domain: stat.ml phys stat

text:
Long Bai
email: bailong@hust.edu.cn

Florian Le Bourdais

Roberto Miorelli

Pierre Calmon

Alexander Velichko

Bruce W Drinkwater

Ultrasonic defect characterisation using the scattering matrix: A performance comparison study of Bayesian inversion and machine learning schemas

Accurate defect characterisation is desirable in ultrasonic non-destructive evaluation as it can provide quantitative information about the defect type and geometry. For defect characterisation using ultrasonic arrays, high resolution images can provide the size and type information if a defect is relatively large. However, the performance of image-based characterisation becomes poor for small defects that are comparable to the wavelength. An alternative approach is to extract the far-field scattering coefficient matrix from the array data and use it for characterisation. Defect characterisation can be performed based on a scattering matrix database that consists of the scattering matrices of idealised defects with varying parameters. In this paper, the problem of characterising small surface-breaking notches is studied using two different approaches. The first approach is based on the introduction of a general coherent noise model, and it performs characterisation within the Bayesian framework. The second approach relies on a supervised machine learning (ML) schema based on a scattering matrix database, which is used as the training set to fit the ML model exploited for the characterisation task. It is shown that convolutional neural networks (CNNs) can achieve the best characterisation accuracy among the considered machine learning approaches and they give similar characterisation uncertainty to that of the Bayesian approach if a notch is favourably oriented. The performance of both approaches varied for unfavourably oriented notches, and the machine learning approach tends to give results with higher variance and lower biases.

I. INTRODUCTION

U LTRASONIC non-destructive testing (NDT) is often used to detect and localise defects in a safety critical structure without causing damage to the component or radiation hazards to personnel [START_REF] Kundu | Ultrasonic Nondestructive Evaluation: Engineering and Biological Material Characterization[END_REF], [START_REF] Song | Development of an ultrasonic phased array system for nondestructive tests of nuclear power plant components[END_REF]. Typically, an input pulse is transmitted into the material under test and the presence of material discontinuities or defects gives rise to the reflected signal. Traditional ultrasonic testing techniques use a single-element transducer to scan the region of interest within a component and detection is achieved by comparing the amplitude of the received signal with a pre-set threshold [START_REF] Brierley | Data fusion for automated non-destructive inspection[END_REF]. The defect size can then be determined by the 6 dB drop approach as the distance over which the amplitude of the defect signal drops by 1/2 [START_REF] Blitz | Ultrasonic methods of non-destructive testing[END_REF]. However, this approach is most effective for relatively large crack-like defects (i.e., sizes over 2 ultrasonic wavelengths) that are favorably oriented, and from which the specular signal can be measured. A comprehensive approach is needed to characterise various types of defects that can occur in practice, such as cracks, volumetric voids, porosity and inclusions.

Ultrasonic arrays provide improved focusing capabilities compared to conventional single-element transducers and are increasingly used for inspection of structural components [START_REF] Drinkwater | Ultrasonic arrays for non-destructive evaluation: A review[END_REF]. A data acquisition scheme known as the full matrix capture (FMC) collects A-scan signals measured from all transmitter-receiver pairs of an array [START_REF] Holmes | Post-processing of the full matrix of ultrasonic transmit-receive array data for nondestructive evaluation[END_REF]. These A-scan data can be properly delayed and summed in post-processing and focusing is achieved at each pixel point of the imaging plane. Besides detection, advanced imaging algorithms such as the total focusing method (TFM) [START_REF] Holmes | Post-processing of the full matrix of ultrasonic transmit-receive array data for nondestructive evaluation[END_REF], virtual source aperture (VSA) [START_REF] Sutcliffe | Multiple virtual source aperture imaging for non-destructive testing[END_REF], and plane-wave imaging (PWI) [START_REF] Montaldo | Coherent plane-wave compounding for very high frame rate ultrasonography and transient elastography[END_REF], [START_REF] Le Jeune | Plane wave imaging for ultrasonic non-destructive testing: Generalization to multimodal imaging[END_REF] also facilitate image-based characterisation of defects. The size of surface-breaking cracks larger than 3 mm were measured with errors less than 8% by identifying the crack tips from an image [START_REF] Kimoto | Image-based sizing of surface-breaking cracks by SH-wave array ultrasonic testing[END_REF]. For internal cracks, the size and orientation angle were shown to be measurable using a rectangular box that covers all image pixels within -6 dB from the peak value [START_REF] Zhang | The use of ultrasonic arrays to characterize crack-like defects[END_REF]. While obtaining the crack propagation path and orientation by adopting TFM, Camacho et al. showed that sizing by tip detection with phase coherence imaging (PCI) was more robust than the 6 dB drop approach [START_REF] Camacho | Ultrasonic crack evaluation by phase coherence processing and TFM and its application to online monitoring in fatigue tests[END_REF]. In addition, signal and image processing techniques were developed which improve the time resolution of the measured A-scan signals [START_REF] Shakibi | Resolution enhancement of ultrasonic defect signals for crack sizing[END_REF], [START_REF] Li | Acoustic microscopy signal processing method for detecting near-surface defects in metal materials[END_REF] and spatial resolution of the ultrasonic images [START_REF] Mosey | Resolution enhancement of ultrasonic B-mode images[END_REF], [START_REF] Merabet | 2-D and 3-D reconstruction algorithms in the Fourier domain for plane-wave imaging in nondestructive testing[END_REF]. However, when the defect size is comparable to the ultrasonic wavelength, the defect indication becomes indistinguishable from that of a point scatterer and image-based characterisation of small defects is fundamentally challenging.

There exist two different types of approaches in literature which have the potential to size and characterise sub-wavelength defects: defect shape reconstruction through super-resolution imaging, and inversion which aims to retrieve key defect parameters based on the availability of a forward model. Super-resolution imaging algorithms such as the time-reversal multiple signal classification (TR-MUSIC) [START_REF] Marengo | Time-reversal MUSIC imaging of extended targets[END_REF] and the factorisation method (FM) [START_REF] Kirsch | Characterization of the shape of a scattering obstacle using the spectral data of the far field operator[END_REF] were used to characterise simulated rough cracks [START_REF] Elliott | Sizing subwavelength defects with ultrasonic imagery: An assessment of super-resolution imaging on simulated rough defects[END_REF] and were shown to be capable of achieving a resolution beyond the diffraction limit in ideal low-noise measurement scenarios. The main issue of these algorithms is their performance degradation for non-ideal defects (such as rough and inclined cracks [START_REF] Elliott | Sizing subwavelength defects with ultrasonic imagery: An assessment of super-resolution imaging on simulated rough defects[END_REF]) and noisy materials [START_REF] Fan | A comparison between ultrasonic array beamforming and super resolution imaging algorithms for non-destructive evaluation[END_REF]. A model-based inversion approach was developed for crack sizing by utilising an analytic expression relating the crack size to the array data [START_REF] Tant | A model-based approach to crack sizing with ultrasonic arrays[END_REF]. Li et al. studied the crack sizing problem in immersion setups and developed flaw-sizing curves that account for the surface condition of a component including curvature, roughness, and coating [START_REF] Li | Sizing small crack-like flaws through non-ideal part surface using ultrasonic measurement model[END_REF]. A type of more general inversion procedures iteratively update the defect geometry according to the difference between the actual and predicted measurements until a satisfactory agreement is achieved [START_REF] Moreau | 3-D reconstruction of sub-wavelength scatterers from the measurement of scattered fields in elastic waveguides[END_REF], [START_REF] Brigante | Numerical algorithm for defect reconstruction in elastic media with a circular ultrasonic scanning[END_REF]. Metamodels (or surrogate models) can speed up the inversion process by enabling fast evaluation of the forward model [START_REF] Miorelli | Database generation and exploitation for efficient and intensive simulation studies[END_REF] and are recently used as tools for propagation of uncertainties [START_REF] Miorelli | Assessing performance of flaw characterization methods through uncertainty propagation[END_REF]. Machine learning (ML) algorithms [START_REF] Bianco | Machine learning in acoustics: Theory and applications[END_REF] offer another possibility to address the inverse problem, in which measurements from a number of known defects are used to train a suitable classifier [START_REF] Bai | Characterization of defects using ultrasonic arrays: A dynamic classifier approach[END_REF] or regressor [START_REF] Salucci | Real-time NDT-NDE through an innovative adaptive partial least squares SVR inversion approach[END_REF], [START_REF] Ahmed | An adaptive sampling strategy for quasi real time crack characterization on eddy current testing signals[END_REF]. Deep learning approaches such as the convolutional neural networks (CNNs) have been applied to 2D image data, including ultrasonic plane wave images [START_REF] Pyle | Deep learning for ultrasonic crack characterization in NDE[END_REF], scanning electron microscopy (SEM) images [START_REF] O'leary | Deep learning for classification of the chemical composition of particle defects on semiconductor wafers[END_REF], and images taken using hand-held cameras [START_REF] Cha | Deep learning-based crack damage detection using convolutional neural networks[END_REF], for crack detection and characterisation. A subwavelength ultrasonic beamforming approach has been developed based on a deep neural network architecture [START_REF] Song | Super-resolution visualization of subwavelength defects via deep learning-enhanced ultrasonic beamforming: A proof-of-principle study[END_REF], and one-dimensional CNNs are employed to address the limited-sample problem with line-level labels [START_REF] Zhang | Two deep learning networks for rail surface defect inspection of limited samples with line-level label[END_REF]. Besides images, measurement data used for inversion can also have different forms including A-scans [START_REF] Cacciola | Computational intelligence aspects for defect classification in aeronautic composites by using ultrasonic pulses[END_REF], FMC data [START_REF] Laroche | An inverse approach for ultrasonic imaging from full matrix capture data. Application to resolution enhancement in NDT[END_REF] or quantities derived from raw signals such as the scattering matrix [START_REF] Bai | Characterization of defects using ultrasonic arrays: A dynamic classifier approach[END_REF]. The scattering matrix is defined for a single defect and encodes all the information about the defect geometry while an FMC dataset normally contains contributions from all defects within a component. Subwavelength defects remain distinguishable in terms of the scattering matrix but image-based inversion methods often fail when the defect size is small [START_REF] Zhang | The use of ultrasonic arrays to characterize crack-like defects[END_REF]. For the above reasons, we note that the scattering matrix is more suitable for characterisation of small defects than ultrasonic images or unprocessed FMC datasets. Beside the deep learning approaches, recently other works take on inversion based on guided wave images applied to Structural Health Monitoring problems using kernel machines for regression and classification tasks [START_REF] Miorelli | Supervised learning strategy for classification and regression tasks applied to aeronautical structural health monitoring problems[END_REF]. It is worth to be mentioned that comparisons between regression tasks based on kernel methods and deep learning approaches for regression are slightly overlooked in the ultrasonic non-destructive testing community. Furthermore, to the best of the author's knowledge, a sensitivity analysis study of machine learning based regression applied to Scattering matrix images has not been investigated in the literature.

