMM):

P(I(v)) = π v P (I(v))
Where v is a voxel and I(v) its intensity value, M is the number of mixed components, π v represents the coefficient of the i th component and P is the Gaussian probability density function that defines the i th component. Here we consider two classes (i=2):

Cl = {Needle (X n ), Background (X b )}.
Given the observed feature I(v), the Bayesian classifier assigns the most likely class as next equation:

C (I(v)) = arg max P(Cl = X |I(v)).
In the case of binary classification, Bayesian classifier is summarized by the followed inequalities:

X n π P(I(v) |Cl = X n ) ≷ π P(I(v)|Cl = X b ). X b
Each voxel of intensity I(v), is classified as a needle member if the posterior probability of X n is larger than that of X b . Several parameters are required to establish this probabilistic model. They vary from GMM to another, even in the same image. Though, each needle localization starts with the prediction of the required parameters.

Expectation Maximization algorithm (EM)

The unknown parameters representing the GMM are estimated by an EM algorithm. This iterative process computes the maximum likelihood of an observed feature for each iteration. The following equations represent the main instructions done iteratively to estimate the parameters.

π = 1 N P X |v , θ µ = v P X |v , θ P X |v , θ σ = v P X |v , θ v -µ P X |v , θ
Where N is the number of voxels and θ is the parameters vector to be estimated at the k th iteration. This work was partly supported by the Investissements d'Avenir programme (Labex CAMI, ANR-11-LABX-0004) and by the FOCUS project (ANR-16-CE19-0011).

Starting from an initial set of values for θ, the algorithm uses the newly derived parameters as the guess for the next iteration. The iteration stops when the parameters values stabilize: |θ -θ | < 𝜀 . A meaningful initialization of θ is necessary for suitable convergence. More details can be found in [START_REF] Younes | Automatic needle localization in 3D ultrasound images for brachytherapy[END_REF].

Needle tip segmentation

The needle visibility depends on its location in the US volume. Before applying Bayesian classification, the VOI is cropped to keep the 40% of its original length which contain the needle extremity. In this way, we ensure that the tip and shaft voxel intensities are similar (i.e. can be considered as belonging to the same class Xn).

The needle tip search is then done in a spherical neighborhood around the estimated needle tip. In that neighborhood, if there is no voxel belonging to Xn, the segmentation fails. Else, the segmented tip is computed as the weighted centroid of the voxels in Xn. The weighting 𝑤 of a voxel p verifies:

𝑤 ∝ 𝑤 𝑥 -𝑤 (𝑦 -𝑦 )

Where 𝑤 , 𝑤 ∈ ℝ are set by the user (resp. 3 and 1 here); 𝑥 is the abscissa of the voxel in the estimated tip frame Ft (see Fig. 1); 𝑦 (resp. 𝑦 ) is the ordinate of the voxel (resp. estimated tip) in the image frame F (see Fig. 1). The needle tip is therefore researched preferentially in the direction of the insertion (𝑥 ), and in the opposite direction of expected tip reverberation artefacts (-𝑦).

RESULTS

The proposed solution has been implemented in the context of automated needle insertions to reach a target. A beveled-tip 24 Gauge Nitinol needle was inserted and rotated into gelatin phantoms and pork tenderloin with a robot. The needle surface was either coated with polyurethane foam or etched with a laser. The ultrasound volumes were acquired at 1 Hz with a 3D end-fire probe 4DEC-9/10 used with the Ultrasonix Sonix RP ultrasound system. The US volume voxels are cubes with 0.4 mm edges. Running on an Intel ® Core ™ I5-8300H CPU, an iteration of the algorithm takes between 20 ms and 30 ms depending on the VOI size. 60 manual needle tip segmentation (0.3 mm repeatability) have been done by a non-expert individual among 4 different needle insertions (2 in pork tenderloin and 2 in gelatin phantoms). The 3D segmentation error Δ was computed as the Euclidean distance between manual and automatic tip segmentation. Axial segmentation errors were computed as the Euclidean distance on each axis between manual and automatic tip segmentation. Results are reported in Table 1.

