Krupali Donda

Yifan Zhu

Aurélien Merkel

Sheng Wan

Badreddine Assouar
email: badreddine.assouar@univ-lorraine.fr

Deep Learning Approach for Designing Acoustic Absorbing Metasurfaces with High Degrees of Freedom

Keywords: Acoustic metasurfaces, Sound absorption, Low frequency regime, Deep learning, Convolutional neural network, Conditional generative adversarial network, Inverse design

The advent of the acoustic metasurfaces offered an unprecedented expansion of our ability to manipulate and structure sound waves for many exotic functionalities. However, the metasurface designs require deep expertise in acoustics and highly intensive iterative computations using Finite Element Method. In this work, we use a two-dimensional convolutional neural network to model complex metasurface absorber structures under oblique wave incidences. The proposed network architecture is capable of computing the absorption spectra with a large number of design degrees of freedom. Further, we implement a conditional generative adversarial network to solve the inverse design problem. When fed with an input set of predefined absorption spectra, the constructed generative network produces candidate designs of metasurface absorbers that match on-demand the spectra with high fidelity. We demonstrate the capability of the implemented network architecture by designing a metasurface absorber operating at 82Hz for oblique wave incidences with a thickness of 𝜆/64. To implement the deep learning methods for acoustic designs, the main challenge is to generate large and high-quality training data set using numerical simulations. To mitigate this issue, we implement data augmentation. The presented approaches open new avenues to automate the design process of metasurfaces and enable a much more generalized and broader scope of optimal designs that go beyond acoustics applications.

Introduction

Acoustic metasurfaces have enabled precise acoustic wave manipulation with strong potential that provides a new route to obtain unprecedented functionalities with compact geometric structure [START_REF] Assouar | Acoustic metasurfaces[END_REF][START_REF] Ma | Acoustic metamaterials: From local resonances to broad horizons[END_REF][START_REF] Qi | Acoustic energy harvesting based on multilateral metasurfaces[END_REF][START_REF] Xie | Wavefront modulation and subwavelength diffractive acoustics with an acoustic metasurface[END_REF]. Such advantages in precision and functionalities come at the cost of tremendous difficulty in finding individual metasurface structures based on specific criteria, which makes design and optimization of metasurface a key challenge in the acoustic field. Current design strategies include theoretical methods [START_REF] Assouar | Acoustic metamaterials for sound mitigation[END_REF][START_REF] Li | Metascreen-based acoustic passive phased array[END_REF][START_REF] Huang | Acoustic perfect absorbers via spiral metasurfaces with embedded apertures[END_REF][START_REF] Zhu | Ultrathin acoustic metasurface-based Schroeder diffuser[END_REF][START_REF] Zhu | Multifunctional acoustic metasurface based on an array of Helmholtz resonators[END_REF], numerical simulations [START_REF] Zhu | Ultrathin acoustic metasurface-based Schroeder diffuser[END_REF][START_REF] Zhu | Multifunctional acoustic metasurface based on an array of Helmholtz resonators[END_REF][START_REF] Li | Reflected wavefront manipulation based on ultrathin planar acoustic metasurfaces[END_REF][START_REF] Xie | Wavefront modulation and subwavelength diffractive acoustics with an acoustic metasurface[END_REF][START_REF] Li | Acoustic focusing by coiling up space[END_REF][START_REF] Fan | Tunable broadband reflective acoustic metasurface[END_REF], optimization algorithms for the inverse designs [START_REF] Cobo | Inverse estimation of the non-acoustical parameters of loose granular absorbers by Simulated Annealing[END_REF][START_REF] García-Chocano | Acoustic cloak for airborne sound by inverse design[END_REF][START_REF] Dong | Design optimization for structural-acoustic problems using FEA-BEA with adjoint variable method[END_REF][START_REF] Noguchi | An acoustic metasurface design for wave motion conversion of longitudinal waves to transverse waves using topology optimization[END_REF][START_REF] Noguchi | Level set-based topology optimization for graded acoustic metasurfaces using two-scale homogenization[END_REF][START_REF] Zhang | Realization of full and directional band gap design by non-gradient topology optimization in acoustic metamaterials[END_REF][START_REF] Noguchi | Topology optimization of acoustic metasurfaces by using a two-scale homogenization method[END_REF], and deep neural networks [START_REF] Ahmed | Deterministic and probabilistic deep learning models for inverse design of broadband acoustic cloak[END_REF][START_REF] Weng | Metaneural-network for real-time and passive deep-learning-based object recognition[END_REF][START_REF] Ciaburro | Artificial neural network-based models for predicting the sound absorption coefficient of electrospun poly (vinyl pyrrolidone)/silica composite[END_REF][START_REF] Zhao | Machine-learning optimized method for regional control of sound fields[END_REF][START_REF] Orazbayev | Far-field subwavelength acoustic imaging by deep learning[END_REF][START_REF] Luo | Probability-density-based deep learning paradigm for the fuzzy design of functional metastructures[END_REF][START_REF] Donda | Ultrathin Acoustic Absorbing Metasurface Based on Deep Learning Approach[END_REF][START_REF] Zhang | SAP-Net: Deep learning to predict sound absorption performance of metaporous materials[END_REF][START_REF] Ding | Deep learning enables accurate sound redistribution via nonlocal metasurfaces[END_REF][START_REF] Ivry | Nonlinear acoustic echo cancellation with deep learning[END_REF][START_REF] Tran | Acoustic cloak design via machine learning[END_REF][START_REF] Sun | Acoustic structure inverse design and optimization using deep learning[END_REF][START_REF] Lai | Conditional Wasserstein generative adversarial networks applied to acoustic metamaterial design[END_REF][START_REF] Cheng | Machine learning inversion design and application verification of a broadband acoustic filtering structure[END_REF][START_REF] Wu | A physicsconstrained deep learning based approach for acoustic inverse scattering problems[END_REF][START_REF] Gurbuz | Generative adversarial networks for the design of acoustic metamaterials[END_REF][START_REF] Gao | Inverse design and experimental verification of an acoustic sink based on machine learning[END_REF][START_REF] Mahesh | Inverse design of a Helmholtz resonator based low-frequency acoustic absorber using deep neural network[END_REF]. Exploring designs beyond the ones that can be theoretically modeled without difficulties requires a cumbersome empirical trial and error method using simulation algorithms, usually based on the Finite Element Method (FEM). Starting from certain initial and boundary conditions, and by discretizing linearized Naiver-Stokes equations, such acoustic simulations solve the design problem. The accurate acoustic response can be calculated by setting up sufficient meshes and iterations. It is frequently needed to fine-tune the structure and perform simulations iteratively to gradually approach the desired responses. This process depends on the experience of the design models and it also takes a long time, especially for low-frequency three-dimensional (3D) calculations. This is due to the narrow bandwidth at low frequencies, which necessitates a high enough resolution to sweep the frequencies for accurate calculations. In addition, large computational times arise in thermoviscous acoustics due to the necessity to fine mesh the boundary layers where momentum and heat transfers occur. With the long simulation time and thus limited exploration of the design space, the optimal solution is often overlooked.

Inverse design problems are even more difficult because they require direct retrieval of the proper structure for the desired acoustic performance, which necessitates a much larger number of Degrees Of Freedom (DOFs) in the design space. Such designs are usually based on optimization methods, most commonly either gradient-based or evolutionary-based [START_REF] Cobo | Inverse estimation of the non-acoustical parameters of loose granular absorbers by Simulated Annealing[END_REF][START_REF] García-Chocano | Acoustic cloak for airborne sound by inverse design[END_REF][START_REF] Dong | Design optimization for structural-acoustic problems using FEA-BEA with adjoint variable method[END_REF][START_REF] Noguchi | An acoustic metasurface design for wave motion conversion of longitudinal waves to transverse waves using topology optimization[END_REF][START_REF] Noguchi | Level set-based topology optimization for graded acoustic metasurfaces using two-scale homogenization[END_REF][START_REF] Zhang | Realization of full and directional band gap design by non-gradient topology optimization in acoustic metamaterials[END_REF][START_REF] Noguchi | Topology optimization of acoustic metasurfaces by using a two-scale homogenization method[END_REF].

These approaches enable researchers to discover non-intuitive and freeform acoustic structures which outperform empirically designed structures. These algorithms, on the other hand, are driven by rules that have iterative searching steps in a case-by-case manner and frequently rely on numerical simulations to generate intermediate results that aid in the modification of the searching strategy. The converging speed using such an approach depends on the solver which is connected to the optimizer. As a result, modeling and characterization tools play an important role in all current design processes. In this context, precise and time-efficient design methods must be thoroughly investigated to design of next-generation acoustic metasurface devices, which are frequently characterized by large numbers of design DOFs and multifunctionalities.

Very recently, deep neural networks (DNNs) have been adopted as a radically new design tool to address forward and inverse acoustic problems. The great advantage of such learning-based models lies in their ability to model physical systems that fall outside of the training dataset.

Due to such generalization capability, DNN can be used as a forward design optimization engine. In the field of acoustics, the DNNs have been implemented using fully connected layers (FCLs) and convolution layers [START_REF] Ahmed | Deterministic and probabilistic deep learning models for inverse design of broadband acoustic cloak[END_REF][START_REF] Weng | Metaneural-network for real-time and passive deep-learning-based object recognition[END_REF][START_REF] Ciaburro | Artificial neural network-based models for predicting the sound absorption coefficient of electrospun poly (vinyl pyrrolidone)/silica composite[END_REF][START_REF] Zhao | Machine-learning optimized method for regional control of sound fields[END_REF][START_REF] Orazbayev | Far-field subwavelength acoustic imaging by deep learning[END_REF][START_REF] Luo | Probability-density-based deep learning paradigm for the fuzzy design of functional metastructures[END_REF][START_REF] Donda | Ultrathin Acoustic Absorbing Metasurface Based on Deep Learning Approach[END_REF][START_REF] Zhang | SAP-Net: Deep learning to predict sound absorption performance of metaporous materials[END_REF][START_REF] Ding | Deep learning enables accurate sound redistribution via nonlocal metasurfaces[END_REF]. Once properly trained with adequate data such deep learning models are very accurate and capable of generating desired acoustic responses on the millisecond timescale. In addition, various deep learning-based models are proposed to solve inverse acoustic problems [START_REF] Ivry | Nonlinear acoustic echo cancellation with deep learning[END_REF][START_REF] Tran | Acoustic cloak design via machine learning[END_REF][START_REF] Sun | Acoustic structure inverse design and optimization using deep learning[END_REF][START_REF] Lai | Conditional Wasserstein generative adversarial networks applied to acoustic metamaterial design[END_REF][START_REF] Cheng | Machine learning inversion design and application verification of a broadband acoustic filtering structure[END_REF][START_REF] Wu | A physicsconstrained deep learning based approach for acoustic inverse scattering problems[END_REF][START_REF] Gurbuz | Generative adversarial networks for the design of acoustic metamaterials[END_REF][START_REF] Gao | Inverse design and experimental verification of an acoustic sink based on machine learning[END_REF][START_REF] Mahesh | Inverse design of a Helmholtz resonator based low-frequency acoustic absorber using deep neural network[END_REF][START_REF] Kollmann | Deep learning for topology optimization of 2D metamaterials[END_REF][START_REF] Zhang | Accelerated topological design of metaporous materials of broadband sound absorption performance by generative adversarial networks[END_REF]. Despite such development in this field, there are still challenges with the existing networks specially built using FCLs. Such kind of networks primarily deal with simple acoustic structures with a limited number of parameters, severely limiting the accessible acoustic design DOFs and because of this, current deep learning models are insufficient as a general design scheme. Also, deep learning approaches require a large amount of training data to understand the latent patterns. For designing the complex acoustics devices, the generation of the training dataset is usually dependent on the numerical simulations which leads to a substantial computational burden. It is extremely challenging to generate highquality annotated data on a large scale using numerical simulations. The tradeoff between the size of the training dataset and the model accuracy is a critical factor to be considered. To solve this issue, we have implemented a data augmentation technique that not only increases the training dataset size but, also increases the diversity in the dataset which makes the network more robust [START_REF] Shorten | A survey on image data augmentation for deep learning[END_REF][START_REF] Perez | The effectiveness of data augmentation in image classification using deep learning[END_REF]. Furthermore, we have implemented the dimensionality reduction technique principal component analysis (PCA). It is a statistical method that allows to significantly lower the dimensionality of data without a significant loss of information in the process which helps to accelerate the learning process [START_REF] Bro | Main component analysis[END_REF].