The scattering matrix was previously used to characterise small slots machined using electrical discharge machining (EDM) [START_REF] Bai | Characterization of defects using ultrasonic arrays: A dynamic classifier approach[END_REF] and ellipses [START_REF] Bai | Ultrasonic characterization of crack-like defects using scattering matrix similarity metrics[END_REF] and satisfactory results were obtained for these idealised defects in low-noise scenarios. These approaches were based on a nearest-neighbour-search scheme and yielded a single characterisation result. However, the signal-to-noise ratio of the time-domain signals (and hence, the extracted scattering matrices) can become very low in practical cases if a defect is unfavourably oriented (e.g., when only the crack tip signal is measurable). Since the actual characterisation result can potentially have large errors in such challenging cases, uncertainty quantification is a prominent part of any defect characterisation procedure. In this paper, we explore the use of Bayesian inference and calculate the posterior probability of different defect parameters given some experimental measurement. Moreover, for a better understanding of the limitations of the Bayesian inference and the achievable characterisation performance using the scattering matrix, we also implemented some popular machine learning algorithms including linear regression, Gaussian process regression as well as CNNs, which take the scattering matrix data as the input and give defect parameters as the output. Importantly, we investigate how the measurement noise will affect the inversion results for different approaches. The main purpose of this paper is to identify the strengths and weaknesses of both approaches and to provide a guideline on the selection of the suitable inversion approach for different measurement scenarios in practical applications, such as ultrasonic array inspection of surface-breaking cracks, porosity, and inclusions.

II. DESCRIPTION OF THE INVERSION PROBLEM

The scattering matrix describes the angular scattering behaviour of a defect and consists of the far-field scattering coefficients for different incident-scattering angle pairs. Considering the 2-D geometry in Fig. 1, the scattering matrix is defined as [START_REF] Bai | Characterization of defects using ultrasonic arrays: A dynamic classifier approach[END_REF]:

s(θ i , θ s , ω) = a s (ω) a i (ω) d λ exp - iωd c . (1)
In Eq. ( 1), θ i , θ s denote the incident and scattering angles, a i , a s are the amplitude of the plane incident wave and the scattered wave, d is the distance between the receiver and the scatterer, and c is the wave speed in the medium. Note that bold upperand lower-case symbols indicate matrices and vectors in this paper, and a wavelength (λ) is used as a normalisation distance in Eq. ( 1). The experimental data used in this paper were obtained through direct contact measurements, and only longitudinal waves are considered hereafter. As the size of a defect becomes comparable to the ultrasonic wavelength, imaging algorithms such as the TFM can no longer provide sufficient information for evaluation of the defect geometry. This is illustrated in Fig. 2, which shows the TFM imaging results of ideal horizontal cracks of sizes 4 mm, 2 mm, 1 mm, and 0.5 mm. A 5 MHz, 64 element array (see Table I) is used in simulation (the wavelength is 1.26 mm at the array centre frequency), and the cracks are located at a depth of 20 mm from the array centre. The FMC array data used to calculate the TFM images in Fig. 2 are obtained using a hybrid model [START_REF] Zhang | The use of ultrasonic arrays to characterize crack-like defects[END_REF] which combines the classic linear systems model of the wave propagation [START_REF] Schmerr | Fundamentals of Ultrasonic Nondestructive Evaluation-A Modeling Approach[END_REF] and the defect scattering model (i.e., the scattering matrix). Fig. 3 shows the scattering matrices of these cracks as a comparison. As shown in Figs. 2(a)-2(b), the two larger cracks can be readily sized from their TFM results. However, the defect indication has a larger dimension in z-axis for the results of the two smaller cracks in Figs. 2(c)-2(d), which can be mistakenly assumed as an indication of cracks with vertical orientations. On the contrary, the amplitude of the scattering matrix drops monotonically as the crack size decreases. Combined with a change in the shape of the scattering matrix, all the cracks are seen to be distinguishable using the scattering matrix. Hence, using the scattering matrix is an efficient approach to characterising defects smaller than 2λ, and image-based characterisation can be used if the defect size exceeds 2λ [START_REF] Zhang | The use of ultrasonic arrays to characterize crack-like defects[END_REF]. I) is used in simulation to generate the FMC array data.

The use of the scattering matrix for inversion is not limited to certain defect types or parameters. To illustrate this, we consider in this paper characterisation of small surface-breaking notches in an Aluminium test specimen [START_REF] Velichko | Ultrasonic defect characterization using parametric-manifold mapping[END_REF]. The FMC data is measured from 8 EDM notches which have different sizes and orientation angles as specified in Table II (test configuration is same for all the notches and is shown in Fig. 4). This experimental configuration allows separation of the defect signal and backwall reflection in the time-domain, thus helps avoid the additional complexity caused by the need to suppress the strong backwall reflections when extracting the defect scattering matrix [START_REF] Zhang | Measurement of the ultrasonic scattering matrices of near surface defects using ultrasonic arrays[END_REF]. The scattering matrices are extracted from experimental FMC data using the sub-array imaging approach [START_REF] Zhang | The use of ultrasonic arrays to characterize crack-like defects[END_REF].

The scattering matrix of an ideal smooth surface-breaking crack can be obtained using a finite element modelling approach [START_REF] Velichko | A generalized approach for efficient finite element modelling of elastodynamic scattering in two and three dimensions[END_REF]. This model is based on an integral representation of the scattered wave field, and generation of the input plane wave and monitoring of the scattered wave are completed at nodes on two concentric surfaces that enclose the defect [START_REF] Velichko | A generalized approach for efficient finite element modelling of elastodynamic scattering in two and three dimensions[END_REF]. Hence, the I) is used in experiments. Measurement noise is an important challenge for the inversion problem. In this paper, we define noise as the difference between the measured scattering matrix and its idealised noise-free version (i.e., those included in the training set). For example, Fig. 5(a) shows the scattering matrix of Notch 1 extracted from experimental FMC data. The scattering matrix of an ideal surface-breaking crack having the same size and angle as Notch 1 is shown in Fig. 5(b), and Fig. 5(c) is the experimental noise. It can be seen from Fig. 5(c) that the actual experimental noise is coherent (as random noise can be removed by simple averaging or filtering operations). In order to simulate multiple realisations of coherent noise for sensitivity analysis, a 2D Gaussian rough surface model (also termed the general coherent noise model in Ref. [START_REF] Velichko | Ultrasonic defect characterization using parametric-manifold mapping[END_REF]) is adopted. The Gaussian rough surface noise model has been applied to different measurement scenarios (e.g., characterisation of small machined notches and elliptical voids) for uncertainty quantification [START_REF] Bai | The effect of distortion models on characterisation of real defects using ultrasonic arrays[END_REF], and parameters of the noise model such as the RMS roughness and correlation lengths can be determined using a maximum-likelihood estimation approach [START_REF] Velichko | Ultrasonic defect characterization using parametric-manifold mapping[END_REF]. In addition, as will be explained in Section III-A, the characterisation result (i.e., the posterior probability of the defect parameter) can be calculated analytically based on the general rough surface noise assumption. Fig. 5(d) shows a random realisation of the simulated noise obtained by running the Gaussian rough surface model. The noise database N db = [n 1 , n 2 , . . . , n M ] contains M =1000 such random noise realisations and is used to evaluate and compare robustness of the two inversion approaches considered in this paper. More details about the Gaussian rough surface model can be found in Ref. [START_REF] Velichko | Ultrasonic defect characterization using parametric-manifold mapping[END_REF].