The mean segmentation error in each direction is generally inferior to the spatial resolution of the probe. The results are therefore very close to voxel size precision and satisfactory for needle tip segmentation. Needle segmentation is more precise in pork than in gelatin. Indeed, in gelatin, the needle can be seen with a better contrast but suffers from greater reverberation artefacts than in pork.

Momentary needle loss occurred in 2 of the pork insertions with no significant impact on automatic segmentation error when the needle becomes visible again.

DISCUSSION

We have proposed an automated needle localization in 3D ultrasound volumes. The solution is composed of interconnected observer and segmentation algorithms. The solution is fully automatic and adapts to the conditions of insertions (i.e. different needles or insertion media). This interconnection allows for precise tip localization despite changing needle visibility in noisy 3D US volumes. It is also robust to momentary needle loss. This makes it suitable for clinically compatible needle steering.


F Defrance
email: fabien.defrance@obspm.fr

M Casaletti

J Sarrazin

M C Wiedner

H Gibson

G Gay

R Lefèvre

Y Delorme

Structured Surface Design to Generate Any Beam Pattern at THz Frequencies

Keywords: phase grating, THz, heterodyne, receiver

An iterative alternate projection-based algorithm is developed to design structured surfaces able to generate any farfield beam pattern at GHz and THz frequencies. To validate the algorithm, two structured profiles (a reflective one and a transmissive one) are designed to generate four beams of similar intensity at 610 GHz. The two prototypes are fabricated and tested to experimentally evaluate their performance. Experimental measurements confirm a good agreement with computer simulations using Feko TM and, therefore, validate the method.

I. INTRODUCTION

Heterodyne receivers at millimeter and submillimeter wavelengths are widely used in astronomy [START_REF] Beaudin | Fundamentals of receivers for terahertz systems[END_REF]. They can achieve very high spectral resolution and are especially interesting for observing spectral lines in our galaxy and in other galaxies. They are usually used to observe the interstellar medium and to determine its physical and chemical properties. In heterodyne receivers, the observed signal, or radio frequency (RF) signal, is down-converted to a lower frequency without losing any amplitude or frequency information. The downconversion is achieved by mixing the RF signal with an artificial monochromatic signal generated by a local oscillator (LO). The down-converted signal at the output of the mixer is the intermediate frequency (IF) signal (fig. 1). The heterodyne mechanism allows to down-convert a RF signal to a lower frequency (e.g. a THz signal can be downconverted to a few GHz), and to process it with a very high spectral resolution. Heterodyne receivers are widely used in astronomy between 100 GHz and 1000 GHz. A new generation of heterodyne receivers are being built for the THz regime.

This frequency range has, so far, been little observed by astronomers and it is one of the last parts of the astronomical spectrum remaining mostly unexplored (fig. 2). Early THz heterodyne receivers in astronomy usually only had one spatial pixel (e.g. HIFI instrument on Herschel satellite [START_REF] Helmich | Herschel-HIFI: The Heterodyne Instrument for the Far-Infrared[END_REF]). Recently, heterodyne receiver arrays have been developed, in order to observe spectra at several positions in the sky simultaneously. Each pixel has a mixer, so, observing with several pixels involves distributing the LO beam to every mixers. Beam splitters are a good solution to divide the LO beam and to specifically illuminate each mixer (fig. 3). Phase gratings [START_REF] Mait | Understanding diffractive optic design in the scalar domain[END_REF] are already used as beam splitters in some astronomical heterodyne receivers (e.g. CHAMP [START_REF] Güsten | CHAMP -The Carbon Heterodyne Array of the MPIfR[END_REF], upGREAT [START_REF] Risacher | The upGREAT Heterodyne Array Receivers for Far Infrared Astronomy[END_REF]). However, the phase gratings currently used in THz astronomical receivers, stepped gratings [START_REF] Dammann | Coherent optical generation and inspection of two-dimensional periodic structures[END_REF], [START_REF] Mait | Design of Dammann gratings for two-dimensional, nonseparable, non-centrosymmetric responses[END_REF] and Fourier gratings [START_REF] Graf | Fourier Gratings as Submillimeter Beam Splitters[END_REF], [START_REF] Heyminck | Array-Receiver LO Unit using collimating Fourier-Gratings[END_REF], have a constrained geometry which limits the beam patterns they can efficiently produce. To overcome this limitation, an iterative alternate projection-based algorithm has been developed to design structured surfaces without any geometrical constraints, and reach a good effi-ciency for any far-field distribution. To validate this method, we have designed, simulated and built two structured surface prototypes (one in transmission and the other in reflection) able to split the LO beam into four beams [START_REF] Defrance | Instrumentation of a 2.6 THz heterodyne receiver[END_REF].