The ultrathin design of acoustic metasurface absorbers for very low frequency remains a major topic in research. The prototypical designs are mostly based on educable guesses which may not always be sufficient to achieve full absorption at extremely low frequency [START_REF] Li | Acoustic metasurface-based perfect absorber with deep subwavelength thickness[END_REF][START_REF] Huang | Acoustic perfect absorbers via Helmholtz resonators with embedded apertures[END_REF][START_REF] Donda | Extreme lowfrequency ultrathin acoustic absorbing metasurface[END_REF][START_REF] Guo | A compact low-frequency soundabsorbing metasurface constructed by resonator with embedded spiral neck[END_REF][START_REF] Kong | Lowfrequency multi-order acoustic absorber based on spiral metasurface[END_REF][START_REF] Shen | Low-frequency anechoic metasurface based on coiled channel of gradient cross-section[END_REF]. In addition, such designs are explored for the normal wave incidence, whereas very little attention has been paid to the oblique incidence conditions [START_REF] Wu | Low-frequency tunable acoustic absorber based on split tube resonators[END_REF][START_REF] Ji | Low-frequency broadband acoustic metasurface absorbing panels[END_REF][START_REF] Liang | A controllable lowfrequency broadband sound absorbing metasurface[END_REF][START_REF] Xu | Tunable low-frequency and broadband acoustic metamaterial absorber[END_REF]. In this work, we present forward and inverse deep learning models for modeling and designing metasurface absorbers, which address the issues and limitations mentioned earlier. To expand the design space, our approach considers a large number of design DOFs and accounts for the two-dimensional (2D) geometrical pattern with complex freeform propagation channels, as well as properties like the thickness of the structure, and angle of incidence. Once fully trained with a sufficient amount of data, the proposed forward prediction network can generate accurate absorption coefficients of complex metasurface absorber structures for the given frequency. Practical applications of the metasurface absorber demand acoustic absorption effect at a particular frequency, which needs to realize design parameters that produce the customer-defined spectral response defined at the input. The goal of inverse design is to generate a working structure directly from the desired acoustic responses, eliminating the lengthy parameter scans or trial-and-error methods.

Due to the extensive amount of design DOFs in non-intuitive metasurface patterns, the design scheme using CNN, such as backpropagation in trained simulator networks, is ineffective in the inverse design of metasurface. Furthermore, the trained CNN will always produce a fixed result for a given input. Meanwhile, multiple solutions may exist for the same target spectra fed to the simulator. To overcome these challenges, we implement a Conditional Generative Adversarial

Network (CGAN). It is a conditional version of the generative adversarial network (GAN)

whose generator and discriminator are conditioned during the training by using some additional information to direct the data generation process [START_REF] Mirza | Conditional generative adversarial nets[END_REF].

Using the proposed DNN methods, we demonstrate a metasurface absorber at 82Hz with nearly omnidirectional performance with a thickness of λ/64. The performance of the resulting metasurface absorber prototypes confirms that the implemented CNN and CGAN architectures achieves two important goals in the field of acoustic designs: (1) fast and accurate performance evaluation of complex structures for the oblique wave incidence at very low frequencies; and

(2) finding non-intuitive designs based on predetermined acoustic response requirements. It is envisioned that the proposed approach also validates the feasibility of object-driven 3D acoustic device design which can be easily extended and generalized to many other acoustic device designs.

Structure Design of Acoustic Metasurface Absorber

The metasurface under evaluation is composed of a resonant cavity consist of freeform of propagation channel and cylindrical aperture covered by a plate of thickness t=1 mm as shown in Figure 1(a). The embedded aperture is appended to the opening of the propagation channel.

Such aperture and the propagation channel provide a feasible approach to tune the acoustic impedance powerfully, thereby exhibiting superior capability and tunability for achieving desired absorption performances. The cell size is fixed at a=58 mm and the thickness of the outer wall of the cavity is 2mm and thus, the lateral dimension of the square cavity is 56mm.

The whole area of the square cavity is then decomposed into the square lattice sites of 2mm×2mm to construct the propagation channel made of PLA (density, ρ = 2700 kg m -3 ) with a thickness of 2mm. The absorption responses of these structures were then calculated with the preset thermoviscous module of COMSOL Multiphysics v5.6. Hard boundary conditions are imposed on the airstructure interface due to a huge impedance mismatch. Over 9000 metasurface structures with different propagation channel shapes, diameter of hole, incidence angle, and thickness were generated within the following ranges (all dimensions are in mm and angles are in degree): thickness, hϵ [65,75], incidence angle 𝜃ϵ[0,60], diameter of hole, dϵ [START_REF] Li | Reflected wavefront manipulation based on ultrathin planar acoustic metasurfaces[END_REF][START_REF] Li | Acoustic focusing by coiling up space[END_REF] since this range includes ample samples generating absorption resonance for the given frequency range. The height of the circular aperture is selected as (h-2) mm to take full advantage of the resonance to achieve a lower frequency absorption. The spectra of interest are chosen as 80-100Hz for the purpose of demonstration. The whole spectrum is down sampled into 201 frequency points with a frequency step of 0.1Hz as the bandwidth of resonance is very narrow at a very low frequency.

Here, the spectral response for the normal incidence is not influenced by the rotation of the structure. Hence, data augmentation is used by rotating 2D images by the random rotation and keeping their spectral response constant to generate more training data. It reduces the probability of the network overfitting the input data and also, increase the quantity and diversity in the training data. Before feeding to the network, the image data, associated properties, and corresponding acoustic response are cleaned and preprocessed (more details are included in the Supplementary Information, Section I).

Forward network architecture and results

We first construct a predicting neural network (PNN) based on a 2D CNN architecture for the forward design. The PNN aims to reveal the hidden relationship between the metasurface absorbers and their spectral response and thus accurately predict absorption responses for the given metasurface design. The proposed approach is not limited to particular shapes of the propogation channel and normal incidence only. We parameterized each metasurface structure's top view (2D pattern that includes the propagation channel and hole diameter), thickness, and angle of incidence. Later, they are concatenated as an input to the PNN. For feeding, the input data to the PNN, a batch generator is used that considerably lowers the RAM requirements. For any data, it randomly picks n rows (n is the batch size) and preprocesses them following which the batch input is fed to the network. As compared to total variables for our case, the training dataset size is small, and hence to help neural networks learn the patterns more easily, PCA is implemented using the 'scikit-learn' library. For our data, we use the first 20 components which explained 96.5% variance thus enabling us to reduce the dimensionality of the target absorption spectra from 201 to 20, i.e. 90% reduction.

Figure 2-Illustration of the predicting network architecture (PNN) for the metasurface absorber design for oblique wave incidence. The input structure is divided into 1D properties and 64×64 pixels 2D images. 1D properties are added to the 2D images as additional channels and both are processed through the same network. The output of the last max-pool layer, M3 is followed by a batch normalization layer before flattening. After being processed with two dense layers (D1 and output), the absorption spectra are ready for evaluation.

In prior work [START_REF] An | Deep learning modeling approach for metasurfaces with high degrees of freedom[END_REF], good results were obtained by processing both 2D images and 1D properties (dimensions of the structure) using the separate networks and later concatenating and processing through the rest of the network. However, in our case, we find that the network shows overfitting when processing both properties separately. Therefore, we proposed a PNN that processes both in the same network by adding 1D properties (thickness and angle of incidence) into 2D binary image composed of 64×64 pixels as additional channels. The combined input is then fed to the PNN as illustrated in Figure 2. The input is passed sequentially through three combinations of convolution and max pooling layers (C1-M1-C2-M2-C3-M3), each convolution layer having a kernel size of 3 and max pooling having the pool size of 2. The total number of filters used in the convolution layers are 16, 32, and 64 respectively. The output of layer M3 is passed through the batch normalization layer. Here, the aim of batch normalization is to accelerate the training process by reducing the internal covariate shift. Output of the batch normalization is passed through the flattening layer, F1 to flatten the output into single long feature vector. Before, feeding to the dense layers, a dropout is implemented to reduce the network overfitting issue. The output is than passed through a dense layer, D1 having 512 neurons followed by the output layer 20 neurons corresponding to components post PCA. At the output, we applied Rectified nonlinear activation function (ReLU) to introduce the nonlinearity in the network.

For the training process, the 9000 generated metasurface structures are split into training, validation, and test datasets, with 88% are used during the training process, 10% for the validation, and remaining 2% are used to evaluate the trained network. Figure 3

𝑀𝑆𝐸 = 1 ! (𝛼 " -𝛼 # ) 2 (1)
where, 𝛼 " and 𝛼 # are the predicted and simulated absorption coefficients, respectively. When the training was finished, the average MSE was 0.00904 for the predicted absorption coefficients in the validation data. During the training process, the weights are continuously optimized to minimize the validation loss (the hyperparameters are included in the Supplementary Information, Section II). We then calculate the predicted spectra and compare the results to numerical simulations to extract the prediction error. To justify the usage of the batch normalization layer, and flatten layer in the PNN, we also carry out an ablation analysis (Supplementary Information, Section III). Results from this study indicates that including these layers improves the speed of convergence and accuracy.

To check the prediction accuracy of the trained model, we have evaluated it on the test dataset. . The total thickness of the whole structure is as small as 1/64 of the working wavelength, which is much smaller than the previously reported acoustic absorbing metasurface designs for the oblique wave incidences [START_REF] Ji | Low-frequency broadband acoustic metasurface absorbing panels[END_REF][START_REF] Liang | A controllable lowfrequency broadband sound absorbing metasurface[END_REF]. The absorption for both structures is about 86% at θ=60° which shows overall good performance at large angles incidence.

Inverse Network Architecture and Results

Practical applications of the metasurface absorber demand the perfect absorption at a particular frequency or frequency range which needs to generate specific design parameters to produce the desired absorption response using the inverse design. Once properly trained, the inverse network should be able to excavate the best possible metasurface absorber design that yields absorption spectra with minimum deviation from the input absorption spectra. The development of such inverse design using the deep neural networks significantly reduces the computational time for design optimization.