III. THE DEFECT CHARACTERISATION APPROACHES A. The Bayesian approach

In this approach, the inversion problem is formulated as calculating P (p|s n ), i.e., the conditional probability of the defect parameter p given the measurement of the scattering matrix s n . Using Bayes' theorem, we have [START_REF] Hastie | The elements of statistical learning[END_REF]:

P (p|s n ) = P (s n |p)P (p) P (s n ) . (2)
In the above equation, the subscript n refers to the scattering matrix containing noise, such that

s n = s p + n, (3)
where s p is the scattering matrix of an ideal defect with parameter p and n represents noise [see for example, Fig. 5(d)]. Without a priori knowledge about the defect or measurement noise, it is reasonable to assume that the ratio of the prior distributions P (p) and P (s n ) is a constant C [START_REF] Velichko | Ultrasonic defect characterization using parametric-manifold mapping[END_REF]. If we further assume that the probability of noise n is independent of the defect parameter p (and hence, the scattering matrix s p ), we can conclude from Eqs. ( 2)-(3) that

P (p|s n ) = CP (n). (4)
Eq.( 4) provides a convenient way of calculating the defect characterisation result. Note that the noise vector n in the noise database follows a multivariate Gaussian distribution and its components become independent in principal-component space [START_REF] Velichko | Ultrasonic defect characterization using parametric-manifold mapping[END_REF]. For the noise database N db prepared in Section II, the principal component analysis (PCA) procedure can be performed in following two steps [START_REF] Jolliffe | Principal component analysis[END_REF]:

Step 1. Calculate the covariance matrix as N cov = (N db -N 0 )(N db -N 0 ) T /(M -1). N 0 = [n 0 , n 0 , . . . , n 0 ] and n 0 is the mean noise vector of the noise database.

Step 2. Perform eigendecomposition of the covariance matrix as N cov = V DV T . The defect characterisation result can then be obtained by calculating the noise n and its probability P (n). Based on the result of the PCA process, P (n) can be expressed as

P (n) = P (n pc ) = 1 (2π) Ns/2 |D| 1/2 exp -n pc V D -1 V T n pc , (5)
where

n pc = V T (s n -s p -n 0 ), (6)
represents the noise vector n in the principal-component space and N s is the dimensionality of the noise vector n pc . In Section IV, we calculated the conditional probability P (p|s n ) using Eqs. ( 4)-( 5) for the defect parameter p that is evenly distributed in the 2D parameter space (i.e., corresponding to the training points in Table III). The result of the Bayesian approach will then be presented in the form of probability maps.

B. Application of the machine learning approaches 1) Overview of machine-learning approaches: Let us consider a second approach for the defect characterisation task, which casts the problem in a supervised machine learning (ML) setting. The problem is set up in the following way: we employ the training and test databases described in Table III and train several machine learning algorithms on the training set to learn the mapping between the scattering matrices and the defect parameters (i.e., size and angle). It should be noted that the trained regression models are then inverse models and can be used directly to obtain estimates of defect parameters from scattering matrices, unlike classical inversion schemes. The learned models are evaluated on the test set and the experimentally measured scattering matrices of machined notches.

A major difference between machine learning approaches and the Bayesian approach described above is that the machine learning models do not account for the existence of noise directly in their formulations. However, an approach for taking into account the influence of noise is developed and applied while characterizing the experimental scattering matrices of Notches 1-8 as will be explained below.

Three machine learning approaches are considered in this paper: linear regression (used as a baseline), a kernel model (Gaussian process regression) that uses a dimensionality reduction technique, and a deep convolutional neural network that implicitly learns a lower-dimensional manifold.

2) Generation of the training and test sets and dimensionality reduction: We define the training and test sets for the machine learning approaches as the collection of input-target pairs:

S T ={(x 1 , y 1 ), . . . , (x N T , y N T )} ≡ (X T × Y T ) N T ,
where T = {tst, trn}.

In the above equation, the i-th couple (x i , y i ) represents the i-th input vector and the associated i-th target (or label) within S trn or S tst sets. Here we assume that x i ∈ R M ×1 and y i ∈ R P ×1 , where M represents the number of the scattering coefficients within a scattering matrix and P is the number of parameters to be retrieved. It can be seen from Table III that the training and test sets were generated with defect parameters on non-overlapping grids. The size of the training set was N trn = 976 samples whereas the test set held N tst = 180 samples. Note that the number of the scattering coefficients in a scattering matrix is equal to M = 64 2 = 4096 when using a 64 element array without the sub-array method. In order to remove such a large cardinality and mitigate the so-called curse of dimensionality issue [START_REF] Bellman | Adaptive Control Processes: A Guided Tour[END_REF], we apply PCA feature extraction on the scattering matrix data. Using PCA, we can significantly reduce the dimensionality of the input features by projecting the original data into principal-component space following the Steps 1 and 2 detailed in Section III-A.

3) Linear and kernel based models for regression: As a baseline, a linear regression (LR) model is used. Each sample in the scattering matrix database is considered as a vector (instead of a 2D image) by collapsing the rows, and the PCA is applied to reduce the dimensionality of the database. A linear model is then fitted using the extracted features and corresponding defect parameters. As a second model, a Gaussian process regression (GPR) [START_REF] Theodoridis | Machine Learning: A Bayesian and Optimization Perspective[END_REF] is applied to the scattering matrix database. Following Ref.

[49], we start with the PCA-extracted features and fit a kriging predictor which yields an approximate solution ŷ of the inverse problem of finding the parameters giving rise to the observation x as a linear combination of the input vectors:

ŷ = Ntrn i=1 λ i (x)y i , (8)
with the weights λ i (x) computed in a stochastic framework. Each parameter y i is modeled as a Gaussian random process, defined by its mean and covariance functions. The coefficients λ i (x) are chosen according to the following linear system of equations:

   k(||x 1 -x 1 ||) • • • k(||x 1 -x Ntrn ||) . . . . . . . . . k(||x Ntrn -x 1 ||) • • • k(||x Ntrn -x Ntrn ||)       λ 1 (x) . . . λ n (x)    =    k(||(x -x 1 )|| . . . k(||(x -x Ntrn )||    . (9)
The covariance function k(•) is unknown and one has to predict it using the training samples. In this work, we have used the rational quadratic covariance model.

4) Convolutional Neural Networks for regression:

Since the seminal 2012 paper on image classification [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF], deep convolutional neural networks (CNN) have shown that they are well adapted to all sorts of image-related tasks. Building upon the previous work [START_REF] Kulakovskyi | Convolutional neural network for automated diagnostic from guided wave imaging in a structural health monitoring context[END_REF], we propose to use a CNN network architecture specifically tailored on the scattering matrix data targeting inversion (regression) tasks. In the following, we refer to this architecture as "SMInvNet". SMInvNet is composed of two CNN stages made of two blocks. The first two blocks contain two convolutional layers and a max-pooling layer. For each block, the convolutional layer and the pooling layer has 32 and 64 feature map channels. The third block composed by a CNN layer and a max-pooling layer having size equal to 128 feature map channels, was concatenated to another CNN layer made by 128 feature map channels. The fully connected part is based on a dropout layer, with drop-out rate equal to 0.4, and two fully-connected layers made of 2048 neurons each. Rectified linear unit activation functions have been employed for all layers and Adam optimizer with learning rate equal to 1e-4 was employed to optimize the network weights and biases. The total number of trainable parameters was approximately equal to 13.8 million. The sketch of the architecture is provided in Fig. 6.

The SMInvNet takes the 2D scattering matrix as its input data. The convolutional layers extract features by convolving the input data with learnable kernels. An output convolutional feature I l+1 at layer l + 1 is computed from I l of the previous layer using the relationship I l+1 = C(I l ). More specifically, on a pixel-by-pixel basis we have

I l+1 m,n = a b w l+1 a,b η(I l m-a,n-b ) + b l+1 m,n , (10)
where m and n are the pixel coordinates of the image, w l+1 a,b are the weights, η is the activation function and b l+1 m,n are the biases. The weights and biases are to be learned on the data while the activation function is set in advance (here, we use a rectified linear unit function). The backpropagation algorithm and ADAM minimization algorithm [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF] are used to adjust the weights and biases by minimizing the loss function, which is defined here as the mean absolute error. IV. RESULTS

A. Results on the test set

In this section, we present the results obtained on the test set. Figs. 7(a)-7(b) show the size and angle results of the Bayesian approach, where the predicted parameter values are plotted against the true values. When implementing the Bayesian approach for the test samples, the noise model introduced in Section II was used for calculating the probability distribution of the defect parameter. The set of defect parameters that gives the maximum probability is then plotted as circles (for simulated noise-free data) and crosses (for experimental data).