II. DESIGN PROCESS

A. Structured surface design

The iterative algorithm is based on alternate projections involving inverse Fourier transforms (IFT) and Fourier transforms (FT). This algorithm is used to generate the phase profile for the two structured surface prototypes (a reflective one and a transmissive one) generating four beams of similar intensity at 610 GHz. More details about the algorithm itself can be found in [START_REF] Defrance | Structured surface reflector design for oblique incidence beam splitter at 610 GHz[END_REF]. Both prototypes are based on the same initial phase profile (fig. 4). The transmissive prototype is made of TPX ® , a plastic quite transparent at THz frequencies, and is illuminated by an orthogonal incident beam. The reflective prototype is made of brass and is illuminated by an oblique incident beam making an angle of 25 • with the normal of the surface.

B. Electromagnetic simulation

The transmissive and reflective structured surface prototypes have been simulated (fig. 5) with a commercial electromagnetic simulation software, Feko TM , which is based on the Method of Moments. In the simulations, the structured surfaces were illuminated by a Gaussian beam and both were able to generate a far-field pattern composed of four beams of similar intensity, as expected. The beams generated by the gratings have the same Gaussian shape as the incoming beam, but a lower intensity. The efficiency predicted by the simulation is 81 % for the reflective prototype and 68 % for the transmissive prototype.

III. TEST OF THE STRUCTURED SURFACE PROTOTYPES

A. Mechanical design

The structured surface prototypes were milled by a 200 µm diameter end-mill. A surface accuracy of 6 µm and an average roughness inferior to 0.4 µm were measured with a profilometer (Dektak8). The dimensions of the two prototypes were set to be larger than 4 times the beam radius of the incoming Gaussian beam (which is 10 mm), in order to ensure that more than 99.9 % of the incoming energy is received by the structured surfaces. A picture of these two prototypes is shown in fig. 6.

B. Efficiency test

The prototypes were tested with a 610 GHz source and the intensity of the beams was measured with a Golay cell power meter (fig. 7). The source and the prototypes were positioned on a rotating platform, and the horizontal radiation pattern generated by the prototypes was measured by rotating the platform. The distance between the Golay cell power meter and the LO was kept constant, so the measured beams had always the same radius at the input of the power meter and were comparable. The measured efficiency of the prototypes (sum of each beam's intensity divided by the intensity of the source beam without any prototype) is 78 ± 4 % for the reflective one and 62 ± 4 % for the transmissive one. These efficiency values are very close to the ones predicted by the electromagnetic simulations.

IV. OTHER BEAM PATTERN POSSIBILITIES

The iterative algorithm can also create structured surfaces able to generate any kind of beam pattern. To demonstrate this capability, two more complex beam patterns were given as targets to the algorithm, the name of our laboratory (LERMA), and the drawing of a butterfly. In both cases, the algorithm managed to calculate and non-periodic structured surfaces able to generate these beam patterns. Due to computational limitations, only the reflective structure has been simulated. The profiles of the structured surfaces and the corresponding far-field beam patterns simulated by Feko TM are shown in figs. 8 and9. These structured surfaces have not been constructed and tested yet. However, we expect simulation results to be reliable, as the two prototypes presented in the previous section exhibit a very good agreement between the simulated and the measured beam patterns. The structured surfaces required to generate these beam patterns are very complex and were obtained because, unlike in classical phase gratings used at THz frequencies, our design procedure can be used to design non-periodic structures.

V. CONCLUSION

We have developed an iterative alternate projection-based algorithm to design structured surfaces that can be used to generate any beam pattern at GHz and THz frequencies. The commercial electromagnetic software Feko TM was used to simulate the design and to have a precise estimation of its efficiency. In order to validate our approach, we designed, manufactured, and tested two prototype structured surfaces for 610 GHz, able to split one incoming beam into four beams. The fabricated prototypes have a measured efficiency of 78 % (for the reflective structured surface) and 62 % (for the transmissive structured surface). These results are very close to the simulations and confirm our algorithm and simulation tool. Therefore, the structured surfaces are ready to be used in the next generation of astronomical array receivers as beam dividers, but also in any other GHz or THz application where it is required to generate a custom beam pattern.