To realize the inverse design, we have constructed the network architecture based on CGAN as shown in Figure 4(a). The aim of this network is to predict the metasurface structure for the prescribed absorption spectra and the thickness for the given incidence angle, θ. The network is divided into two parts: a generator (G) and a discriminator (D). Both are convolutional neural networks with small differences in their detailed architecture. The generator is conditioned to produce images of new structures as a function of angle of incidence, thickness, and desired absorption spectra. The inputs to the generator are the incidence angle θ, thickness h, desired absorption spectra A, and latent input space, z which comprises the number (latent size corresponding to the 2D pattern) of randomly chosen values between -1 to 1. Before feeding to the generator, the inputs are concatenated and passed through the dense layer for reshaping. The detailed architecture of the generator is shown in Figure 4(b). The reshaped input is passed through four transpose convolution layers CT1, CT2, CT3, CT4 with varying number of filters 128,32,16,1 and strides of 2,2,1,1 respectively. Such layers can be understood as the opposite of a pooling layer i.e. it increases the dimensions of the input and performs convolution after that to learn appropriate weights. Nonlinear activation function ReLU is applied after each the output of layers CT1, CT2, CT3, and sigmoid activation function is applied at the output of the layer CT4. A sigmoid activation is appropriate at the output of the layer as we want the discriminator's output to be 1 or 0. As a demonstration of the proposed CGAN framework's overall competence, we use the absorption spectra, A, and other properties; thickness h, and angle of incidence θ, randomly selected from the test data as input and ask the network to look for proper metasurface patterns based on these spectra. We denote this test set as s, the absorption spectra of each s as A. Once these spectra and other properties passed through the network, a 2D pattern is retrieved which is denoted by s′. To verify the result, we simulate the generated set s′ in COMSOL Multiphysics and denote the output spectra as A′. In Figure 5

Conclusion

In this research, a deep neural network based metasurface absorber designing approach for the oblique wave incidence is proposed and demonstrated. In particular, we have implemented CNN and CGAN for accurate forward and inverse design, respectively. Compared to the existing works, our approach can handle a large set of input parameters, including 2D patterns that contain freeform propagation channel and hole diameter, and other properties like thickness and angle of incidence for the complex metasurface absorber structure. Using the presented CNN approach, we have conceived a metasurface absorber having nearly omnidirectional absorption at 82Hz with a thickness of 𝜆/64. We have further implemented CGAN which can be adapted for designing metasurfaces with prescribed absorption spectra. The proposed deep learning approaches can be generically used for fast and accurate modeling and design process of the complex physical systems with minimum human intervention. We envision widespread and increasing use of deep learning methods in the field of physics, allowing scientists and engineers to focus more on truly creative ideas to solve the complex design problems that have yet to be explored by the machine, rather than on tedious trial and error processes.

Figure 1 -

 1 Figure 1-(a) Schematic of metasurface absorber (cross-area=a×a and whole thickness h+t) for the oblique angle incidence. The full system includes a resonant cavity consisting of a freeform propagation channel (thickness h) and cylindrical aperture (diameter d). The cavity is covered by a plate with a centered hole (diameter d, thickness t). (b) Top view of the system generated from the simulation software which is used as an input to the deep learning network.

  (a) shows the learning curves (for the training and validation data), as a function of epochs. During the training phase, both the training and validation errors decreased and converged after 500 epochs. The network performance is quantified in term of mean squared error (MSE) which is given by,

Figure 3 (

 3 Figure 3 (b)-(g) displays the example of some best fits in order to show the capability of the network. The results are shown for two different metasurface structures with different shapes of the propagation channel (inset in the plot (b) and (e)) with three different angles of incidence.For each case, the mean square error is listed in the inset. As indicated by the MSE, the absorption spectra predicted by the PNN (red dashed curves) agreed well with the FEM simulations (blue curves) for the variety of different propagation channels and different incidence angles. From the figures 3(b) to 3(d), it can be observed that the absorption peak is obtained at a very low frequency of 82Hz for the metasurface structure shown in the inset of the plot (b). The total thickness of the whole structure is as small as 1/64 of the working

Figure 3 -

 3 Figure 3 -(a) Learning curves as a function of epochs. (b)-(g) Comparison of PNN predicted absorption spectra (red dashed curves) with the simulated one (blue curves) for two different example metasurface structures with different shapes of the propagation channel (inset in the plot (b) and (e)). For each example, the comparison is shown for 0°, 30°, and 60°. MSE of each case is included as insets.

Figure 4 -

 4 Figure 4-Illustration of the conditional generative adversarial network for the inverse metasurface absorber design for oblique wave incidence. (a) Schematic of the implemented network. (b) Detailed structure of the Generator (G). (c) Detailed structure of the Discriminator (D).

  (a), the first row shows the sample from the test data, s and the second row show the corresponding samples s′ generated by the CGAN. Each pair in two rows matches very well. Figure 5(b)-(e) show some best fits in order to demonstrates the capability of the network. Target designs of various metasurface absorbers and the corresponding suggested designs show good agreement.

Figure 5 :

 5 Figure 5: (a) 2D patterns s from the test dataset are shown in the top row and the corresponding generated patterns by CGAN are shown in the second row. (b) Comparison of the absorption spectra, A from the test set (black curve) and FEM simulated absorption spectra, A′ of the retrieved pattern s′ shown in the second row.

Acknowledgement

This work was supported by the Air Force Office of Scientific Research under Award No. FA9550-18-17021. The authors also acknowledge the support from la Region Grand Est.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.


Sheng Wan

Liyun Cao

Yifan Zhu

Mourad Oudich
email: mourad.oudich@univ-lorraine.fr

Badreddine Assouar
email: badreddine.assouar@univ-lorraine.fr

Nonreciprocal sound propagation via cascaded time modulated slab resonators

Keywords: Acoustic nonreciprocity, Temporal modulation, Acoustic metamaterials, Frequency conversion

In this work, we demonstrate nonreciprocal acoustic wave transmission in a two cascaded Fabry-Perot like slab resonators undergoing time modulation of their effective density. Phase difference is introduced in the time modulation between the two coupled resonators to produce spatial bias in order to access unidirectional wave propagation. A theoretical model based on plane wave expansion and transfer matrix method is developed to study the Fabry-Perot-based system. The theoretical model allows rapid and precise characterization of the acoustic dispersion, with results in excellent agreement with finite element based numerical simulations. We demonstrate acoustic nonreciprocity behavior for the fundamental acoustic mode, and further show that with proper optimization, the coupled Fabry-Perot based device can achieve nonreciprocal acoustic frequency conversion. This work could inspire designing simple and compact nonreciprocal acoustic devices for efficient wave control.

I. Introduction

During the last decade, a growing area of research is concerned with the possibility of breaking the reciprocity for waves to realize unidirectional wave propagation. Particularly, an acoustic nonreciprocal device, that enables one-way transmission of sound, is highly desirable for numerous technological applications such as unidirectional sound barrier, improved medical ultrasound devices, and SAW based communication systems [START_REF] Fleury | Nonreciprocal Acoustics[END_REF][START_REF] Rasmussen | Acoustic Nonreciprocity[END_REF]. Reciprocity is a fundamental property of wave phenomena in any linear time-invariant system [START_REF] Casimir | On Onsager's Principle of Microscopic Reversibility[END_REF][START_REF] Strutt | Some General Theorems Relating to Vibrations[END_REF]. It can only be broken using proper biasing of space or time quantities, meticulously introduced into the propagation medium. Various systems were constructed to achieve acoustic nonreciprocity and demonstrate different wave behaviors in opposite propagation directions. Early attempts for an acoustic nonreciprocity realization were successfully conducted by combining nonlinear medium with a phononic crystal (PC) [START_REF] Liang | Acoustic Diode: Rectification of Acoustic Energy Flux in One-Dimensional Systems[END_REF][START_REF] Liang | An Acoustic Rectifier[END_REF]. Later-on, geometrical design strategies were proposed to achieve asymmetric wave dispersion, though these systems do not break reciprocity but rather affect the waves packets path [START_REF] Li | Tunable Unidirectional Sound Propagation through a Sonic-Crystal-Based Acoustic Diode[END_REF][START_REF] Wang | Broadband Acoustic Diode by Using Two Structured Impedance-Matched Acoustic Metasurfaces[END_REF]. The P-T symmetry can also lead to asymmetric wave reflection by introducing loss and gain elements in a linear waveguide [START_REF] Wen | Asymmetric Frequency Conversion with Acoustic Non-Hermitian Space-Time Varying Metamaterial[END_REF].

An exotic way of achieving nonreciprocity is the application of space-time modulated (STM) mediums in order to break the time reversal symmetry. The idea was proposed in 1959 in photonics and electrical circuits, where permittivity, permeability or impedance is harmonically modulated both in space and time to create unidirectional band gaps for waves [START_REF] Li | Tunable Unidirectional Sound Propagation through a Sonic-Crystal-Based Acoustic Diode[END_REF][START_REF] Wang | Broadband Acoustic Diode by Using Two Structured Impedance-Matched Acoustic Metasurfaces[END_REF][START_REF] Wen | Asymmetric Frequency Conversion with Acoustic Non-Hermitian Space-Time Varying Metamaterial[END_REF]. The medium can be seen as keeping its intrinsic properties spatially modulated and moving at a certain velocity. This thus breaks the time reversal, allowing different dispersions of waves traveling in opposite directions along the same path. This route, however, faded until this last decade where researchers reinvigorated their interest on the realization of such space time modulated systems, all thanks to the technological progress in fabrication and characterization [START_REF] Hadad | Space-Time Gradient Metasurfaces[END_REF][START_REF] Taravati | Self-Biased Broadband Magnet-Free Linear Isolator Based on One-Way Space-Time Coherency[END_REF][START_REF] Taravati | Nonreciprocal Electromagnetic Scattering from a Periodically Space-Time Modulated Slab and Application to a Quasisonic Isolator[END_REF][START_REF] Taravati | Aperiodic Space-Time Modulation for Pure Frequency Mixing[END_REF][START_REF] Taravati | Giant Linear Nonreciprocity, Zero Reflection, and Zero Band Gap in Equilibrated Space-Time-Varying Media[END_REF][START_REF] Taravati | Programmable Nonreciprocal Meta-Prism[END_REF][START_REF] Taravati | Pure and Linear Frequency-Conversion Temporal Metasurface[END_REF]. Drawing inspiration from photonics, multiples STM based designs were proposed in acoustics [START_REF] Fleury | Subwavelength Ultrasonic Circulator Based on Spatiotemporal Modulation[END_REF][START_REF] Shen | Nonreciprocal Acoustic Transmission in Space-Time Modulated Coupled Resonators[END_REF][START_REF] Zhu | Non-Reciprocal Acoustic Transmission via Space-Time Modulated Membranes[END_REF][START_REF] Oudich | Space-Time Phononic Crystals with Anomalous Topological Edge States[END_REF] and elastodynamics [START_REF] Trainiti | Non-Reciprocal Elastic Wave Propagation in Spatiotemporal Periodic Structures[END_REF][START_REF] Wang | Observation of Nonreciprocal Wave Propagation in a Dynamic Phononic Lattice[END_REF][START_REF] Trainiti | Time-Periodic Stiffness Modulation in Elastic Metamaterials for Selective Wave Filtering: Theory and Experiment[END_REF][START_REF] Yi | Active Metamaterials with Broadband Controllable Stiffness for Tunable Band Gaps and Non-Reciprocal Wave Propagation[END_REF][START_REF] Sugino | Nonreciprocal Piezoelectric Metamaterial Framework and Circuit Strategies[END_REF][START_REF] Farhat | Spacetime Modulation in Floating Thin Elastic Plates[END_REF]. In those designs, the mechanical properties of the designed mediums where varied both in space and time to achieve nonreciprocity. Besides, acoustic circulator [START_REF] Fleury | Sound Isolation and Giant Linear Nonreciprocity in a Compact Acoustic Circulator[END_REF] was proposed as a nonreciprocal device, which contains circulating flow insides three ports to break the time reversal.