An important difference between the Bayesian and machine learning approaches is that the noise model is not explicitly used in the machine learning approaches for defect characterisation. Based on the training data, we have tuned the LR and the GPR models by employing a 5-fold cross validation (CV) procedure. The CV has been employed to optimise the number of PCA components for the LR and GPR models, as well as the model hyper-parameters of the GPR covariance matrix. The number of principal components selected was equal to four, and this corresponds to 99% of the explained variance.

Figs. 7(c)-7(f) show the results obtained with the LR and GPR models. The baseline linear regression model performs poorly in this context and shows many severe outliers (sizing errors greater than 0.2λ and/or angle errors greater than 20 • , see Table IV), with high variance and bias in predictions. This implies that the relationship between the input (the scattering matrix) and the output (defect size and angle) is non-linear, which necessitates the use of more sophisticated machine learning algorithms. The GPR model handles defect size estimation better, even though a considerable variance is found in the predicted data (see Fig. 7(e)). In addition, high bias and variance have been observed for the angle results (see Fig. 7(f)). From the results of the GPR model, one can infer that it generalises to experimental data as well as it does on the synthetic simulation data. To quantify the performance of different inversion approaches, we use standard metrics on the predictions, namely mean absolute error (MAE), root mean square error (RMSE) as well as R 2 coefficient of determination (see Table V). In contrast to the LR and GPR models above where the feature extraction was performed via PCA, SMInvNet learns its own feature descriptors without the need for explicit dimensionality reduction. As can be seen in Figs. 7(g)-7(h), the SMInvNet provides notably better predictions than the LR and GPR models. Indeed, both bias and variance in the predicted results remain low. Moreover,on the same plots, one can notice that SMInvNet is able to generalize well on real experimental data.

Overall, the Bayesian approach is seen to have achieved the best characterisation performance on the simulated data. The SMInvNet model gives comparable characterisation accuracy for most test samples, but it produced increased number of outliers as can be found in Table IV. It is suggested that the use of (the optimised) noise model has contributed to the high accuracy of the Bayesian approach. Interestingly, the overall sizing performance is seen to be poorer for larger defects. The main contribution to this outcome comes from unfavorably oriented large cracks. Because of the small amplitude of the tip diffracted signal, characterisation of these unfavourably oriented cracks is more challenging and their sizes are more likely to be underestimated. For the smallest (0.55λ) cracks in the test set, the sizing results given by the SMInvNet model (mean:0.564λ, standard deviation: 0.038λ) are found to be more accurate than those of the Bayesian approach (mean:0.597λ, standard deviation: 0.044λ). II) using the Bayesian inversion approach. The colour bars show the normalised probability.

B. Uncertainty analysis

Figs. 8(a)-8(h) are the characterisation results of Notches 1-8 in the test specimen, obtained using the Bayesian inversion approach. The red squares in these results correspond to the true defect parameters, and they are in general close to the maximum probability points. Although the maximum probability points give large errors for Notches 2 and 6, the true parameter points are seen to be near the secondary peaks in Figs. 8(b) and 8(f), which demonstrates the robustness of the Bayesian approach.

As explained in the previous section, the ML model outputs a single prediction of defect size and angle for each test scattering matrix. In the context of defect characterisation, it is however of paramount interest to evaluate the uncertainty associated with the inversion procedure. Here we propose a methodology to empirically evaluate the characterisation uncertainty caused by the measurement noise for the SMInvNet model. This approach is similar in spirit to the one in Ref. [START_REF] Miorelli | Assessing performance of flaw characterization methods through uncertainty propagation[END_REF].

To add uncertainty estimates to our analysis, we use the noise database N db prepared in Section II. This database allows us to prepare experimental scattering matrices with additive noise. The noisy scattering matrices are obtained by adding one of the noise realisations to the noise-free data as follows:

∃k, x noisy = x i + n db k . (11)
For sake of brevity and due to the higher accuracy shown by the SMInvNet model, we focus only on the evaluation of this model. Each noisy scattering matrix yields a single characterisation result (i.e., a point in the two-dimensional defect parameter space), and Fig. 9 shows the 2D histogram of all 1000 characterisation results (bin centres of the histogram are the same as the grid points of the probability map in Fig. 8). As can be seen in Fig. 9, the probability distribution is quite peaked for Notches 4, 7 and 8 and thus implies high confidence. For Notches 1 and 3, the probability distribution is either very spread out or bi-modal, implying low confidence in the prediction. Notch 2 seems to be in-between since it is peaked but spreads out much into the defect parameter space.

The characterisation results shown in Figs. 8 and9 for the Bayesian and SMInvNet model are compared in Table VI, where the characterisation performance is evaluated qualitatively based on the bias and variance of the results. It can be concluded from Figs. 89and Table VI that the characterisation results of both approaches are indeed very similar for favourably oriented defects (Notches 4, 7, and 8). The Bayesian approach has produced more accurate results with lower uncertainty for Notches 1 and 5. The SMInvNet model is seen to give a higher variance in most cases, but it avoided the large sizing errors of the Bayesian approach for Notches 2 and 6. With the development of ultrasonic array transducers and data acquisition technology, the NDT industry now has access to larger amount of measurement data than ever before. Therefore, the selection of appropriate data processing technique (among many options that are available) has become a key issue for tasks such as defect characterisation. In this paper, we conducted a comparative study of the characterisation performance of the Bayesian inversion and machine learning approaches. The studied problem is the characterisation of surface-breaking cracks. The training and test sets consist of simulation data that cover the size range between 0.5λ and 2λ, and the angle range between -60 • and 60 • . In addition, experimental data measured from 8 machined notches are also used for testing. Among the three machine learning approaches considered in this paper, the SMInvNet (a convolutional neural network used for regression) achieves the best characterisation performance on the test set. The Bayesian inversion approach gives similar results for most test samples, and it also reduces the number of outliers compared to the SMInvNet model (number of large size and angle errors reduced by 5.3% and 3.2%, respectively). The characterisation uncertainty of both approaches is seen to be similar for favorably oriented defects. For unfavourably oriented defects, there is no clear "winner" and the performance of both approaches varied with different test samples. Overall, the results of the Bayesian approach exhibited higher bias while the SMInvNet model yielded higher variance, and hence, one faces the bias-variance tradeoff when choosing the appropriate characterisation approach for unfavourably oriented defects.

It is also interesting to compare the two approaches across other dimensions. Firstly, the Bayesian inversion approach is a specialized method that was developed specifically to address the defect characterisation problem based on the extraction of the scattering matrix, while the SMInvNet neural network reuses an architecture that could be applied to any type of images. Hence, depending on the background of the user, one may be more readily usable than the other. In particular, the acoustic domain knowledge is necessary for using the correct noise parameters in the Bayesian inversion method (in our case, 4 parameters are needed), which rely on the understanding of the underlying scattering matrix noise distribution. It is also desirable to develop a comprehensive noise model which takes into account all possible sources of noise for a given test configuration, and the characterisation result is expected to improve by adopting more accurate noise models. That being said, some sort of domain knowledge is also necessary to design the SMInvNet network. Here the machine-learning domain knowledge is required to choose the right architecture such as number of layers, number of of filters, and the size of pooling layers. Another dimension of interest to the end user is how different approaches deal with the uncertainty associated to the inversion problem. The Bayesian inversion approach naturally maps the confidence in the defect parameter to the probability. This is not the case for the SMInvNet model, and additional steps were adopted to allow for an uncertainty analysis. As we have shown in this paper, these two different paths led to comparable results.

A closely related question is how to interpret the uncertainty results. The Bayesian inversion method provides an explicit interpretation of the different terms that make up the probability, while the SMInvNet neural network is a "black box" with no straightforward way of interpreting the different model components. There are ongoing works which aim at developing tools that facilitate better interpretation of the internal structure of neural networks such as the one used in this paper. Hence, it is reasonable to expect that the "black box" issue can be resolved in the future by making use of the technology advances in the field of data science.

Both approaches are compatible with real-time use, since the learning (or fitting) step can be accomplished offline. The Bayesian approach would be restricted to problems of small cardinality since it is based on a grid-like evaluation of the posterior probability, while the neural network is not explicitly limited by this aspect of the inversion problem. This is a consequence of the curse of dimensionality when applied to grid methods.

At last, we note that turning to problems with a larger number of unknowns would affect the two approaches differently: the Bayesian model and its noise part could be fitted based on the summary statistics of the scattering matrix database, while the neural network would need to find a different path through the high-dimensional space in which it is trained using backpropagation.