Fig. 1 .

 1 Fig. 1. Down-conversion of the RF signal in a heterodyne receiver.

Fig. 2 .

 2 Fig. 2. Transmission of the Earth atmosphere depending on the frequency.

Fig. 3 .

 3 Fig. 3. Use of a beam splitter in a multi-pixel heterodyne receiver.

Fig. 4 .

 4 Fig. 4. Structured surface profile calculated by the iterative algorithm.

Fig. 5 .

 5 Fig. 5. Far-field beam patterns calculated with electromagnetic simulations in (a) reflection and in (b) transmission.

Fig. 6 .

 6 Fig. 6. Pictures of the manufactured (a) reflective and (b) transmissive prototypes.

Fig. 7 .

 7 Fig. 7. Schematic of the experimental set-up to measure the radiation beam pattern of the structured surface reflector.

Fig. 8 .

 8 Fig. 8. (a) Structured surface reflector profil generating the (b) normalized far-field, simulated with Feko TM .

Fig. 9 .

 9 Fig. 9. (a) Structured surface reflector profil generating the (b) normalized far-field, simulated with Feko TM .

ACKNOWLEDGMENT This work was funded by the French Space Agency (CNES) and the Laboratory for Studies of Radiation and Matter in Astrophysics (LERMA).


Rad Chetan Joshi

Wideband Diamo

Keywords:

This paper presents a low diamond dipole antenna backed with an A Conductor. The paper addresses the proble broadside gain due to the split in the radiati particular frequencies within the operati solution is proposed by using a hybrid A Conductor that can effectively cancel this ef antenna is characterized by an overall thickne corresponds to a tenth of wavelength at th frequency.

INTRODUCTION

A printed diamond dipole is an interestin of its wide operating bandwidth. When coup plane, the gain of the antenna is mainly broadside direction. The problem with class made of electrical conductors like copper is be placed at a distance of a quarter wave promote constructive interference in the broa magnetic conductor is a solution to this p does not cause a phase reversal of the inci thereby allowing the antenna to be placed ground plane. An Artificial Magnetic Cond been traditionally designed using the Sieven printed surfaces [1]. The dipoles in proxim been previously studied in various works. In pattern splits close to the resonant frequency is corrected by changing the periodicity of t it is observed that radiation pattern splits in in the bandwidth. For applications that requ broadside direction of the antenna, appear split radiation pattern inhibit the functioni This paper presents a UWB antenna desig tackles the problem of splitting of radiation hybrid AMC.

II. DESIGN

The antenna structure comprises of th backed with an artificial magnetic conduct printed on CuClad Substrate of thicknes dielectric constant, ε r = 2. ris, France France

Reflector.

MC Reflector.

The dimensions of the unit cell in the mm, g = 1 mm, where w is the length of the is the distance between two consecutive pa that there is no via in the structure. The AMC is FR4 Epoxy (h = 3.2 mm, ε r = 4.1, above dimensions allow a zero reflection f GHz. At this frequency, the patches beha conductor. As the number of unit cells of A bandwidth of the antenna increases. The si thus a compromise between optimum bandw shown in Fig. 2, an 8 x 8 cells configura designed using unit cell as defined above is The size of FR4 substrate is 75 mm x 75 m antenna substrate is also taken identical substrate. The AMC is placed at a distance antenna. The overall thickness of the antenna

III. RESULTS AND DISCUSS

The antenna has been simulated using Studio (Transient solver). The antenna is input port into the gap S. The bandwidth i array behaves as a magnetic conductor is 5.84 GHz) around the zero reflection phase he magnitude of the e backed with a ndwidth defined by |S11| < -10 dB is 2 GHz (4.2 between 6.8 and 7 GHz. On ob various frequencies in the ban radiation lobe splits up around 5 in blue trace by plotting the broadside direction. The max bandwidth is observed to be 7.4

Using the technique detai currents which lead to destru radiation are located in the la AMC. In order to change the metallic strip as shown in F respectively. Thus, the AMC magnetic and electrical cond reflector. When the antenna is impedance bandwidth of the n GHz (4.2 -6 GHz) with a sec GHz, as seen plotted in red t consideration both bands, ba improved.