However, in most of the works applied to classical waves and especially in acoustics, the physics of nonreciprocity is based on shifting the Bragg bandgap in the frequency domain for opposite directions to prohibit wave propagation unidirectionally [START_REF] Trainiti | Non-Reciprocal Elastic Wave Propagation in Spatiotemporal Periodic Structures[END_REF][START_REF] Wang | Observation of Nonreciprocal Wave Propagation in a Dynamic Phononic Lattice[END_REF][START_REF] Trainiti | Time-Periodic Stiffness Modulation in Elastic Metamaterials for Selective Wave Filtering: Theory and Experiment[END_REF]. This mechanism was achieved by spatiotemporally modulating the effective parameters of the constituent materials (stiffness) along several periods of the structure (at least 10 periods). Consequently, the periodic parts of the system have to be precisely synchronized which makes the experimental realization quite challenging, and thus limit the device practicability. For instance, in acoustics, although dynamically changing the effective density of air is locally possible [START_REF] Chen | A Tunable Acoustic Metamaterial with Double-Negativity Driven by Electromagnets[END_REF], a precise harmonic modulation in both space and time along several periods remains challenging. In contrast, nonreciprocity based on STM becomes relatively accessible in elastodynamics by modulating the effective stiffness of the material [START_REF] Trainiti | Time-Periodic Stiffness Modulation in Elastic Metamaterials for Selective Wave Filtering: Theory and Experiment[END_REF][START_REF] Yi | Active Metamaterials with Broadband Controllable Stiffness for Tunable Band Gaps and Non-Reciprocal Wave Propagation[END_REF][START_REF] Sugino | Nonreciprocal Piezoelectric Metamaterial Framework and Circuit Strategies[END_REF]. However, this process involves introducing piezoelectric elements connected to electrical circuits to introduce effective negative capacitance with the complexity of synchronizing the modulations. The whole nonreciprocal device becomes cumbersome, limiting practical applications in the future. Besides, systems such as coupled modulated Helmholtz resonators [START_REF] Shen | Nonreciprocal Acoustic Transmission in Cascaded Resonators via Spatiotemporal Modulation[END_REF], air cavities [START_REF] Shen | Nonreciprocal Acoustic Transmission in Space-Time Modulated Coupled Resonators[END_REF], membranes [START_REF] Zhu | Non-Reciprocal Acoustic Transmission via Space-Time Modulated Membranes[END_REF], and even Fabry-Perot based photonic slabs [START_REF] Chegnizadeh | Non-Reciprocity Using Quadrature-Phase Time-Varying Slab Resonators[END_REF] inspired us for a simple design compared to the aforementioned systems. For instance, Shen et al. [START_REF] Shen | Nonreciprocal Acoustic Transmission in Cascaded Resonators via Spatiotemporal Modulation[END_REF] showed that using two cascaded time-modulated Helmholtz resonators (HR), nonreciprocal wave behavior can be realized via frequency conversion with high order modes appearing differently for positive and negative propagation. However, nonreciprocity is achieved far from the resonance frequency of the HR as the wave is completely reflected at this frequency. Under this restriction, the fundamental mode with the frequency of the incident wave is always accompanied with high order modes in the transmission whatever the direction of propagation is. Nevertheless, the choice of cascaded slabs is more advantageous since it allows high wave transmission at the Fabry-Perot resonance. Consequently, the slabs-based-system has the potential to create the situation of unidirectional wave conversion for instance, hence offering more control over nonreciprocity via mode conversion.

In this work, we demonstrated nonreciprocal acoustic transmission in a finite system based on coupled Fabry-Perot resonators with time modulation of their effective properties. A good nonreciprocity is achieved in our system with a quasi-lossless transmission in one direction and a nearly-zero transmission of the fundamental mode in opposite direction. In contrast to the previous works that used Bragg band gap combined with modulating the effective parameters of the materials along several periods of the structure, our system is composed of two coupled fluid slabs that behave as coupled Fabry-Perot (FP) resonators whose density, however, is temporally modulated. A spatial bias is introduced to realize nonreciprocal wave propagation by forcing a phase difference on the time modulated density between the two slabs. This kind of modulation could be easily achieved by periodically compressing the fluid for instance [START_REF] Chen | A Tunable Acoustic Metamaterial with Double-Negativity Driven by Electromagnets[END_REF][START_REF] Wang | Acoustic Asymmetric Transmission Based on Time-Dependent Dynamical Scattering[END_REF]. In such design, the physical mechanism is mainly based on frequency conversion where the acoustic energy is transferred from the fundamental mode into higher order ones [START_REF] Jeon | Nonlinear Acoustic Metamaterial for Efficient Frequency Down-Conversion[END_REF][START_REF] Schneider | Frequency Conversion through Nonlinear Mixing in Acoustic Waves[END_REF].

The plane wave expansion (PWE) method [START_REF] Riva | Generalized Plane Wave Expansion Method for Non-Reciprocal Discretely Modulated Waveguides[END_REF] is first adopted to construct the scattering matrix describing the system, which analytically solves the acoustic transmission. The analytical result matches well with the finite-element-method (FEM) simulation results. Then, we took advantage of this model to seek for parameters and configurations that yield a good nonreciprocity. Furthermore, under the premise of high nonreciprocity, we also discussed the possibility of unidirectional frequency conversion. The latter is an important wave phenomenon that enables generating waves at desired frequencies from an incident wave at different frequencies. Recently, the application of frequency conversion is expanded to acoustic domain, such as directional loudspeakers and nondestructive evaluation [START_REF] Westervelt | Parametric Acoustic Array[END_REF][START_REF] Moussatov | Frequency Up-Conversion and Frequency down-Conversion of Acoustic Waves in Damaged Materials[END_REF]. We first considered an adiabatic single slab resonator as shown in Fig. 1. The background medium is air, whose density is 𝜌 ! and sound speed is 𝑐 ! , with bulk modulus 𝜅 ! = 𝜌 ! 𝑐 ! " . The thickness of the slab resonator is 𝑑. The medium of the slab resonator is a fluid whose static density is higher than air. In this model, we assume that the density of the slab can be modulated in time:

II. Design and theory

𝜌 # = 𝛿𝜌 ! [1 + 2 • 𝑀 𝑐𝑜𝑠(Ω𝑡 + 𝜙)].
As we consider the bulk modulus to be fixed throughout all this study (𝜅 ! = 141.2𝑘𝑃𝑎), the speed of sound in the slab resonator is consequently modulated as

𝑐 # = :𝜅 ! 𝜌 # ⁄ which yields 𝑐 # = <𝜅 ! =𝛿𝜌 ! (1 + 2 • 𝑀 𝑐𝑜𝑠(Ω𝑡 + 𝜙))> ⁄ . Without temporal modulation,
the static speed of sound in the slab resonator is

𝑐 # $ = 𝑐 ! √𝛿 ⁄ .
In practice, this kind of modulation can for instance be achieved by driving a piston with rotating elements to induce changing compression in time. Here, 𝛿 is the static density ratio, 𝑀 is the modulation depth, Ω is the angular frequency of modulation, and 𝜙 is the initial phase of the modulation. Considering a forward incident wave at an angular frequency 𝜔 , and using the Floquet-Bloch theorem, the acoustic pressure and the particle velocity fields in the slab yield,

𝑝(𝑥, 𝑡) = D 𝑝 % (𝑥)𝑒 &(()%*),
).

%/0.

(

) v(𝑥, 𝑡) = D 𝑣 % (𝑥)𝑒 &(()%*), 1
).

%/0.

(

) 2
where 𝑝 % (𝑥) and 𝑣 % (𝑥) are the Fourier components of the pressure and velocity, respectively, in the time domain for both forward and backward propagating waves. The time modulated density can be rewritten in the same way,

𝜌 # (𝑡) = 𝛿𝜌 ! D 𝜌 H % 𝑒 &1*, 𝑒 &2 !
).

1/0.

(

) 3
According to the adopted harmonic modulation for 𝜌 # , we have here 𝜌 H ! = 1, 𝜌 H ±# = 𝑀 , and 𝜌 H 1 = 0 for 𝑚 ≠ 0, ±1, and 𝜙 ±# = ±𝜙. By substituting Eqs. ( 2) and ( 3) into the known acoustic equation 𝜕 4 𝑝 = -𝜌(𝑡)𝜕 , 𝑣, we get the following relation,

(b) (a) 𝜕𝑝(𝑥, 𝑡) 𝜕𝑥 = -𝛿𝜌 ! D 𝜌 H 1 𝑒 &1*, 𝑒 &2 !
).

1/0.

D 𝑗(𝜔 + 𝑛 5 Ω)𝑣 % " (𝑥)

).

% " /0.

𝑒 &6()% " *7, (4)
Then we develop Eq. ( 4) by combining the two summations,

𝜕𝑝(𝑥, 𝑡) 𝜕𝑥 = -𝑗𝛿𝜌 ! D (𝜔 + 𝑛 5 Ω)𝜌 H 1 𝑣 % " (𝑥)𝑒 &8()6% " )17*9, 𝑒 &2 # % " ,1 (5)
By eliminating the index script 𝑚 by replacing it with 𝑛 -𝑛 5 , we can develop Eq. ( 5) into the following equation,

𝜕𝑝(𝑥, 𝑡) 𝜕𝑥 = -𝑗𝛿𝜌 ! D(𝜔 + 𝑛 5 Ω)𝜌 H %0% " 𝑣 % " (𝑥)𝑒 &2 !$! " 𝑒 &(()%*), % " ,% (6)
By substituting Eq. ( 1) into Eq. ( 6) and using the orthogonality of =𝑒 &%*, > % , we have,

𝜕𝑝 % (𝑥) 𝜕𝑥 = -𝑗𝛿𝜌 ! D(𝜔 + 𝑛 5 Ω)𝜌 H %0% " 𝑒 &2 !$! " 𝑣 % " (𝑥) % " (7)
The time dependent term 𝑒 &(, is eliminated.

Substituting Eqs. ( 1), ( 2) and ( 3) into the known acoustic equation 𝜕 4 𝑣 = -𝜅 ! 0# 𝜕 , 𝑝 and following the same steps as above, we get,

𝜕𝑣 % (𝑥) 𝜕𝑥 = -𝑗(𝜔 + 𝑛Ω) 1 𝜅 ! 𝑝 % (𝑥) (8)
After further developments by defining a new variable 𝜉 = 𝑥Ω 𝑐 # $ ⁄ , Eq. ( 7) and ( 8) become,

𝜕𝑝 % (𝜉) 𝜕𝜉 = -𝑗𝑍 # $ D Y 𝜔 Ω + 𝑛 5 Z 𝜌 H %0% " 𝑒 &2 !$! " 𝑣 % " (𝜉) % " (9)
and

𝜕𝑣 % (𝜉) 𝜕𝜉 = -𝑗 1 𝑍 # $ Y 𝜔 Ω + 𝑛Z 𝑝 % (𝜉) ( 10
)
where

𝑍 # $ = 𝛿𝜌 ! 𝑐 # $ is the static impedance in the slab.
Thus, a series of first order partial differential equations is constructed based on Eqs. ( 9) and [START_REF] Oliner | Guided Waves on Sinusoidally-Modulated Reactance Surfaces[END_REF], which depends on the number of decompositions 𝑁 we adopt to approximate the solution,

𝑨 ; • ^𝒑< 𝒗 < a ; -𝑰 • 𝜕 = ^𝒑< 𝒗 < a ; = 0 ( 11
)
where

𝒑 < = ⟨𝑝 0< , … , 𝑝 ! , … , 𝑝 < ⟩ > , 𝒗 < = ⟨𝑣 0< , … , 𝑣 ! , … , 𝑣 < ⟩ > , 𝑨 ; = ^0 𝑨 ; ? 𝑨 ; @ 0
a, I is the identity matrix and the subscript 𝑠 indicates the domain index. For example, in Fig. 2, 𝑠 = 1 refers to the slab resonator and 𝑠 = 2 refers to the next air medium in contact. 𝑨 ; ? and 𝑨 ; @ are both square matrices with size 2𝑁 + 1, 𝑨 ;,(%,% " ) ?