An interesting research question based on this work would be asking how to combine the different types of imaging methods that use ultrasonic data coming from FMC data acquisition to enhance defect sizing performance. For instance, reconstruction algorithms such as TFM and plane wave imaging can help characterize wavelength-sized scatterers while Scattering Matrix images can help identify sub-wavelength reflectors. Toward this end, the approach proposed by [START_REF] Pyle | Deep learning for ultrasonic crack characterization in NDE[END_REF] could probably be investigated.

Fig. 1 .

 1 Fig. 1. Definition of the scattering matrix. The incident angle θ i and scattering angle θs are defined with respect to the vertical direction, and are positive if measured clockwise.

Fig. 2 .

 2 Fig. 2. TFM imaging results of horizontal cracks of sizes (a) 4 mm, (b) 2 mm, (c) 1 mm, and (d) 0.5 mm. A 5 MHz, 64 element linear array (see TableI) is used in simulation to generate the FMC array data.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Scattering matrices of horizontal cracks, where the crack sizes are: (a) 4 mm, (b) 2 mm, (c) 1 mm, and (d) 0.5 mm.

Fig. 5 .

 5 Fig. 5. (a) Experimentally measured scattering matrix of Notch 1, (b) scattering matrix of an ideal surface-breaking crack with the same size and angle as Notch 1, (c) experimental noise, and (d) a random noise realisation obtained from the noise model.

Fig. 6 .

 6 Fig. 6. Sketch of the SMInvNet architecture.

Fig. 7 .

 7 Fig. 7. Test set results of different inversion approaches in terms of defect size (left column) and angle (right column), where (a)-(b) Bayesian, (c)-(d) PCA-LR, (e)-(f) PCA-PGR, and (g)-(h) SMInvNet are used, respectively.

Fig. 8 .

 8 Fig.8. Uncertainty analysis for the experimental measurements of machined notches (see TableII) using the Bayesian inversion approach. The colour bars show the normalised probability.

Fig. 9 .

 9 Fig. 9. Uncertainty analysis for the experimental measurements of machined notches (see Table II) using the SMInvNet model. The colour bars show the normalised probability.

TABLE I SPECIFICATIONS

 I OF THE ARRAY TRANSDUCERS USED FOR IMAGING (5 MHZ) AND SCATTERING MATRIX EXTRACTION (2.5 MHZ).

	Number of Element Element Element Central
	elements	width	pitch	length	frequency
		(mm)	(mm)	(mm)	(MHz)
	64	0.53	0.63	10	5
	64	0.35	0.5	15	2.5

TABLE II SIZES

 II (WAVELENGTHS) AND ANGLES (DEGREES) OF THE EDM NOTCHES. WHEN THE FREQUENCY IS 2.5 MHZ, λ = 2.5 MM IN ALUMINIUM.

	Notch	1	2	3	4	5	6	7	8
	Size	0.80 0.83 0.92 1.13 1.20 1.24 1.39 1.70
	Angle	0	15	30	45	0	15	30	45

TABLE III RANGES

 III AND SAMPLING INTERVALS OF THE DEFECT PARAMETERS FOR THE TRAINING AND TEST SETS.

	Dataset	Size (wavelengths) Range Interval	Angle (degrees) Range Interval
	Training from 0.50 to 2.00	0.1	from -60 to 60	2
	Test	from 0.55 to 1.95	0.1	from -55 to 55	10

spatial size of the modelling domain around the defect is small (i.e., local to the defect) and it allows fast evaluation of the forward model. Scattering matrices used as the training and test examples are simulated by adopting this approach and more details about the training and test sets can be found in Table

III

.

TABLE IV NUMBER

 IV OF OUTLIERS (SIZING ERRORS GREATER THAN 0.2λ AND/OR ANGLE ERRORS GREATER THAN 20 • ) OF DIFFERENT INVERSION APPROACHES OBTAINED ON THE TEST SET.

	Parameter Bayesian PCA-LR PCA-GPR SMInvNet
	size	11	130	43	21
	angle	1	115	46	7

TABLE V ERROR

 V METRICS [SIZE (λ) AND ANGLE (DEGREES)] FOR PERFORMANCE EVALUATION OF THE STUDIED INVERSION ALGORITHMS: BAYESIAN, LINEAR REGRESSION (LR), GAUSSIAN PROCESS REGRESSION (GPR) AND DEEP CONVOLUTIONAL NEURAL NETWORK (SMINVNET).

	Metric	Bayesian size angle size angle size angle size angle PCA-LR PCA-GPR SMInvNet
	MAE	0.06 2.10 0.33 25.18 0.14 12.75 0.08 4.88
	RMSE 0.09 6.39 0.39 28.79 0.20 18.61 0.12 9.54
	R2	0.95 0.97 0.16 0.29 0.77 0.71 0.92 0.92

TABLE VI COMPARISON

 VI OF THE CHARACTERISATION RESULTS OF THE BAYESIAN AND SMINVNET MODELS FOR THE EXPERIMENTAL MEASUREMENTS OF

		MACHINED NOTCHES.
	Notch	Bayesian	SMInvNet
	1	High bias / small variance High bias / high variance
	2	High bias / high variance High bias / high variance
	3	Low bias / high variance	Low bias / high variance
	4	Low bias / low variance	Low bias / low variance
	5	Low bias / low variance	Low bias / high variance
	6	High bias / high variance	Low bias / low variance
	7	Low bias / low variance	Low bias / low variance
	8	Low bias / low variance	Low bias / low variance
		V. DISCUSSIONS AND CONCLUSIONS

This work was supported by National Natural Science Foundation of China under grant number 52005205, European Union's Horizon 2020 Research and Innovation Programme under grant agreement number 755500, and the Engineering and Physical Sciences Research Council (UK, EPSRC) under grant number EP/L022125/1. Long Bai is with the State Key Laboratory of Digital Manufacturing Equipment and Technology,


Yiran Zhao

Yoan Léger

Julie Descamps

Neso Sojic
email: sojic@u-bordeaux.fr

Gabriel Loget
email: gabriel.loget@cnrs.fr

Off-Grid Electrogenerated Chemiluminescence with Customized p-i-n Photodiodes

Keywords: photoelectrochemistry, silicon, luminol, photodiode, electrochemiluminescence

published or not. The documents may come

Introduction

Electrochemiluminescence (ECL) is the generation of light triggered by an electrochemical reaction at an electrode surface. This process requires an electrolyte containing a dissolved luminophore that is electrochemically promoted to an excited state and relaxes through the emission of a photon. [1][START_REF] Sojic | Analytical electrogenerated chemiluminescence: from fundamentals to bioassays[END_REF][START_REF] Liu | [END_REF] ECL is a sensitive analytical technique that is widely employed for immunoassays, clinical diagnosis [4][5][6][7][8][9][10] and imaging. [11][12][13] Photoinduced ECL (PECL) is a type of ECL that has been developed as an original light conversion strategy. [14][15][16][17] In PECL, a semiconductor immersed in the ECL electrolyte is used as a photoelectrode and, under illumination with the appropriate wavelenght (λexc), generates ECL (λexc is converted into the emission wavelength, λem) via the transfer of photogenerated charge carriers. It has been demonstrated that this approach can be reliably employed for Stokes (reminiscent to fluorescence, λexc < λem) [18] and anti-Stokes (reminiscent to upconversion, λexc > λem) [15,19,20] light conversion on the photoelectrode surface. The reported examples of wireless ECL [START_REF] Li | Analytical Electrogenerated Chemiluminescence: From Fundamentals to Bioassays[END_REF][START_REF] Chow | [END_REF][23] (and PECL) [24] still require the use of electrical power sources. [25][26][27] Latest research on PECL resulted in the concept of all-optical ECL (AO-ECL), [28] in which ECL is generated by a semiconductor device interfaced with electrocatalysts immersed in the appropriate electrolyte. AO-ECL can be easily activated remotely and does not require an electrical power source and electrical connections, it thus constitutes the simplest ECL configuration existing. AO-ECL has been so far reported only once, with specifically tailored nanostructured Au-pnn ++ Si-Pt junctions [28] produced by a combination of dopant implantation, nanosphere lithography, and reactive ion etching. Despite the simplicity of the AO-ECL concept compared with ECL, the complexity, and the specific expertise required for manufacturing these complicated junctions prevent AO-ECL from being widely used. Due to its narrow bandgap of 1.1 eV allowing its activation by near-IR or solar illumination, the use of Si seems particularly adapted for AO-ECL, especially because it affords low-cost and efficient Si components whose reliability and efficiency have been optimized for decades and that are readily available on the market. In particular, the p-i-n junction is a variant of the well-known p-n junction in which an intrinsic layer (i-type layer) is located between the p-type and the ntype layers. One advantage of p-i-n over p-n homojunction is the wider depletion region that allows a better conversion efficiency for near-IR excitation, making them one of the most commonly-used photodetectors. [START_REF] Sze | Physics of Semiconductor Devices[END_REF] Besides junction considerations, a strategy for promoting AO-ECL is to tailor the electrolyte composition with reactants enabling high reactivity at low overpotentials. In this respect, the anodic ECL 3-aminophthalhydrazide (luminol)-H2O2 system is particularly well adapted as it is known to emit at low anodic overpotentials. [15,[START_REF] Zhao | [END_REF] In addition, it contains H2O2, which is not only the ECL co-reactant (essential to promote an efficient emission) but is also a species that is electrocatalytically reduced at low cathodic overpotentials on an appropriate catalyst. [31] In this communication, we report a photoelectrochemical device, inspired from monolithic photoelectrochemical water splitting cells, [32][33][34][35][36][37][38][39][40][START_REF] Lewis | Proc. Natl. Acad. Sci[END_REF], [START_REF] Lee | [END_REF][43][44] based on a functionalized p-i-n Si photodiode operating in the photovoltaic mode, which produces spontaneous AO-ECL of the luminol-H2O2 aqueous system (see Figure 1a). This device generates a bright light emission at 425 nm without any external source of current but under near-IR or sunlight excitation. This approach is highly promising for future applications in analytical chemistry and clinical diagnosis. Our photoelectrochemical devices are based on commercially available, low-cost (~1 €/unit) pi-n photodiodes, shown in Figure 1b. These components are made of a photosensitive area comprising the p-i-n Si junction (black square in Figure 1b) encapsulated in a transparent polymer and two metal leads (grey pads in Figure 1b) which are classically employed for component integration on a printed circuit board. These p-i-n photodiodes are well-suited to our needs because of their low price and small dimensions (photosensitive area = 4x5 mm², lead area = 0.7x5 mm²). As depicted in Figure 1c, the photoactive junction is composed of intrinsic (i.e., undoped) Si (i-Si) sandwiched between p-type Si (p-Si) and n-type Si (n-Si), the p-Si and the n-Si layers are connected to the metal leads via Ohmic contacts. [45] Upon illumination of the photosensitive area (from the p-Si side) with photons from a monochromatic or a polychromatic source, having higher energy than the Si bandgap (i.e., when λexc < 1.13 µm), e - /h + pairs are generated in the conduction/valence band of Si, and a photovoltage is spontaneously produced between the anode and the cathode leads (Figure 1c). If the opencircuit photovoltage (Voc) is sufficiently high, electrochemical reactions promoted by the e -and h + photogenerated in the depletion region should occur at the cathode and the anode leads, respectively, as shown in Figure 1a. If one of these reactions produces ECL (i.e., generates λem photons) this device should be suitable for spontaneous λexc-to-λem light conversion (Figure 1d).