  Fig. 1. Diamond Dipole.

Fig. 4 .

 4 Fig. 4. Magnitude of reflection coefficient versus freq

  Fig. 4 also shows t does not have an adverse effec antenna. The introduction o distribution of the currents on destructive contribution in the This restores the radiation in br trace in Fig. 5. This antenna pattern in its entire impedance of the antenna is 8.3 dB. IV. CO In this paper, a wideband a discussed. The problem of spl placed over AMC is investigate by cancelling the contribution the magnetic reflector. The re by an overall thickness of a te operating frequency. This ante bandwidth and ensures a broa whole bandwidth. V. ACKNO This work is supported by by the IDEX Paris-Saclay, ANR REFER [1] D. Sievenpiper, L. Zhang, R. Yablonovitch, "High-impedanc forbidden frequency band," Mic Transactions on , vol. 47, no. 11, [2] R. M. Mateos, C. Craeye and G. antenna," Microwave and Optic pp. 2615-2619, 2006. [3] L. Akhoondzadeh-Asl, D. Kern Dipoles on Electromagnetic Ba Propagation, IEEE Transactions 2007. [4] A. C. Lepage, J. Sarrazin an Antennas with High Impeda Millimeter Wave Circuits and S pp. 69-73. -6.2 GHz) with a second band bserving the radiation patterns on ndwidth, it is seen that the main d 5.7 GHz. This is shown in Fig. e realized gain of the antenna in ximum gain in the impedance 4 dB. iled in [4], it is found that the uctive interference in broadside ast two rows of patches in the distribution of these currents, a Fig. 3 replaces the two rows reflector is now comprised of ductor and is called a hybrid coupled to the hybrid AMC, the new device is observed to be 1.8 cond band between 6.3 and 6.9 trace in Fig 4. When taking in andwidth has been marginally that changing the AMC pattern ct on the input impedance of the of metallic strip changes the n AMC in order to cancel the e broadside radiation direction. roadside direction as seen in red thus has a broadside radiation bandwidth. The maximum gain ONCLUSION antenna with a hybrid reflector is lit radiation patterns for dipoles ed and the problem was resolved of destructive currents found on esulting antenna is characterized enth of wavelength at the lowest enna offers a larger impedance adside radiation pattern over its OWLEDGEMENT the NanoDesign project funded R-11-IDEX-0003-02. RENCES F. J. Broas, N. Alexopolous and E. ce electromagnetic surfaces with a crowave Theory and Techniques, IEEE , pp. 2059-2074, 1999. Toso, "High-gain wideband low-profile cal Technology Letters, vol. 48, no. 12, n, P. Hall and D. Werner, "Wideband andgap Ground Planes," Antennas and on, vol. 55, no. 9, pp. 2426-2434, Sept nd X. Begaud, "Wideband Directive ance Surfaces," in Microwave and ystems, John Wiley & Sons,Ltd., 2013,


Fabien Defrance
email: fabien.defrance@obspm.fr

Massimiliano Casaletti

Julien Sarrazin

Martina Wiedner

Yan Delorme

Grégori Gay

Roland Lefevre

Non-periodic phase gratings

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.

I. INTRODUCTION

Early heterodyne systems concentrated on high spectral resolution and large bandwidth (i.e. many spectral channels), but usually only one spatial pixel. Recently, arrays of heterodyne receivers have been developed to simultaneously measure spectra at several positions in the sky. Heterodyne receivers at terahertz frequencies do not have many pixels (1 pixel for HIFI (1.9THz), 1 pixel for the first version of GREAT (2.5THz), arrays of 7 pixels in development for upGREAT (2.5THz and 4.7THz)). Pumping each pixel with enough LO power is a challenge. Usually, one LO is used to pump several pixels and its beam is divided optically into several beams using phase gratings.