= -𝑗𝑍 # $ Y 𝜔 Ω + 𝑛 5 Z 𝜌 H %0% " 𝑒 &2 !$! " (12)
𝑨 ;,(%,% " )

@ = -𝑗 1 𝑍 # $ Y 𝜔 Ω + 𝑛Z 𝛿 %0% " (13)
where 𝛿 %0% " is the Kronecker function. As 𝑨 ; is function of 𝜉, Eqs. ( 11) have a solution in this form,

^𝒑< 𝒗 < a ; = D 𝑪 ;,A ) 𝑽 ;,A ) 𝑒 0B %,' ( (=(4)0= ) ) )< A/0< + D 𝑪 ;,A 0 𝑽 ;,A 0 𝑒 0B %,' $ (=(4)0= ) ) )< A/0< (14)
where 𝑽 ;,A ± and 𝜆 ;,A ± are the 𝑖 ,C eigenvector and eigenvalue associated to the matrix 𝑨 ; respectively, corresponding to forward/backward wave propagation associated with the layer 𝑠 , 𝑪 ;,A ± are unknown coefficients which correspond to wave amplitudes. By considering the boundary conditions (continuity of pressure and velocity) between the layers in contact labeled s and s+1, and performing some algebraic manipulations, we can easily obtain the transfer matrix 𝑴 ;→;)# which is defined by,

^𝑪;)#

) 𝑪 ;)# 0 a = 𝑴 ;→;)# ^𝑪;

)

𝑪 ; 0 a ( 15
)
And the definition of 𝑪 ; ± is explained in Fig. 2. 𝑴 ;→;)# can be developed from equation ( 14) and the boundary conditions,

𝑴 𝒔→𝒔)𝟏 = ⟨𝑽 ;)# ) |𝑽 ;)# 0 ⟩ 0# ⟨𝑽 ; ) |𝑽 ; 0 ⟩ ^𝜢; 𝑂 𝑂 𝜢 ; 0# a ( 16
)
where 𝜢 ; = ^𝜢;

) 𝑂 𝑂 𝜢 ; 0 a , 𝜢 ;,(%,% ′ ) ± = 𝑒 0B %,!$*$+ ± -% . / % 𝛿 %0% ′ and 𝑽 ; ± = 〈𝑽 ;,0< ± , … , 𝑽 ;,! ± , … , 𝑽 ;,< ± 〉 . 𝑙 ; is
the thickness of layer 𝑠, 𝑐 ; is the speed of sound in layer s. From Eq. ( 15), we constructed the transfer matrix M describing the complete system presented in Fig. 2,

𝑴 = q 𝑴 (G0A)→(G)#0A) G A (17)
The above solution is based on PWE and the precision of the solution is related to the considered number 𝑁 of plane waves. In our work, the convergence of the result is guaranteed with 𝑁 ≥ 4.

Finally, we obtained the scattering matrix 𝑺 from the transfer matrix 𝑴 (see Appendix 1).

Using the vector describing the incident wave of positive direction

𝑱 ) = ⟨𝛿 < , … , 𝛿 ! , … , 𝛿 0< |𝑶 < ⟩ > ,
we can deduce the corresponding reflection and transmission coefficients, 𝛿 % is Kronecker function,

v 𝒓 <,) 𝒕 <,) y = 𝑺 • 𝑱 ) (9)
where

𝒓 <,) = 〈𝑟 < ) , … , 𝑟 ! ) , … , 𝑟 0< ) 〉 > and 𝒕 <,) = 〈𝑡 < ) , … , 𝑡 ! ) , … , 𝑡 0< ) 〉 > .
Similarly, the reflection and transmission coefficients for negative incidence can be obtained using the vector associated to negative incident wave:

𝑱 0 = ⟨𝑶 < |𝛿 0< , … , 𝛿 ! , … , 𝛿 < ⟩ > , v 𝒕 <,0 𝒓 <,0 y = 𝑺 • 𝑱 0 (19)
where 𝒕 <,0 = 〈𝑡 < 0 , … , 𝑡 ! 0 , … , 𝑡 0< 0 〉 > and 𝒓 <,0 = 〈𝑟 < 0 , … , 𝑟 ! 0 , … , 𝑟 0< 0 〉 > are the corresponding transmission and reflection coefficients, respectively. These equations allowed us to evaluate the transmission and reflection coefficients for each frequency order. We have performed a series of calculations using our theoretical model to determine the optimized design that enables unidirectional propagation and frequency conversion. The involved parameters for our optimization process are the incident angular frequency 𝜔, the modulation frequency Ω , the static density ratio 𝛿 and the phase difference between the modulations of the slabs Δ𝜙 = 𝜙 # -𝜙 " .

III. Results and discussion

III.1 Single slab resonator

In the present study, the considered intrinsic parameters for air at ambient temperature are: 𝜌 ! = 1.2 𝑘𝑔 𝑚 H ⁄ and 𝑐 ! = 343 𝑚 𝑠 ⁄ . Since the acoustic equations are linear in this study, we considered normalized incident frequency 𝐹 A% = ω (2𝜋 𝑓 I $ ) ⁄ and normalized modulation frequency

𝐹 1 = Ω (2𝜋 𝑓 I $ ) ⁄
, where

𝑓 I $ = 𝑐 # $ (2𝑑 ⁄
) is the resonance frequency of the slab without temporal modulation. Figure 3 shows the transmission results for the case of 𝛿 = 5, 𝐹 1 = 1.12 and 𝑁 = 5. In such configuration, since the modulation rate is relatively low, the conversion from the fundamental mode to higher order harmonics (|𝑛| ≥ 3) can be omitted. So only 𝑛 = 0, ±1, ±2 are shown in the figure. Fig. 3(a) displays the transmission curves for the static case without modulation (black curve) and the transmission coefficients for the temporally modulated slab for orders 𝑖 = 0, ±1, ±2 as function of the incident frequency. We can see that under temporal modulation, part of the energy of the fundamental mode is converted to higher orders modes located at 𝐹 &* ± 𝑛𝐹 ) . It indicates that if the system is properly tuned, the frequency conversion could be stronger, then the nonreciprocity is possible with some spatial bias involved. Further, we confronted these results with numerical ones from simulations based on finite elements (FE) method via the commercial software COMSOL Multiphysics v5.5. The comparison is shown in Fig. 3(b) for the chosen case of the monochromatic incident wave with frequency 𝐹 A% = 10.5. Because of the existence of timevarying parameters, the FE simulations were performed in the time domain. The length of input signal has to be much longer than the modulation period 𝑇 1 = 2𝜋 Ω ⁄ in order to guarantee the stability and accuracy of the result. In Fig. 3(b), the response of the time modulated slab is analyzed via Fourier transform. Good agreement between theory (flat symbols) and simulation (blue solid lines) was observed.

III.2 Bilayer slab resonator

As shown in Fig. 1 (b), spatial bias is introduced into the system by adding a second time-modulated slab resonator with different initial phase. We consider the two slabs with the same thickness 𝑑, located at 𝑥 = 0 and 𝑥 = 𝐿 + 𝑑, with an air medium of length 𝐿 between them. The inner air medium can be seen as a resonator as well whose resonance frequency is 𝑓 J = 𝑐 ! 2𝐿 ⁄ . In order to achieve good nonreciprocity with this system, we tune the incident frequency 𝐹 A% , the modulation frequency Ω, the static density ratio 𝛿 and the phase difference Δ𝜙 between the modulations of the two slabs, within certain ranges. Our strategy to achieve nonreciprocity is based on two-level tuning. Firstly, we tune 𝐹 A% and 𝐹 1 with properly preselected Δ𝜙 and 𝛿. By analyzing the diagrams of transmittance asymmetry difference

!𝑇 0 + -𝑇 0 -! (with 𝑇 ! ) = |𝑡 ! ) | " and 𝑇 ! 0 = |𝑡 ! 0 | " ),
) ! " -(∑ 𝑇 𝑖 - 2 32
)

! " !, we were able to select a proper set of 𝐹 A% and 𝐹 1 . Afterwards, a fine optimization was performed by tuning Δ𝜙 and 𝛿 with fixed 𝐹 A% and 𝐹 1 . To acquire the nonreciprocity at a specific frequency, we can scale the slab dimension 𝑑 with the tuned parameters.

The coupling behavior of the resonators in static state (without modulation) has a significant effect on nonreciprocity. A definite static coupling behavior reduces variable elements in the tuning process. To characterize the coupling behavior, we introduce coupling factor 𝛼 = 𝑓 J 𝑓 I $ ⁄ , the ratio between the resonance frequency of the inner air waveguide 𝑓 J = 𝑐 ! 2𝐿 ⁄ and the static slab resonance frequency 𝑓 I $ . In what follows, we used the configuration of 𝛼 = 1 (see Appendix B).

The static density ratio 𝛿 determines the effective sound speed and the effective density in the slab medium, thus, its static impedance. Therefore, it affects directly the scattering behavior on the boundary between slab and air, because this density change leads to mismatched impedance on the boundary which impacts the wave reflection and transmission. As 𝛿 is larger, this scattering becomes stronger, which contributes to the wave asymmetry under spatial bias. However, large 𝛿 weakens the feasibility of the system. To be more realistic, we set 𝛿 = 5. The thickness of slab is set as 𝑑 = 0.05𝑚 and the length of the air medium 𝐿 is determined by 𝐿 = √𝛿𝑑 𝛼 ⁄ .

The density of the two cascaded slabs is time-modulated as

𝜌 # = 𝛿𝜌 ! [1 + 2 • 𝑀 cos(𝛺𝑡)] and 𝜌 " = 𝛿𝜌 ! [1 + 2 • 𝑀 cos(𝛺𝑡 + 𝛥𝜙)],
where the phase difference Δ𝜙 is taken into account. In what follows, we utilized 0.5𝜋 as the preset value of Δ𝜙 which leads to spatiotemporal bias and breaks the time reversal symmetry. When Δ𝜙 = 0, the system complies with time reversal symmetry as the bilayer slab resonator system is symmetric. When Δ𝜙 = 𝜋 , although the time reversal symmetry is broken, it only leads to phase inversion between the wave transmitted in opposite directions while the amplitudes are kept the same. As asymmetry appears in Δ𝜙 ∈ (0, 𝜋) while we get no asymmetry at both limits of this range, an extremum exists. Consequently, we have first chosen Δ𝜙 = 0.5𝜋 where we expect to have the maximum effect of asymmetry (see Appendix C).

The modulation depth 𝑀 is directly related to the external energy brought into the system actively, so practically we tend to adopt relatively small values for the modulation depth. A weak modulation not only enables the feasibility of the system, but also prevents the transmitted wave from being overamplified by the modulation. Here the modulation depth is chosen as 𝑀 = 0.06. Then, we first searched for nonreciprocal wave behavior by sweeping the incidence frequency 𝐹 &* and the modulation frequency 𝐹 ) . Here we are interested in the scattering properties of the fundamental mode. ) -𝑇 ! 0 | > 0.9 and |10 𝑙𝑜𝑔 #! (𝑇 ! ) /𝑇 ! 0 )| > 20𝑑𝐵 in a wide range of 𝐹 A% , which correspond to yellow zones that are wide on the 𝐹 A% axis in the As shown in Fig. 4(c), it is difficult to achieve high level asymmetry of total transmission in this system, so the nonreciprocity here is mainly due to frequency conversion. We observe that Fig. 4(a) and(b) show some periodicity on the 𝐹 A% axis. The nonreciprocity zones tend to appear at multiples of the resonance frequency of the slab, for example, the marked point where 𝐹 A% = 8 corresponds to the actual frequency 8𝑓 I $ . In fact, far away from resonances, the reflection is dominant and the effect of frequency conversion on the transmitted waves becomes weak. Another interesting phenomenon reflected in the Fig. 4 is that when 𝐹 1 = 1, all the asymmetries disappear. In this case, the actual modulation frequency is equal to the resonance frequency of the slab, 𝑓 1 = 𝑓 I $ . In this case, the spatial biasing effect from phase difference becomes weak.