Results and discussion

We first discuss the photovoltaic component of the device, that is, the (unmodified) p-i-n junction, studied in the air. The external quantum efficiency (EQE) spectrum, shown in Figure 1e, reveals that the junction has the highest photoconversion efficiency in the near-IR region with an EQE of 87% at 850 nm. To further assess its photovoltaic properties, current density/bias curves were recorded in the dark (Figure 1f, black curve) and under 850 nm illumination (Figure 1f, colored curves) at different illumination power densities (2.5 < PLED < 18 mW cm -2 ). The black curve shows that the p-i-n junction does not produce significant forward current before 0.3 V. The colored curves reveal that it produces a short-circuit photocurrent density (jsc) in the mA cm -2 range, which varies linearly with PLED (Figure S1) as well as Voc values varying from 0.45 to 0.53 V with PLED increasing from 2.5 to 18 mW cm -2 .

The maximum jsc values determined from these measurements are relatively in good agreement with the EQE data of Figure 1e (with an average jsc variation of +16% attributed to experimental and component-to-component variations). Next, the electrochemical behavior of the device is discussed. Since the p-i-n junction is partially shielded within an insulating matrix, the only surfaces that can promote solid-liquid charge transfer are the two conductive metal leads, shown in the photograph of Figure 1b and depicted in grey in Figure 1a and Figure 2a. Energy dispersive X-ray spectroscopy (EDS) measurements revealed that the lead surface is composed of Sn (Figure 2b, black spectrum), a transition metal subject to anodic degradation at high pH, [START_REF] Pourbaix | Atlas of electrochemical equilibria in aqueous solutions[END_REF] which has a poor electrocatalytic activity for H2O2 reduction. To overcome these limitations, the surface of the leads was specifically modified with materials known for their stability, efficient ECL response, and good electrocatalytic properties. The electrochemical behavior of the unmodified (Electrode 1 in ) at a low onset potential of -0.4 V and an anodic current at an onset potential of 0.17 V. To decrease both overpotentials required to reduce HO2 -and generate ECL, and to increase the reaction kinetics, the leads were modified as depicted in Figure 2a. Carbon is the best material for ECL generation due to its inertness, its good electron transfer performance toward ECL reactants, and its high ECL intensity. [START_REF] Valenti | [END_REF] It was therefore naturally considered for our device. For that, a commercial carbon paint coating, composed of carbon particles dispersed in a fluoroelastomer matrix was directly applied on the Sn lead. Scanning electron microscopy (SEM) (Figure S2) and EDS analyses (Figure 2b, blue spectrum) indicate that the layer was homogeneous and mainly composed of C, F, and Al (Al-rich particles (likely Al2O3) were identified in the commercial carbon paint, see Figure S3 and Section 3.1 in the SI). The blue CV of Figure 2c reveals that the carbon coating increases considerably the oxidation kinetics, in addition to protecting the underlying Sn lead from previously mentioned corrosion. If Sn is expected to be more stable under cathodic bias, [START_REF] Pourbaix | Atlas of electrochemical equilibria in aqueous solutions[END_REF] it presents poor electrocatalytic activity, as confirmed by its low onset potential for HO2 -reduction (Figure 2c). Pt is among the best electrocatalysts for H2O2 oxidation, [31] and is also very efficient at alkaline pH. [48] For that reason, Pt was selected as the cathodic catalyst and it was spontaneously deposited on the cathode lead, by a simple wet chemical process, to improve drastically its electrocatalytic performance. For that, the Sn lead was soaked for 20 min in a solution containing H2PtCl6, in which Pt replaced Sn atoms at the lead surface by galvanic displacement (for more information, see Section 3.2 in the SI). [49] SEM (Figure S4) and EDS analyses (Figure 2b, red spectrum) confirm that the chemical modification produced a Pt coating on the Sn lead. The red voltammogram of Figure 2c shows that the Pt coating considerably decreased the overpotential for HO2 -reduction, i.e., the onset potential was increased by +300 mV, resulting in a value of -0.1 V, coherent with previous literature. [48] Based on the difference between the anodic and cathodic onset potentials determined from the CVs, we can estimate that a bias of 0.27 V (indicated in purple in Figure 2c) is required to generate both oxidation and reduction reactions simultaneously. This was confirmed by voltammetry performed in a two-electrode configuration, depicted in the Inset of Figure 2d. In this experiment, a bias was applied using a potentiostat between the carbon-modified anode (Electrode 2) and the Pt-modified cathode (Electrode 3). The voltammogram of Figure 2d (corroborated by impedance measurements, Figure S5) shows that indeed, current starts to flow around 0.27 V. These measurements suggest that a photoactive junction generating a Voc > 0.27 V should be able to trigger the reactions at the modified electrodes. Since Figure 1f revealed that illuminated photovoltaic pi-n junction produced a Voc values comprised between 0.45 and 0.53 V, it indicates that these systems can be coupled for AO-ECL generation without any electrical power sources. The potential/current operation conditions can be estimated from the intercept of the current-bias curves recorded with the p-i-n junction and the electrochemical system. [50] These points, represented as colored stars in Figure 2d, show that the current expected to flow into the coupled device varies from to 11 to 18 µA (when PLED is increased from 2.5 to 18 mW cm -2 ), which is sufficient to generate bright ECL. Additional experiments, presented in Figure S6, demonstrate that the control over simple geometrical parameters of the electrodes can be used to considerably improve the light-to-electricity conversion efficiency of the coupled system, as we were able to operate very close to the maximum power of the photovoltaic p-i-n junction with adapted electrode geometries. However, for cost and miniaturization reasons, the focus of this study is given on the smallest modification of the leads, therefore geometrical aspects are not further considered in the following.