Dammann gratings [START_REF] Dammann | Coherent optical generation and inspection of two-dimensional periodic structures[END_REF] have been widely used in the past decades because they are simple to design and to manufacture. A Dammann grating is a binary phase grating realized using steps able to shift the phase of the signal by π rad. Fourier gratings (or mirrors, when they are used in reflection) [START_REF] Urs | Fourier Gratings as Submillimeter Beam Splitters[END_REF] use a spatial phase modulation given by Fourier series expansion. They are smoother than Dammann gratings and can usually reach higher efficiency.

II. SIMULATIONS

To design a more general kind of gratings, we decided to start from the desired far field beam pattern and to use an iterative method based on far-field to near-field transformation. At each iteration the following steps are performed: -the actual radiation pattern is mapped into the aperture field on the grating via an inverse FFT; the physical realization constraints are imposed (as for example the minimum step size of the grating); the modified aperture field is mapped into the corresponding radiation pattern via an FFT; the constraints on the radiation pattern are imposed (shape of the beam, Side Lobe Level, etc.). These iterations are repeated until the convergence to a physically realizable profile with the desired radiation properties. This method produces a phase grating design that is not periodic, unlike the Fourier and Dammann gratings. The main advantage is that the grating can be totally adapted to the shape (intensity and phase profiles) of the incoming beam and can reach a very high efficiency. As this method allows us to adapt the grating to the incoming beam, it should be possible to focus and divide the beam directly at the output of the LO with a single grating.

III. PRELIMINARY RESULTS

An interesting test is to calculate a grating profile able to divides the LO beam in 4 symmetric beams at 1.4THz. To do so, we considered, first, the grating in one dimension, then in two dimensions.

A. 1D grating

We consider a phase grating able to divide an incoming beam into 2 output beams at an angle of +20° and -20°, respectively. Without imposing any physical constrains, we can obtain side lobes below -50dB, as shown in fig 1 (the desired beams are the peaks at 0dB). The desired output beams contain 99.99% of the input LO power. The phase profile (fig 2 ) gives us the profile of the grating we need to machine to obtain this far field beam pattern. To study a more realistic case (with a limited resolution of the machining), we set a minimum step size of 50µm for the x axis and a step of π/2 for the y axis of the phase profile. This produced a discrete phase profile (fig 4). As a result, the corresponding far field beam pattern (fig 3) presents side lobes up to -21dB. The two wanted output beams contain 97.15% of the input LO power. The efficiency is worse than with a continuous phase profile, but still quite good. Changing some of the input parameters should enable us to improve these results.

B. 2D grating

As the method, in general, converges after a few iterations, the computational time needed in the case of 2D gratings will be very short. Thus, we apply the same algorithm to simulate the phase profile needed to divide the LO beam in 4 symmetric beams as shown in fig 5. The corresponding grating phase profile resembles that of figure 2, but in two dimensions. Such a grating would produce 4 beams as seen in figure 5. Without imposing any physical constrains to the phase profile, the power in the 4 beams corresponds to 99.99% of the input LO power, according to the simulations. The highest side lobes are below -50 dB. When we impose the same physical constraints as in 1D, the power in the 4 beams corresponds to 88.60% of the input LO power. The highest side lobes are at -21 dB.

IV. PERSPECTIVES The next step is to design and manufacture a prototype phase grating. The required phase profile can be obtained in transmission with a transparent material (HDPE, Silicon) or in reflection with a mirror. Different manufacturing methods and materials have to be tested to realise a prototype. This prototype will then be tested and the measurements will be compared to our simulations.

V. CONCLUSION

Non periodic phase gratings promise a powerful tool for dividing a single LO beam into multiple beams for future heterodyne array receivers. They can be designed and optimised for many different situations, for example:

• Turn a non Gaussian beam (i.e. QCL) into a Gaussian beam • Refocus and divide a beam with one element • Generate a custom made beam to adapt to any output configuration

Figure 1 :Figure 2 :

 12 Figure 1 : Far field beam pattern

Figure 3 :Figure 4 :

 34 Figure 3 : Far field beam pattern

Figure 5 :