For an optimized case of nonreciprocity, we selected 𝐹 A% = 8 and 𝐹 1 = 0.76, where |𝑇 ! ) -𝑇 ! 0 | = 0.956 and |10 𝑙𝑜𝑔 #! (𝑇 ! ) /𝑇 ! 0 )| = 25.98dB (Black dot indicated in Fig. 4). Under this configuration, the positive and negative transmission coefficients of the fundamental mode are, 𝑡 ! ) = 0.049 and 𝑡 ! 0 = 0.979. While fixing the same values 𝐹 A% and 𝐹 1 , we performed a 2 nd level fine tuning where we swept for 𝛿 and Δ𝜙 (see Appendix D). A better result is achieved with 𝛥𝜙 = 0.46𝜋 and 𝛿 = 5. With this fine optimization, the transmission coefficients become 𝑡 ! ) = 0.043 and 𝑡 ! 0 = 0.979, so nonreciprocal transmission is slightly improved in comparison with the previous result. For this configuration, we plot in Fig. 5 the transmission coefficient in both propagation directions with incident monochromatic wave at the frequency of 𝐹 A% = 8 (Fig. 5(a) and (b)). We can clearly see that the 0 th order mode is dominant in the negative propagation direction while it is almost depleted in the positive direction with strong frequency conversion at higher order modes. We also plot the fundamental mode transmission in both propagation directions in Fig. 5(c To further illustrate the nonreciprocity behavior, we plot in Fig. 6 the total pressure as function of space and time based on transit FEM simulation. The figure shows the total pressure on the whole structure within a chosen duration of time. In the negative direction (Fig. 6(a)), the incident wave can pass through our structure and the wave form is little distorted, indicating the transmission of the fundamental mode with relatively low reflection in contrast to that of the positive direction. However, in the positive direction (Fig. 6(b)), we observe a different wave propagation phenomenon where the pattern of the fundamental mode is no longer dominant, and frequency conversion to higher frequency orders is observed. The modulation wave behavior shown in Fig. 6(b) corresponds to the superposition of several harmonic waves (±1 order, ±2 order, etc.).

III.3 Unidirectional frequency conversion

Since the nonreciprocity in our system is achieved owing to frequency conversion, we further exploited the unidirectional frequency conversion under the nonreciprocity premise. As schematically shown in Fig. 7, for a harmonic excitation, if the modulation frequency is chosen so that the first order corresponds to one of the resonance frequencies of the slab, this first order mode can be highly transmitted. There are numerous possibilities for frequency conversion, so we consider for instance two of them: from 0 th to -1 st and from 0 th to +1 st which are shown by arrows vertically orienting. In the example of Fig. 7, if we choose a specific frequency of modulation so that the -1 st order corresponds to one of the resonance frequencies of the slab, this order could be highly transmitted while the fundamental mode and the +1 st order being weakly transmitted. Further, if high transmission exists at both fundamental mode and one of the ±1 st order modes due to large transmission bandwidth, we can achieve both unidirectional frequency conversion and nonreciprocity by manipulating the modulation.

Figure 7. Schematic of the frequency conversion. The -1 st order mode is at the resonance frequency of the slab instead of the fundamental mode and the 1 st order mode. Thus, -1 st order mode may be transmitted with higher amplitude. To illustrate this idea, we consider the configuration of 𝐹 1 = 0.85, 𝛥𝜙 = 0.589𝜋 and 𝛿 = 12 where strong asymmetries of all the ±1 st and 0 th order modes are achieved. We plot in Fig. 8 (a) the transmission coefficient in the frequency domain for the 0 th and ±1 st orders. We can observe that the ±1 st orders modes are both dominant in the transmission for one of the directions of propagation.

At 𝐹 A% = 8.85 for instance, the -1 st orders mode (𝐹 A% -𝐹 1 ) is at a resonance frequency (8𝑓 I $ ). Figures 8 (b) and (c) display the transmission coefficient with incident monochromatic wave at 𝐹 A% = 8.85. In the positive direction, the amplitude of the -1 st order reaches 0.61 and becomes the dominant mode and the amplitude of the fundamental mode is 0.18, while in the negative direction, the fundamental mode's amplitude is 0.78 and the amplitudes of all the neighboring modes are low. In this case, unidirectional frequency conversion appears.

Regarding the experimental implementation of the system, for instance, it can be built using successive layers of air and gas with different densities, separated by an airtight thin membrane to avoid being mixed. Then, the effective density can be modulated in time using pistons that can change the volume of each cavity layer. The thickness of the layers can be chosen to operate at frequency ranges far from the flexural resonances of the membranes. The existence of the membranes will only slightly reduce the bandwidth of nonreciprocity which can still be achieved by fine tuning the parameters of the system.

Conclusion

In this work, we have demonstrated nonreciprocal acoustic transmission and unidirectional frequency conversion in cascaded fluid slab resonators based on temporal modulation of their effective densities, with phase difference of the time modulations between the slabs. A theoretical transfer matrix method based on plan waves expansion has been developed for efficient characterization of the acoustic dispersion of the system, while numerical simulations based on FEM have been conducted to assess the theoretical findings. Optimizations over the system's parameters was carried out to achieve broadband nonreciprocity and frequency conversion in the two-slab system. An acoustic nonreciprocity behavior has been demonstrated for a monochromatic incident wave, with almost 97.9% transmission in the positive direction while only 4.3% is transmitted in the negative direction. Furthermore, unidirectional frequency conversion has been achieved. Over half of the transmitted acoustic energy has been converted into one mode with higher static density ratio 𝛿 . The proposed design of nonreciprocal acoustic system displays promising functionalities, with a simple compact configuration to achieve unidirectional wave propagation and frequency conversion. Though the present system is purely theoretical and far from representing a realistic device for application, it is possible to imagine a solid system where we only consider longitudinal waves which are described with the same Helmholtz equation, thus behaves the same as acoustic waves in fluid. In this case, our approach could also be conducted by considering the time modulation of the effective stiffness of the materials using piezoelectric elements instead of density. Further, the time-modulation capability of the design is to be investigated especially when the frequency of modulation is comparable to the incident frequency of the system, our design offers a platform that could inspire futuristic devices for multiple applications, including biomedical ultrasound devices, improved energy harvesting and communication system.

Figure 1 .

 1 Figure 1. Schematic of a single-slab (a) system and a two-cascaded-slab (b) system with time modulated effectives densities. Harmonics appear in the transmitted and reflected waves due to the modulation.

Figure 2 .

 2 Figure 2. Schematic of the transfer matrix 𝑴 !→!#$ between layers. 𝑪 ! ± is the complex amplitude of forward/backward wave at the right boundary of layer 𝑠.

Figure 3 .

 3 Figure 3. Transmission through a single time modulated slab. (a) Calculated transmission in the static case (black solid line) and transmission components |𝑡 & | for the 𝑖 '( order up to the second order. The modulation frequency in this case is 𝐹 ) = 1.12. (b) Transmission spectrum for an incident monochromatic wave at frequency Fin=10.5. Blue solid curve stands for the spectrum obtained from time dependent finite element simulation (FEM), while red flat marks stand for theoretically predicted transmission coefficients for the 0 th , ±1 st and ±2 nd orders (TMM).

Figure 4 .

 4 Figure 4. Transmission analysis between right and left propagations as function of the normalized incident frequency 𝐹 &* and the normalized modulation frequency 𝐹 ) , for ∆𝜙 = 0.5𝜋 and 𝛿 = 5. (a) Asymmetry

  difference of the 0 th order mode transmittance, standing for |𝑇 + # -𝑇 + , |. (b) Asymmetry ratio of 0 th order mode transmittance, standing for |10 𝑙𝑜𝑔 $+ (𝑇 + # /𝑇 + , )|. (c) Total transmission difference, standing for all the modes are included. The black dot in these figures marks the point where 𝐹 &* = 8 and 𝐹 ) = 0.76.

Fig. 4 (

 4 a) and (b) show the quantities |𝑇 ! ) -𝑇 ! 0 | and |10 𝑙𝑜𝑔 #! (𝑇 ! ) /𝑇 ! 0 )| for the fundamental mode, respectively, as function of the normalized modulation frequency 𝐹 1 and incident frequency 𝐹 A% . Fig.4(c) plots the normalized total transmission difference for the convenience of observation. The criterions for finding nonreciprocity are |𝑇 !

Figure 5 .

 5 Figure 5.(a) (b) Transmission coefficient for the positive direction (a) and the negative direction (b) with 𝐹 &* = 8, 𝐹 ) = 0.76, 𝛥𝜙 = 0.46𝜋 and 𝛿 = 5. (c) Theoretically calculated transmission coefficient curve

  mode (0 th order) for both directions as function of 𝐹 &* , for 𝐹 ) = 0.76, 𝛥𝜙 = 0.46𝜋 and 𝛿 = 5. The black dash lines and arrow mark the bandwidth.

  ) where we can deduce acoustic nonreciprocity around the incident frequency. The bandwidth of nonreciprocity is defined as|𝑇 ! ) (ω) -𝑇 ! 0 (ω)| > 0.5 × Max (|𝑇 ! ) -𝑇 ! 0 |).The normalized bandwidth of nonreciprocity ∆𝐹 is approximately 0.51 in this case.

Figure 6 .

 6 Figure 6. Pressure field pattern for the case reported in figure 5(a) and (b), from 30ms to 31ms. 𝜆 &* = 1 𝑓 &* ⁄ is the wavelength of the incident wave and 𝜆 ) = 2𝜋 𝛺 ⁄ is the modulation wavelength. (a) Negative direction; (b) positive direction.

Figure 8 .

 8 Figure 8. (a) Transmission coefficient in the frequency domain for the 0 th , ±1 st and ±2 nd orders with 𝐹 ) = 0.85, 𝛥𝜙 = 0.589𝜋 and 𝛿 = 12. Transmission spectrum coefficient in the positive direction (a) and the negative direction (b) at the monochromatic incident frequency 𝐹 &* = 8.85, (flat marks for theory and solid blue line for FEM based simulation).

Figure 12 .

 12 Figure 12. Results of the theoretical calculation sweeping ∆𝜙 from 0 to 𝜋 and 𝛿 from 3 to 7, with 𝐹 &* = 8 and 𝐹 ) = 0.76 inherited from the last calculation. (a) Asymmetry difference of 0 th order mode transmittance. (b) Asymmetry ratio of 0 th order mode transmittance. (c) Total transmission difference. The black dot in these figures marks the point where ∆𝜙 = 0.46𝜋 and 𝛿 = 5.

Appendix A: Conversion from transfer matrix to scattering matrix.

The schematic of the scattering system is shown is Fig. 9. The transfer matrix 𝑴 and the scattering matrix 𝑺 are defined as,

. Schematic of the system for matrix conversion

where

𝑴 $$ , 𝑴 $. , 𝑴 .$ and 𝑴 .. are matrices with 2𝑁 + 1 orders. With some algebraic manipulations, we can easily obtain the elements of the scattering matrix from the transfer matrix,

Appendix B: Determination of the coupling factor

To illustrate the static coupling behaviors, we plotted in Fig. 10

Appendix C: Preset value of the initial phase difference

In order to verify the consistency of the preset phase difference 𝛥𝜙 = 0.5, a series of calculations were performed upon 𝛥𝜙. Figure 11 presents the absolute value of difference between the right and left transmission coefficients of the fundamental mode,

•, as function of 𝛥𝜙 for the considered modulating frequencies 𝐹 1 = 0.7, 0.9, and 1.1. For all these three cases, high level of transmission difference, indicating strong nonreciprocity, appears where Δ𝜙 is between 0.4 𝜋 and 0.6 𝜋. To conclude, it is rational to assume strong asymmetry at 𝛥𝜙 = 0.5.

Appendix D: Second-level fine tuning

Under 𝐹 A% = 8 and 𝐹 1 = 0.76 , we performed a refinement of the optimization for stronger nonreciprocity.