Finally, spontaneous AO-ECL emission with the modified p-i-n photodiode is reported. As presented in Figure 3a, after functionalization of the leads, the device was immersed in a quartz cuvette filled with the previously-described ECL electrolyte. The excitation source, namely the near-IR LED (λexc = 850 nm) or a solar simulator (reference AM 1.5G spectrum is shown in grey in Figure 1d) was placed in front of the photosensitive area and pictures of the anode lead were acquired using a smartphone camera. During experiments, the liquid phase was vigorously stirred to remove the bubbles produced at the solid/liquid interface due to the generation of O2 and N2 by H2O2 and luminol decomposition. [START_REF] Zhao | [END_REF] An ambient-light photograph of the anode is presented in the left panel of Figure 3b. In this image, the black surface (whose contours are indicated by a dotted white line) is the electrochemically active (1.9 mm²) carbon area, and the blue part on the left is the insulating resin that was used to define the active surface area. Figure 3b also presents photographs of the anode in measurement conditions (taken in a dark box where only the excitation light illuminates the photosensitive area of the p-i-n diode). Indeed, ECL measurements for analytical applications are typically recorded in a dark environment. [1- 10] No light is emitted when λexc is off (middle panel) whereas, when λexc is applied, bright blue AO-ECL is generated (right panel) and easily visible with naked eyes at λem = 425 nm (Figure 1d, blue spectrum), demonstrating the AO-ECL emission with this device. As shown in the bottom row of Figure 3b, these devices can be used for a long time. Chopped light measurements indicated that AO-ECL responds instantaneously to the illumination (Figure S7). Even if bubble generation at the anode was found to alter AO-ECL intensity and homogeneity, stable AO-ECL could be recorded after 1 hour of operation without changing the device nor the electrolyte. In order to avoid electrode fouling caused by bubble generation, the addition of a surfactant, Triton X-100 (0.1 %) was found efficient to improve the stability of the emission. Interestingly, the overall AO-ECL intensity was found to vary as a function of experimental parameters. AO-ECL light intensity increases progressively with illumination power density, PLED (Figure 3c). This behavior is consistent with the anticipated behavior predicted by the current-bias responses of the illuminated p-i-n junction and the electrochemical system (Figure S8). This shows that the intensity of the AO-ECL intensity can be controlled by the illumination power of the LED. Figure 3d presents the evolution of the AO-ECL intensity as a function of H2O2 concentration, with a notable linear correlation (R² = 0.965) in the 30-to-230 mM range. The dependency is caused by the fact that H2O2 (more precisely, its deprotonated form, HO2 -), participates in the emission mechanism of the ECL reaction (Figure 1a). Furthermore, H2O2 concentration also affects the counter-reaction kinetics. This effect is particularly interesting because it suggests that our AO-ECL approach has the potential to be employed for bioanalysis, as H2O2 is an important biological marker, and is known to play a role in many ECL bioassays. [51][52][53][54] In addition, Figure S9 shows that the concentration of H2O2 affects the AO-ECL emission lifetime, due to its consumption over time. Finally, our approach was assessed using sunlight as an excitation source instead of near-IR light. This is relevant because, due to the narrow bandgap of Si (pink dashed line in Figure 1d), our device is active within a large portion of the solar spectrum (grey spectrum in Figure 1d) with a good photoconversion efficiency (Figure 1e). In these experiments, the same experimental setup was employed except that a solar simulator (AM 1.5G, 100 mW cm -2 ) was used as the excitation source (Figure 3a). As the excitation (the photosensitive area) site is spatially decoupled from the emission site (the anode lead), removal of the noise originating from the excitation can be easily performed. To do so, a mask was placed outside the cuvette to isolate the anode lead from the illumination source. Figure 3e shows that this approach is very efficient since bright AO-ECL light was recorded, using the smartphone camera, under illumination of the photoactive area of the photodiode with simulated sunlight. This remarkable result demonstrates the possibility to generate AO-ECL emission with sunlight, opening the way to the development of portable bioanalytical methods and point-of-care testing with extremely simple devices; without the need for power generator and specific electrochemical setups.

Conclusion

In this article, we have reported off-grid ECL generation with devices that can be manufactured by any laboratory. Using this approach, ECL light can be emitted without any external electric power sources, electrical wires, and connectors. The photoelectrochemical devices are based on cheap and small (~0.27 cm²) commercial p-i-n Si photodiodes that integrate, in a single component, the photovoltaic junction and the reactive electrodes, tailored directly onto the component contact leads by wet simple processes. Under near-IR and solar illumination, the p-i-n Si junction provides the driving force (photovoltage) required to achieve simultaneously the ECL reaction of luminol-H2O2 at the carbon-coated anode and the counter-reaction at the Ptmodified cathode. These devices can be operated under monochromatic or polychromatic excitation. In monochromatic illumination mode, we have shown that the device allows converting λexc = 850 nm (invisible to the human eye) in λem = 425 nm, resulting in an anti-Stokes shift of -425 nm. This illumination mode is beneficial for measurements with ultralow background signal. In polychromatic mode, we have shown that these devices can be used with simulated sunlight, which opens a promising perspective for outdoor, solar-powered AO-ECL sensing devices. Compared with the previously reported approach, [28] which involved a more complicated junction, here, the spatially-decoupled emission and excitation sites are particularly adapted for removing background optical noise originating from the excitation. The AO-ECL intensity depends on the illumination power density, thus, the device can be used as a photodetector. In addition, AO-ECL intensity depends on H2O2 concentration, which makes this device suitable for bioanalytical detection. In follow-up studies, the Pt coating could be replaced by other inorganic or biocatalysts (e.g., Prussian blue layers or Horseradish peroxidase), thereby further reducing cost and improving efficiency of the device. Although these devices have an overall light-to-light efficiency that is too low to be used for displays, they offer major advantages in the field of detection. In addition to opening fundamental new doors in luminescence, this system, because of its size, cost, and ease of use, also holds promise for remote and portable bioanalysis.

Off-grid all-optical ECL is generated with devices based on cheap and small commercial p-i-n Si photodiodes that integrate, in a single component, the photovoltaic junction and the reactive electrodes, integrated directly onto the contact leads by wet simple processes. Because of their size, cost, and ease of use, these systems are promising for remote and portable bioanalysis.

Characterization

Scanning electron microscopy (SEM) was performed using a JSM 7100F (JEOL). Energy-dispersive Xray spectroscopy was done using a Silicon Drift Detector (SDD) -X-Max (Oxford Instruments) and the AZtecEnergy software. Electrochemical measurements were performed with a SP300 (Biologic) potentiostat, the spectra were acquired with a Flame (Ocean Optics) spectrometer. The cyclic voltammetry study was performed with an electrolyte degassed with Ar. EQE measurements were performed in air at a bias of 0 V between the leads with a CIMPS-QE IPCE 3 workstation (Zahner) comprising a TLS03 tunable light source controlled by a PP211 potentiostat (Zahner). The AO-ECL See Supplementary Note 3.1. for an explanation of this data. As shown by the above figure, at low bias, the system studied in Figure 2d can be qualitatively modeled by the equivalent circuit shown in the inset of panel a), which comprises two constant phase elements (CPEs) in parallel with resistances (representing qualitatively the charge transfer processes occurring at both electrodes), and a series resistance. Although the fit seems to match relatively well the Nyquist and the Bode plots of Figure S5a, the low exponent values of the CPEs (~0. 7-0.8) indicate that this equivalent circuit gives only an indicative overview of the system and is not suitable for extracting physical parameters from the impedance data. A more complicated and detailed analysis taking into account all interfaces, their heterogeneity and roughness may be performed to enable a more precise analysis. An interesting result that can be observed from the Bode plots of Figure S5c is that the semicircle decreases when the bias increases, implying a decrease of the charge transfer resistance R1 and R2, in good agreement with the voltammogram presented in Figure 2d.

Figure 1 .

 1 Figure 1. Principle and photovoltaic characterization of the p-i-n junction. (a) Scheme showing the band diagram of the device consisting of a p-i-n Si photodiode interfaced with a carbon and a catalyst layer, immersed in the ECL electrolyte comprising luminol and H2O2. (HO2 -is the deprotonated form of H2O2) Ef, Ec, and Ev are the Fermi level, the conduction, and the valence band edges, respectively. (b) Photograph of a p-i-n Si photodiode (the ruler on the top edge indicates 1 line per mm). (c) Scheme showing the structure of a p-i-n photodiode. (d) Normalized spectra of the incident near-IR LED (brown curve), the reference AM 1.5G solar spectrum (grey curve), and AO-ECL emission of luminol/H2O2 (blue curve), the Si bandgap is indicated by a pink dashed line. (e) External quantum efficiency (EQE) spectrum of the p-i-n photodiode. Inset: Scheme of the photodiode components. (f) Photovoltaic current density-bias characteristics of the p-i-n photodiode at different IR power densities. The short-circuit photocurrent density (jsc) and the photovoltage (Voc) for the red curve are indicated by arrows.

Figure 2 .

 2 Figure 2. Characterization of the reactive surfaces and of their electrochemical properties. (a) Schemes of Electrode 1: unmodified Sn lead, Electrode 2: lead coated by carbon paint and an insulating polymer, and Electrode 3: lead modified by Pt by spontaneous deposition. (b) EDS analysis of the three electrodes. (c) Cyclic voltamograms recorded (using a potentiostat and a three-electrode cell configuration) with working Electrodes 1-3 indicated in panel (a) in the ECL electrolyte. (d) Linear sweep voltammogram recorded (using a potentiostat and a twoelectrode cell configuration) with Electrode 2 as an anode and Electrode 3 as a cathode in the ECL electrolyte (scheme in inset). Colored stars plot the operating current-bias conditions (estimated from intersect of the voltammogram (black curve) and the current-bias characteristics of the p-i-n photodiode (colored curves of Figure 1f)) of the coupled electrolysis/illuminated p-i-n junction with PLED = 2.5 (dark blue), 6 (clear blue), 9 (green), 14 (orange), and 18 mW cm -2 (red). ECL electrolyte composition: 10 mM luminol, 33 mM H2O2 in 0.1 M KOH. Scan rate: 50 mV s -1 .