 5 Figure 5 : Far field beam pattern with 4 pixels


Yenni Pinto
email: pintobal@telecom-paristech.fr

Julien Sarrazin

Anne Claire Lepage

Xavier Begaud

Nicolas Capet

Yenny Pinto

Y Pinto

Design and measurement of a thin and light absorbing for space applications

Introduction

Absorbing material can be used to reduce the reflection of electromagnetic waves on a surface. For space applications, these materials may for example be placed on the satellite to reduce interference between antennas. Conventional design methods consist in the insertion of losses on the surface of the material. The Salisbury screen [START_REF] Salisbury | Absorbent body for electromagnetic waves[END_REF] is an example of this approach in which a resistive layer is placed on top of a metal surface at a distance equal to a quarter wavelength. The major drawback of this resonant structure is to operate on a narrow band of frequencies. The Jauman absorber [START_REF] Du Toit | The design of Jauman absorbers[END_REF], consisting of several resistive layers spaced approximately by a quarter wavelength, operates over a wide band. However, this technique greatly increases the thickness of the structure.

In 2002, Engheta proposed to introduce metamaterials in the design of absorbers [START_REF] Engheta | Thin absorbing screens using metamaterial surfaces[END_REF]. This approach has represented a technological breakthrough as it allows reducing drastically the thickness. Thus, in [START_REF] Costa | Analysis and design of ultra thin electromagnetic absorbers comprising resistively loaded high impedance surfaces[END_REF] the use of a high impedance surface (HIS) associated with a resistive material as absorber is presented. This type of structure, called resistive high impedance surface (RHIS), consists of a frequency selective surface (FSS) over a grounded dielectric slab. The FSS is usually a periodic array of printed patterns loaded with resistors or resistive sheets to achieve absorption. Such a material has the advantage of being lightweight and thin.

In this article, RHIS is optimized for good absorption performance (magnitude of the reflection coefficient |Γ | < -15 dB) in the band (2-2.3 GHz) (14 %) at normal incidence, while fulfilling specific constraints for space applications as a low mass density (<2.5 kg/m 2 ) and a thickness less than 50 mm. This structure is also analyzed at oblique incidence for TE and TM polarizations.

The paper is organized as follows: in the next Section, the design of the RHIS structure is described, in Sect. 3, the experimental validation and results are presented, and finally in Sect. 4, conclusions are given.

RHIS design

The RHIS structure is composed of square patches interconnected by resistors on top of a grounded dielectric slab. Figure 1 shows the unit cell of the RHIS structure. It is composed of square copper patches over Rogers RO4003 substrate (ε r = 3.38 ± 0.05). Below, are a honeycomb layer (ε r = 1.08 ± 0.05) and the ground plane, consisting of a copper film. Interconnecting patches, resistors are formed with TICER The structure is simulated and optimized using the CST Microwave Studio (frequency solver) [START_REF] Linot | Characterization of a loaded high impedance surface[END_REF]. For simulations, only one cell is considered and periodical boundaries are applied.

Simulation results

The structure is optimized to obtain a reflection coefficient lower than -15 dB (|Γ | < -15 dB) in a wider band [(1.6-2.3 GHz) 36 %] than required [(2-2.3 GHz) 14 %] for normal incidence, to compensate, to some extent, the expected performance degradation when the incidence angle increases. This structure is also analyzed at oblique incidence for TE and TM polarizations. The simulations results are given in Figs. The total size of the prototype is 578 × 428 mm and the mass density is about 2.4 kg/m 2 .

Measurement setup

The measurement of the reflection coefficient of the absorbing material is carried out in three stages:

• Measurement with a reference metallic plane (Γ _PEC ) • Measurement with the RHIS structure (Γ _RHIS ) • Calculation of the reflection coefficient of material under test using [START_REF] Salisbury | Absorbent body for electromagnetic waves[END_REF].

|Γ | = Γ _RHIS Γ _PEC , ( 1
)
The reflection coefficient is measured using two horn antennas. These are positioned in front of the material under test (MUT). The two horn antennas are connected to a network analyzer E5071C Agilent. The measurement setup at normal incidence is shown in Fig. 5. The antennas are positioned (Fig. 6) at a distance (d am ) of MUT and are separated by a distance (d a ). The material is positioned at a height of 1.15 m above the ground.

The Figure 6 illustrates the measurement setup at oblique incidence. The distance between the antenna and MUT is d am = 107 cm. MUT is fixed and the antennas are displaced according to incidence angle (θ i = θ r ).