Puthiya Veetil Nidheesh

Clément Trellu

Hugo Olvera Vargas

Emmanuel Mousset

Oladejo Soliu

Mehmet A Ganiyu

Oturan

Soliu Oladejo Ganiyu

Mehmet A Oturan
email: mehmet.oturan@univ-eiffel.fr

Electro-Fenton process in combination with other advanced oxidation processes: Challenges and opportunities

Keywords: Electro-Fenton, Advanced oxidation processes, Combined process, Hydroxyl radical

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

Introduction

Even though electro-Fenton (EF) process has several advantages like its capability to generate hydrogen peroxide in-situ, higher mineralization efficiency, enhanced ferrous ion regeneration capacity and lesser catalyst requirement; the disadvantages like longer time requirement for complete mineralization of pollutants, scaling up issues, passivation of electrodes etc. made a barrier for its real field implementation [START_REF] Nidheesh | Trends in electro-Fenton process for water and wastewater treatment: An overview[END_REF][START_REF] Nidheesh | An overview on the removal of synthetic dyes from water by electrochemical advanced oxidation processes[END_REF][START_REF] Ganiyu | Electrochemical advanced oxidation processes for wastewater treatment: Advances in formation and detection of reactive species and mechanisms[END_REF][START_REF] Brillas | Electro-Fenton process and related electrochemical technologies based on Fenton's reaction chemistry[END_REF][START_REF] Titchou | An overview on the elimination of organic contaminants from aqueous systems using electrochemical advanced oxidation processes[END_REF][START_REF] Monteil | A review on efficiency and cost effectiveness of electro-and bio-electro-Fenton processes: Application to the treatment of pharmaceutical pollutants in water[END_REF]. Another drawback of EF process is related to the acidic pH (pH 3) requirement for its effective performance. A few heterogeneous catalyst was found much effective even in neutral pH [START_REF] Gopinath | Supported catalysts for heterogeneous electro-Fenton processes: Recent trends and future directions[END_REF]. However, electrolytic generation of hydrogen peroxide at neutral pH is a challenging task, as proton is needed for the two-electron reduction of oxygen and subsequent generation of hydrogen peroxide. Therefore, in general, higher mineralization rates in EF process are achievable at acidic conditions and neutralization of EF treated effluent is needed for further treatment or its discharge.

Several modified versions of EF process like photo-Fenton, peroxicoagulation, and anodic-Fenton were tested to solve the issues associated with EF process [2, [START_REF] Nidheesh | Removal of organic pollutants by peroxicoagulation[END_REF][START_REF] Brillas | A review on the photoelectro-Fenton process as efficient electrochemical advanced oxidation for wastewater remediation. Treatment with UV light, sunlight, and coupling with conventional and other photo-assisted advanced technologies[END_REF][START_REF] Nidheesh | Recent advances in electro-Fenton process and its emerging applications[END_REF]. However, the drawbacks of EF process are marginally solved by these modified processes. Another method to improve the performance of the EF process is to combine it with other advanced oxidation processes (AOPs). Thus, the process efficiency can be improved in terms of pollutant degradation kinetics, mineralization efficiency, biodegradability enhancement, toxicity reduction etc. Passivation of electrodes can be reduced by combing with sonochemical processes [START_REF] Hassani | Recent progress on ultrasound-assisted electrochemical processes: A review on mechanism, reactor strategies, and applications for wastewater treatment[END_REF].

The present article, analyse the performance of EF process in combination with other AOPs.

Performance enhancement is detailed along with advantages and disadvantages of combined systems.

Coupling EF process with different AOPs

Implementation of the EF process requires [4,12]:

 suitable cathode material for H2O2 formation and Fe 3+ reduction to Fe 2+ ;  a homogeneous (dissolved in the bulk) or heterogeneous (solid) iron source ;  pH adjustment, particularly for homogeneous EF (optimal pH is 3) ;  current supply.

Several combinations have then been investigated for promoting the removal of organic compounds. The idea is based on the establishment of suitable operating conditions for the implementation of another AOP in the same reactor. The objective is to increase the production of oxidant species (to enhance degradation/mineralization efficiency) or reduce energy consumption (to enhance cost effectiveness) in such a hybrid reactor.

The choice of the anode material offers a first opportunity. For example, non-active materials such as BDD or TiOx materials allow for production of large amounts of hydroxyl radicals from water oxidation at the anode surface [START_REF] Oturan | Comparative performance of ten electrodes in electro-Fenton process for removal of organic pollutants from water[END_REF][START_REF] Trellu | Comparative study on the removal of humic acids from drinking water by anodic oxidation and electro-Fenton processes: Mineralization efficiency and modelling[END_REF][START_REF] Oturan | Sub-stoichiometric titanium oxide as a new anode material for electro-Fenton process: Application to electrocatalytic destruction of antibiotic amoxicillin[END_REF]. Using porous anode materials in flow-through configuration also provides solutions for improving mass transport conditions [START_REF] Trellu | Integration of sub-stoichiometric titanium oxide reactive electrochemical membrane as anode in the electro-Fenton process[END_REF].

Addition of suitable chemical reagents gives the opportunity to enhance the formation of oxidant species. Ferrate (Fe VI ), a powerful oxidant, was recently used as an iron source (instead of conventional Fe II or Fe III ) for the EF process (EF-Fe VI ) [START_REF] Kateb | Ferrate(VI) pre-treatment and subsequent electrochemical advanced oxidation processes: Recycling iron for enhancing oxidation of organic pollutants[END_REF]. Addition of sulfite or persulfate and subsequent electrochemical activation is another way to generate oxidant species such as SO4 •-, SO3 •-and • OH [18][START_REF] Song | Mechanistic Insight into the Heterogeneous Electro-Fenton/Sulfite Process for Ultraefficient Degradation of Pollutants over a Wide pH Range[END_REF][START_REF] Long | Removal of diclofenac by three-dimensional electro-Fenton-persulfate (3D electro-Fenton-PS)[END_REF]. Sulfite is able to accelerate the regeneration of Fe 2+ [START_REF] Song | A novel electro-Fenton process coupled with sulfite: Enhanced Fe 3+ reduction and TOC removal[END_REF].

Introduction of O3 also improves the oxidation of organic compounds through the electroperoxone process via reaction of O3 with H2O2 [START_REF] Wang | The electro-peroxone technology as a promising advanced oxidation process for water and wastewater treatment[END_REF].

Using irradiations from artificial or solar light allow for implementation of various processes linked to the EF processes. The (solar) photo-EF (PEF) process is based on the use of UV irradiations for promoting the formation of • OH and the regeneration of Fe 2+ through (i) rupture of the metal-carboxylate ligand bond, (ii) photodecarboxylation of Fe(III)-oxalate complexes, (iii) photoreduction of Fe(OH) 2+ and (iv) photolytic rupture of H2O2 [START_REF] Brillas | A review on the photoelectro-Fenton process as efficient electrochemical advanced oxidation for wastewater remediation. Treatment with UV light, sunlight, and coupling with conventional and other photo-assisted advanced technologies[END_REF][START_REF] Brillas | Benchmarking recent advances and innovative technology approaches of Fenton, photo-Fenton, electro-Fenton, and related processes: A review on the relevance of phenol as model molecule[END_REF]. Addition of a photocatalyst (e.g. TiO2) has been investigated for implementation of the photocatalytic EF(EF-PC) process [START_REF] Brillas | A review on the photoelectro-Fenton process as efficient electrochemical advanced oxidation for wastewater remediation. Treatment with UV light, sunlight, and coupling with conventional and other photo-assisted advanced technologies[END_REF][START_REF] Mousset | A new 3D-printed photoelectrocatalytic reactor combining the benefits of a transparent electrode and the Fenton reaction for advanced wastewater treatment[END_REF]. Development of photo-anodes by coating a semiconductor photocatalyst on a conductive support allows for implementing the photoelectrocatalytic EF (EF-PEC) process [9, [START_REF] Xu | FTO-TiO2 photoelectrocatalytic degradation of triphenyltin chloride coupled to photoelectro-Fenton: A mechanistic study[END_REF][START_REF] Orimolade | Coupling cathodic Electro-Fenton with anodic photoelectrochemical oxidation: A feasibility study on the mineralization of paracetamol[END_REF], which is the focus of a great attention over the last few years [START_REF] Orimolade | Towards visible light driven photoelectrocatalysis for water treatment: Application of a FTO/BiVO 4 /Ag2S heterojunction anode for the removal of emerging pharmaceutical pollutants[END_REF].

Ultrasound irradiation is another way to improve the EF process (SEF) efficiency, owing to (i) formation of additional • OH from water by cavitation, (ii) faster regeneration of Fe 2+ from Fe-O2H 2+ and (iii) mass transport enhancement [START_REF] Babuponnusami | Advanced oxidation of phenol: A comparison between Fenton, electro-Fenton, sono-electro-Fenton and photo-electro-Fenton processes[END_REF][START_REF] Oturan | Sonoelectro-Fenton process: A novel hybrid technique for the destruction of organic pollutants in water[END_REF][START_REF] Ghanbari | Insights into paracetamol degradation in aqueous solutions by ultrasoundassisted heterogeneous electro-Fenton process: Key operating parameters, mineralization and toxicity assessment[END_REF].

Performance of EF process coupled with other AOPs

Coupling between the EF process and other AOPs always can enhance degradation/ mineralization efficiency, reduce solution toxicity and operational cost compared to EF alone due to the enhanced generation of oxidants (mainly  OH), which accelerate the degradation and mineralization of the contaminants and reduce treatment time [START_REF] Ganiyu | Electrochemical advanced oxidation processes for wastewater treatment: Advances in formation and detection of reactive species and mechanisms[END_REF][START_REF] Brillas | Electro-Fenton process and related electrochemical technologies based on Fenton's reaction chemistry[END_REF][START_REF] Nidheesh | Recent advances in electro-Fenton process and its emerging applications[END_REF][START_REF] Ganiyu | Coupling of membrane filtration and advanced oxidation processes for removal of pharmaceutical residues: A critical review[END_REF]. For example, the apparent rate constant (kapp) for the degradation of erythrosine B dye at 20 mA cm -2 was reported to be 0.2461, 0.5222 and 0.4539 min -1 for EF, PEF (100 W) and SPEF (570 W m -2 ) after 2 h of treatment [START_REF] Clematis | Electro-Fenton, solar photoelectro-Fenton and UVA photoelectro-Fenton: Degradation of Erythrosine B dye solution[END_REF]. Degradation rate of 0.076 and 0.322 min -1 and total organic carbon (TOC) removal of 37% and 65% were observed for EF alone and SPEF, respectively, during treatment of 50 mg L -1 of amoxicillin at 150 mA cm -2 [START_REF] Garza-Campos | Air diffusion electrodes based on synthetized mesoporous carbon for application in amoxicillin degradation by electro-Fenton and solar photo electro-Fenton[END_REF]. UVA-PEF treatment of bezafibrate showed higher degradation (92%) compared to visible-light PEF (55%) and EF (42%) owing to higher absorbance of Fe catalyst within UV region [START_REF] Ye | Mechanism and stability of an Fe-based 2D MOF during the photoelectro-Fenton treatment of organic micropollutants under UVA and visible light irradiation[END_REF]. EF-sulfite showed enhanced degradation of anthracene, achieving 85.9% degradation compared to 75.3% obtained in EF alone [18] and carbamazepine, 100% compared to 83.7% observed in EF alone [START_REF] Song | Mechanistic Insight into the Heterogeneous Electro-Fenton/Sulfite Process for Ultraefficient Degradation of Pollutants over a Wide pH Range[END_REF]. Enhanced degradation was observed during EF-persulfate treatment of cefotaxime (100%) [START_REF] Lei | Degradation of aqueous cefotaxime in electrooxidation -electro-Fenton -persulfate system with Ti/CNT/SnO2-Sb-Er anode and Ni@NCNT cathode[END_REF], and diclofenac (96.3%) [START_REF] Long | Removal of diclofenac by three-dimensional electro-Fenton-persulfate (3D electro-Fenton-PS)[END_REF]. PEC-EF always perform better than either photocatalysis or EF alone [START_REF] Ganiyu | Renewable energies driven electrochemical wastewater/soil decontamination technologies: A critical review of fundamental concepts and applications[END_REF][START_REF] Ganiyu | The use of renewable energies driving electrochemical technologies for environmental applications[END_REF]. The kapp for the degradation of amaranth were 0.157, 0.171 and 0.211 h -1 for PEC, EF and PEC-EF respectively [START_REF] Thor | Explicating the importance of aeration and pH for Amaranth degradation and electricity generation in a viable hybrid system of photocatalytic fuel cell and electro-Fenton process[END_REF]. The degradation kinetics of triphenyltin chloride was observed to follow the sequence PEC-EF (0.274 min -1 ) > PEF (0.219 min -1 ) > PEC (0.011 min -1 ) [START_REF] Xu | FTO-TiO2 photoelectrocatalytic degradation of triphenyltin chloride coupled to photoelectro-Fenton: A mechanistic study[END_REF].