Figure 2a )

 2a Figure 2a) and modified leads (Electrodes 2 and 3 in Figure 2a) was studied by voltammetry in the ECL aqueous electrolyte (containing luminol, H2O2, and KOH as the supporting electrolyte) with a conventional three-electrode setup. The measured cyclic voltammograms (CVs) are shown in Figure 2c. The black CVs indicate that the unmodified lead produced a cathodic current that corresponds to the reduction of HO2 -(i.e., the deprotonated form of H2O2 , pKa = 11.65) at a low onset potential of -0.4 V and an anodic current at an onset potential of

Figure 3 .

 3 Figure 3. All-optical ECL. (a) Scheme showing the experimental setup. (b) Top row: photographs of the anode under ambient light (left) and in dark conditions without (middle)/with (right) near-IR excitation of the p-i-n photodiode (λexc = 850 nm, PLED = 18 mW cm -2 ). Bottom-row: photographs of the anode under near-IR illumination (λexc = 850 nm, PLED = 18 mW cm -2 ) during operation at 1 (left), 30 (middle), and 60 min (right). The active surface of the anode lead is delimited by a white dotted line. Electrolyte composition: 10 mM luminol, 66 mM H2O2, 0.1 M KOH. (c) AO-ECL intensity as a function of PLED, [H2O2] = 33 mM, under near-IR illumination (λexc = 850 nm). (d) AO-ECL intensity under near-IR illumination (λexc = 850 nm) as a function of H2O2 concentration, PLED = 18 mW cm -2 . (e) Photographs of the anode under ambient light (left) and in dark condition without (middle)/with (right) solar illumination of the photodiode. Electrolyte composition: 10 mM luminol, 33 mM H2O2, 0.1 M KOH. In b) and e), the active surface of the anode lead is delimited by a white dotted line.

  Y. Zhao, Y. Léger, J. Descamps, N. Sojic,* G. Loget *Off-grid electrogenerated luminescence with customized p-i-n photodiodestransparent square, in contact with the cuvette. The ECL emission pictures were recorded exactly as same as the case of infrared LED. The aqueous electrolyte was composed of 10 mM luminol and H2O2 in 0.1 M KOH (measured pH = 12.85).

  images and videos were analyzed with the open-source ImageJ software, which allowed the determination of the AO-ECL intensity values. AO-ECL experiments of Figure 3 were performed under stirring to generate convection and remove the bubbles produced at the lead surface.

Figure S2 .

 S2 Figure S2. Scanning electron microscopy (SEM) pictures of a) a photodiode lead partially coated with carbon paint (scalebar = 100 µm), b) The uncoated area (Sn) (scalebar = 10 µm), and, c) The coated area (carbon paint) (scalebar = 10 µm).

Figure

  Figure S3. a,b) SEM pictures (scalebars = 1 µm) and c-f) corresponding energy dispersive X-ray spectroscopy (EDS) maps (scalebars = 2.5 µm) of carbon paint applied to a photodiode lead. a) SEM picture. b) Same SEM picture with locations for chosen Al-rich areas (dashed circles) and O-rich areas (purple circles), determined from the EDS maps. c) Aluminum EDS map. d) Carbon EDS map. e) Fluorine EDS map. f) Oxygen EDS map.

Figure

  Figure S5. a) Nyquist plot showing the impedance measurements performed with the 2-electrode setup shown in the inset of Figure 2d with a bias of 0.2 V. Disks are experimental data points, the line is a fit obtained with the equivalent circuit shown in inset. b) Bode plot showing the impedance measurements performed with the 2-electrode setup shown in the inset of Figure 2d with a bias of 0.2 V. Disks are experimental data points, lines are fits obtained with the equivalent circuit shown in the inset of panel a). c) Nyquist plots obtained at different bias values. Disks are experimental data points; lines are fits obtained with the equivalent circuit shown in inset of panel a).

Figure S6 .

 S6 Figure S6. Linear sweep voltammograms (blue curves) recorded with a two-electrode setup using a large Pt cathode and a carbon paint anode with a surface increasing from 0.14 to 1.5 cm². The voltammogram recorded with electrode 2 as an anode and electrode 3 as a cathode (presented in Figure 2d), is also plotted (thick blue line). i-bias characteristics of the PIN photodiode (green curve) at 9 mW cm -2 and corresponding power-bias curve (purple curve). Electrolyte composition: 10 mM luminol, 33 mM H2O2 in 0.1 M KOH; Scan rate: 50 mV s -1 .

Figure S7 .

 S7 Figure S7. Graph showing the normalized AO-ECL intensity as a function of time during intermittent 850 nm excitation. Electrolyte composition: 10 mM luminol, 100 mM H2O2, 0.1% Triton X-100. in 0.1 M KOH. In this experiment, Triton X-100 was added to the electrolyte improve the bubble release from the surface.

  Figure S8. Curve showing the variation of the expected value of i as a function of PLED. i was extracted as the intercepts of the voltammogram presented in Figure 2d and the i-bias characteristics of the illuminated p-i-n junction.

Figure S9 .

 S9 Figure S9. Graphs showing the AO-ECL intensity as a function of time for two independent devices during constant illumination (λexc = 850 nm) for two H2O2 concentration, 33 and 100 mM. Electrolyte composition: 10 mM luminol, 0.1% Triton X-100. in 0.1 M KOH. In this experiment, Triton X-100 was added to the electrolyte improve the bubble release from the surface.

Acknowledgements

This work is funded by ANR (LiCORN -ANR-20-CE29-0006). Joudi Dabboussi is acknowledged for the help with the photodiodes. Dr. Yann Leroux (CNRS ISCR) is acknowledged for discussions on carbon coatings. Dr. Cassandre Quinton is acknowledged for discussion and spectra acquisition. Trung-Nghia Nguyen (ISCR) is acknowledged for his help with the camera setup. Loic Joanny (ScanMat/CMEBA) is acknowledged for SEM and EDS measurements. Prof. Klaus Meerholz and Dr. Dirk Hertel are acknowledged for valuable discussions.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.

Supporting Information

Off-grid electrogenerated luminescence with customized p-i-n photodiodes

Yiran Zhao, Yoan Léger, Julie Descamps, Neso Sojic,* Gabriel Loget*

Experimental details 1.Materials and reagents

The p-i-n diode (BPW34, Vishay) was purchased from RS Components. The mounted 850 nm light emitting diode (LED, M850L3) and the 750 nm shortpass filter (FESH0750) were purchased from Thorlabs. Luminol (97%), hydrogen peroxide (30%, VLSI), and Triton X-100 (laboratory grade) were purchased from Merck. Potassium hydroxide (analytical reagent grade) was purchased from Fisher Scientific. Dihydrogen hexachloroplatinate(IV) hydrate (99.9%, H2PtCl6, xH2O) and carbon paint (DAG-T-502) were purchased from Alfa Aesar and Ted Pella, respectively. The ultrapure water had a resistivity of 18.2 MΩ cm (Purelab Classic UV).

Device preparation

For ease of manipulation and to maintain the AO-ECL device in the cell, the backside of the p-i-n diode was first fixed on a glass capillary "holder" using epoxy resin (Loctite 9460, Henkel) which was cured at 90°C. The anode of the p-i-n diode was covered with a layer of carbon paint and dried in air for 5 min.

The size of the electroactive area on the anode was defined by covering the carbon with an insulator varnish, which was then dried in air for 10 min. The modification of the cathode of the p-i-n diode was done by immersing the lead in an aqueous solution of 10 mM H2PtCl6 for 20 min, followed by rinsing with ultrapure water and drying under Ar flow. The modified p-i-n diode was placed into a quartz cuvette and the glass capillary "holder" was maintained with a pincer inside a black box. The infrared LED was placed at a distance of 2 cm from the p-i-n diode and pictures and movies of the anode were recorded with the camera of a smartphone (OPPO R11s, expert mode, white balance: 4800 K, exposure compensation: 0, sensitivity: 1000, focus parameter: 1 and exposition time of 1 s) covered by the 750 nm shortpass filter and a 24x macro lens, to capture ECL emission. The experiments with the solar illumination are slightly different from that with the near IR LED in order to isolate ECL emission from residual sunlight. The modified p-i-n diode was placed into a quartz cuvette and the Si square was in contact with the wall of the cuvette. A mask made with black electric tape was placed on the cuvette wall, leaving a transparent square of 3x3 mm 2 in front of the photoactive area of the p-i-n diode. A light beam (3 mm diameter) generated by a solar simulator (LS0106, LOT Quantum Design) equipped with an AM 1.5G filter was transmitted through an optical fiber whose extremity was placed on the See Supplementary Note 3.2. for the explanation of this data.