Measurement results

Measurements were realized in two different environments. The first environment is an anechoic chamber where the conditions are suitable to eliminate external interferences and absorb reflections of electromagnetic waves. Second measurements are realized in a laboratory without ideal conditions. The objective is then to estimate if the RHIS struc- ture is efficient in a perturbed environment like a laboratory. The Fig. 7 shows results at normal incidence comparing the measurements from two different environments and simulation. A good agreement between results is obtained, however, a shift in frequency band is observed between simulation and measurements. The influence of the environment is noticeable. Indeed, measurement in laboratory presents more oscillations than measurements in anechoic chamber.

To explain the oscillations presence in both laboratory and anechoic chamber results, different measurements in anechoic chamber, where the distance between antennas and material is changed are performed. First, the distance (d am ) is changed from near field (d am = 35 cm) to far field (d am = 130 cm). The results are displayed in Fig. 8. Differences in absorption levels are observed when measurements are realized in the near field. For all cases, the percentage of the frequency shift (according to simulation result) is about 10 %. The oscillations are present in all cases. Then, the influence of the separating distance between the antennas (d a ) is shown in Fig. 9.

The oscillations amplitude increases when the value of d a decreases. Thus the coupling between the antennas is probably the cause of the oscillations.

For oblique incidence, the measurements are performed for angles 45 • and 60 • . The results are plotted in Figs. 10 and 11 for TE polarization. All the following measurements have been realized in the laboratory (Figs. 10, 11, 12 and 13) with d am = 107 cm. There is an agreement, but differences in frequency and level are observed. At oblique incidence for 45 • and 60 • , requirements are not reached. In fact, after 35 • , the bandwidth shifts to low frequencies and the level increases.

Measurements for TM polarization are presented in Figs. 12 and13.

Shifts in frequency and level can be explained by different ways:

-The imperfections of realization. -The simulation does not take into account the presence of the double-sided adhesive film.

123

-The resistance value is not accurate and has a tolerance of ±10 %. -The permittivity value of honeycomb is not accurate.

Conclusion

The design, realization and measurement of a thin lightweight absorbing material for space applications have been presented. Absorber design is based on HIS loaded with resistors and known as a RHIS. The behavior of RHIS has been analyzed at normal and oblique incidences for TE and TM polarizations. Simulation results have been compared to measurement. Measurements were realized in two different environments: anechoic chamber and laboratory. There is a good agreement, despite some differences in frequency and level. Shifts in frequency and level can be explained by the imperfections of realization, tolerances of materials and the adhesive film not taken into account in the simulation. Further investigation will be driven to highlight and reduce all these uncertainties.

Final design has a reflection coefficient less than -15 dB in S-Band (2-2.3 GHz) at normal incidence and till an angular dispersion of 40 • for waves in TE polarization, and 35 • for waves in TM polarization.

Fig. 1

 1 Fig. 1 Unit cell of RHIS

  2 and 3. A good absorption performance (|Γ | < -15 dB) in the targeted band (2-2.3 GHz) is observed at normal incidence for both polarizations. When the angular dispersion increases, the absorption level decreases and the frequency band shifts. Consequently, a reflection coefficient lower than -15 dB in the 2-2.3 GHz band is limited to an angular dispersion of 40 • for waves in TE polarization, and 35 • for waves in TM polarization.

Figure 4

 4 Figure 4 shows the prototype. It is composed of three layers. The first one is a Rogers RO4003 substrate with the patches and the resistive sheet (TICER 100 /square) above it. This layer has a thickness of 0.203 mm. The second layer consists in a honeycomb with a 25-mm thickness, and the third one

Fig. 2 Fig. 3 Fig. 4

 234 Fig. 2 Simulated reflection coefficient at normal and oblique incidence. TE polarization (BW required bandwidth)

Fig. 5

 5 Fig. 5 Measurement setup at normal incidence

Fig. 6 Fig. 7

 67 Fig. 6 Measurement setup at oblique incidence

Fig. 8 Fig. 9

 89 Fig. 8 Influence of the measurement distance (d am ) on the reflection coefficient (normal incidence, d a = 8 cm)

Fig. 10

 10 Fig. 10 Comparison between simulation and measurements results at oblique incidence (45 • ). TE polarization