The effluents treated by coupled processes PEF, SPEF, PEC-EF and EF-persulfate always have reduced toxicity compared to EF treatment alone because of the enhanced degradation and mineralization of pollutants by the coupled technologies. For example, EF-persulfate treated fruit processing wastewater exposed to Daphnia magna showed increased EC50 (48 h) from 17% (5.8 TU) to 20.4% (4.9 TU) for EF compared to 28.9% (3.5 TU) obtained with EFpersulfate at 50 mA cm -2 [START_REF] Rodrigues | Treatment of fruit processing wastewater by electrochemical and activated persulfate processes: Toxicological and energetic evaluation[END_REF].

Coupled EF-sulfite/persulfate requires does not require additional energy to EF other than chemical cost. Lower energy consumption was always observed for PEF, SPEF PEC-EF compared to EF, but the operating cost of PEF and PEC-EF is always higher owing to the cost and energy consumption of the lamps. The energy consumption during treatment of acid blue 29 by EF and SPEF were 257 and 57 kWh (kg TOC) -1 , respectively [40], whereas the PEF treatment of brown HT dye using UVA lamp with wavelength of 365 nm (75 mW cm -2 ) requires 225 kWh (kg COD) -1 , thus, not much suitable for large-scale applications, even the process has significant additional dye removal efficiency than EF, electrochemical oxidation and ozonation [START_REF] Corona-Bautista | Discoloration of azo dye Brown HT using different advanced oxidation processes[END_REF]. SPEF has lower energy/cost requirement because of abundant and availability of sunlight as natural energy source.

Advantages of coupling EF with other AOPs

The combination between EF and other AOPs can bring interesting synergies and therefore provides some advantages. It can enhance the treatment efficiency of wastewater treatment by increasing the number of • OH active sites within the reactor, and can generate other strong radicals (SO4 •-, Cl • , etc.) that contribute additional removal of organic pollutant [START_REF] Brillas | A review on the photoelectro-Fenton process as efficient electrochemical advanced oxidation for wastewater remediation. Treatment with UV light, sunlight, and coupling with conventional and other photo-assisted advanced technologies[END_REF][START_REF] Salmerón | Solar photo-assisted electrochemical processes applied to actual industrial and urban wastewaters: A practical approach based on recent literature[END_REF].

Moreover, an increase of regeneration rate of Fe(II) from photoreduction of Fe(III) in SPEF has been established [START_REF] Brillas | Benchmarking recent advances and innovative technology approaches of Fenton, photo-Fenton, electro-Fenton, and related processes: A review on the relevance of phenol as model molecule[END_REF][START_REF] Mousset | Photoelectrochemical reactors for treatment of water and wastewater: a review[END_REF], which not only permit to increase the • OH production, but also allows reducing the amount of sludge generation and the associated management cost [START_REF] Mousset | Cost comparison of advanced oxidation processes for wastewater treatment using accumulated oxygenequivalent criteria[END_REF][START_REF] Sirés | Upgrading and expanding the electro-Fenton and related processes[END_REF].

Furthermore, the flexibility of electrochemical systems [46] allows combining EF with AOPs in single hybrid reactor [START_REF] Mousset | A new 3D-printed photoelectrocatalytic reactor combining the benefits of a transparent electrode and the Fenton reaction for advanced wastewater treatment[END_REF][START_REF] Mousset | Photoelectrochemical reactors for treatment of water and wastewater: a review[END_REF], which decrease the footprint and promote synergies of reactions. The fact that the working electrode is the cathode in EF process [START_REF] Du | Nanostructured electrodes for electrocatalytic advanced oxidation processes: From materials preparation to mechanisms understanding and wastewater treatment applications[END_REF], the anode can also favor reactions according to their properties [START_REF] Santos | Testing the role of electrode materials on the electro-Fenton and photoelectro-Fenton degradation of clopyralid[END_REF]. For instance, using transparent anode which allows the passage of UV radiation go through the material and interact both with anode surface and bulk solution following photoelectrocatalysis mechanisms [START_REF] Mousset | A new 3D-printed photoelectrocatalytic reactor combining the benefits of a transparent electrode and the Fenton reaction for advanced wastewater treatment[END_REF][START_REF] Ding | Design of a visible light driven photoelectrochemical/electro-Fenton coupling oxidation system for wastewater treatment[END_REF].

In the meantime, such coupling could modify operating and capital costs; therefore, a cost feasibility study is required when comparing these combinations in order to determine the optimal cost-effective conditions [START_REF] Mousset | Cost comparison of advanced oxidation processes for wastewater treatment using accumulated oxygenequivalent criteria[END_REF]. A significant rise of cost with the coupling could represent drawbacks, which are discussed in the following section.

Disadvantages of EF coupled with AOPs: Side reactions and non-radical pathways

During the coupling of EF with other AOPs, the excess of certain chemicals entails side reactions consuming the oxidant species in detriment of pollutant degradation. Such undesired effects are exacerbated when several chemicals/oxidant promoters are combined in the same reactor. The main side reactions affecting EF efficiency are related to the excess of Fe 2+ and/or H2O2 consuming  OH according to Eqs. 1 and 2, respectively. Additionally, high current densities are known to affect the degradation/mineralization efficiency due to further H2O2 reduction to H2O (Eq. 3) and the evolution of H2 (Eq. 4) [START_REF] Brillas | Electro-Fenton process and related electrochemical technologies based on Fenton's reaction chemistry[END_REF]. When EF is combined with anodic oxidation (EF-AO) using non-active anodes (BDD, TiOx…), the degradation efficiency is also decreased at high current densities due to O2 evolution entailed by recombination of surface • OH (Eq. 5), and H2O2 anodic decomposition (Eq. 6) [START_REF] Olvera-Vargas | Electro-Fenton treatment of real pharmaceutical wastewater paired with a BDD anode: Reaction mechanisms and respective contribution of homogeneous and heterogeneous OH[END_REF][START_REF] Olvera-Vargas | Dynamic crossflow electro-Fenton process coupled to anodic oxidation for wastewater treatment: Application to the degradation of acetaminophen[END_REF]. In the case of the combined EFsulfite and EF-persulfate processes, the excess of H2O2 can consume  OH and SO4 -(Eq. 1 and 7, respectively), decreasing the degradation efficiency [18,[START_REF] Song | Mechanistic Insight into the Heterogeneous Electro-Fenton/Sulfite Process for Ultraefficient Degradation of Pollutants over a Wide pH Range[END_REF]. Excessive amounts of SO3 2-in EF-sulfite can also be detrimental for • OH availability (Eq. 8). In the case of the EF-persulfate process, the excess of S2O8 2-can entail waste reactions as well, reducing the amount of oxidants available for pollutant degradation. Such reactions include the consumption of SO4 -by S2O8 2- (Eq. 9) and the recombination of SO4 -(Eq. 10) once such radicals are formed by Fe 2+ activation of S2O8 2-(Eq. 11). The surplus of Fe 2+ also reduces the amount of SO4 -according to the waste reaction of Eq. 12 [START_REF] Long | Removal of diclofenac by three-dimensional electro-Fenton-persulfate (3D electro-Fenton-PS)[END_REF]. Accordingly, the experimental conditions of combined approaches must be carefully optimized to obtain the highest degradation/mineralization efficiencies avoiding parasitic reactions. Advantages and disadvantages of various EF coupled with AOPs are provided in Table 1.

H2O2 +  OH  H2O + O2 -+ H + (1) Fe 2+ +  OH  Fe 3+ + OH - (2)
H2O2 + 2H + + 2e - 2H2O (3) 2H + + 2e - H2 (4) 2BDD( • OH)  2BDD + O2 + 2H + + 2e - (5)
H2O2

 O2 + 2H + + 2e - (6) H2O2 + SO4 - HO2  + SO4 2-+ H + (7) SO3 2-+ • OH  SO3 •-+ OH - (8)
S2O8 2-+ SO4 - SO4 2-+ S2O8 - (9) SO4 -+ SO4 - S2O8 2- ( 10
) S2O8 2-+ Fe 2+  Fe 3+ + SO4 2-+ SO4 - (11)
SO4 -+ Fe 2+  Fe 3+ + SO4 2- ( 12
)
Regeneration of catalyst is a major concern in all the AOPs. The catalyst used in AOPs may affect the efficiency of subsequent process or it may be a threat to the environment, if it is discharged. Hence, recovery and reusability of catalyst used in all AOPs is a critical parameter for its practical application.

Conclusions

Combination of EF process with other AOPs is an effective way of pollutant removal in terms of oxidative degradation/mineralization efficiency, biodegradability enhancement and toxicity reduction. Even though, the combined system has several advantages, a few disadvantages such as improvements in the rate of side reactions, increase in the process cost (in a few systems), and promotion of non-radical pathways (in a few systems) etc., constitute the challenges.

Overall, the combined systems provide better results than individual AOPs. However, more studies in higher scales are needed for understanding the real issues before their implementation in the real field. -High mineralization yields.

-High cost of BDD, the most powerful material for AO.

-Lack of powerful anodes for AO available in the market, others than BDD.

-Formation of ClO3

H2O2 + Fe 2+  • OH + Fe 3+ + OH - Fe 3+ + SO3 2- SO3 •-+ Fe 2+ SO3 •-+ O2  SO5 •- SO5 •-+ HSO3 - HSO5 -+ SO3 •- Fe 2+ + HSO5 - Fe 3+ + SO4 •- SO5 •-+ SO3 2- SO5 2-+ SO3 •- SO5 •-+ SO3 2- SO4 •-+ SO4 2- SO5 •-+ SO5 •- 2SO4 •-+ O2 -Production of SO4 •-to enhance pollutant degradation.
-Wide pH range of operation.

-SO3 2-is inexpensive and nontoxic.

-SO3 2-is transformed into SO4 2- after treatment.

-Side reactions decreasing efficiency when high SO3 2- doses are used.

-Higher concentration of SO4 S2O8 2-+ e - SO4 2-+ SO4 •- -Selective SO4 -are produced to enhance pollutant degradation.

-Possibility to operate the system at wider pH values.

-Occurrence of several undesired reactions reducing the efficiency, [START_REF] Long | Removal of diclofenac by three-dimensional electro-Fenton-persulfate (3D electro-Fenton-PS)[END_REF] Bulk solution:

H2O2 + Fe 2+  • OH + Fe 3+ + OH - S2O8 2-+ Fe 2+  Fe 3+ + SO4 2-+ SO4 •- Anode : SO4 2- SO4 •-+ e - SO4 •-+ SO4 •- S2O8 2- S2O8 2-+ • OH  SO4 •-+ HSO4 -+ 0.5 O2
Non-radical activation of S2O8 -Need for development of large-scale photoreactors.

-Intermittent solar irradiation / cost of artificial light irradiation

  Photoelectro-Fenton -Photoelectrocatalysis (PEF -PEC)

Table 1 :

 1 Main combined EF-AOPs: Main reactions, advantages and challenges

	Combined process	Main reactions	Advantages	Challenges	Refs.
	EF-Anodic	oxidation	Cathode:	-Production of  OH at the
	(EF-AO)		H2O2 + Fe 2+  • OH + Fe 3+ + OH -	cathodic and anodic sides for
			Fe 3+ + e - Fe 2+	enhanced pollutant degradation
				and higher mineralization
			Anode:	current efficiency.
			M + H2O  M( • OH) + H + + e -
			Bulk solution:
			H2O2 + Fe 2+  • OH + Fe 3+ + OH -