Aishwarya Venkataramanan

Pierre Faure-Giovagnoli

Cyril Regan

David Heudre

Cécile Figus

Philippe Usseglio-Polatera

Cédric Pradalier

Martin Laviale
email: martin.laviale@univ-lorraine.fr

Usefulness of synthetic datasets for diatom automatic detection using a deep-learning approach

Keywords: synthetic dataset, images, diatoms, automatic detection, deep learning

Benthic diatoms are unicellular microalgae that are routinely used as bioindicators for monitoring the ecological status of freshwater. Their identification using light microscopy is a time-consuming and labor-intensive task that could be automated using deep-learning. However, training such networks relies on the availability of labelled datasets, which are difficult to obtain for these organisms. Herein, we propose a method to generate synthetic microscopy images for training. We gathered individual objects, i.e. 9230 diatoms from publicly available taxonomic guides and 600 items of debris from available real images. We collated a comprehensive dataset of synthetic microscopy images including both diatoms and debris using seamless blending and a combination of parameters such as image scaling, rotation, overlap and diatom-debris ratio. We then performed sensitivity analysis of the impact of the synthetic data parameters for training state-of-the art networks for horizontal and rotated bounding box detection (YOLOv5). We first trained the networks using the synthetic dataset and fine-tuned it to several real image datasets. Using this approach, the performance of the detection network was improved by up to 25% for precision and 23% for recall at an Intersection-over-Union(IoU) threshold of 0.5. This method will be extended in the future for training segmentation and classification networks.

Introduction

Diatoms are unicellular microscopic algae found in all aquatic environments, where they play a fundamental role in ecosystem functioning [START_REF] Lobo | Diatoms as bioindicators in rivers[END_REF]. They are widely used in water quality monitoring due to their high sensitivity to water quality (e.g., temperature, pH, nutrients, micropollutants) and habitat conditions [START_REF] Lobo | Diatoms as bioindicators in rivers[END_REF]. This is especially the case for freshwater benthic diatoms (i.e, diatoms living attached to substrates in rivers and lakes), in the context of the European Water Framework Directive implementation [START_REF] Coste | Improvements of the biological diatom index (bdi): Description and efficiency of the new version (bdi-2006)[END_REF]. Diatom-based biological indices are based on the occurrence of indicator species in a natural assemblage. Diatom species are identified based on morphological features (e.g., size, ornamentation) by human experts using light microscopy. This task can be time-consuming and subject to multiple biases due to the quality of the instrument or the level of expertise in diatom taxonomy. Therefore, attempts to automate the process have been ongoing since the early 90s [START_REF] Du Buf | Diatom identification: A double challenge called adiac[END_REF][START_REF] Pedraza | Automated diatom classification (part b): a deep learning approach[END_REF][START_REF] Pedraza | Lights and pitfalls of convolutional neural networks for diatom identification[END_REF][START_REF] Kloster | Deep learning-based diatom taxonomy on virtual slides[END_REF].

Automatic diatom identification consists of two sequential steps: detecting diatoms among other objects in microscopy images, followed by classification of the detected diatoms into their respective taxa. Processing typical microscopy images for diatom identification is difficult for several reasons. Firstly, individual diatoms may be difficult to distinguish from the background due to a lack of contrast. Moreover, diatoms are rarely clearly separated from each other, and debris is often observed, resulting in complex overlays and stacking, making diatoms difficult to recognize and isolate. These difficulties make the detection step challenging to automate.

Object detection procedures using microscopy images rely on both segmentation and bounding box detection. Previous work on microorganism segmentation includes methods such as thresholding, gradient-based, deformable models and feature-based approaches [START_REF] Verikas | Phase congruency-based detection of circular objects applied to analysis of phytoplankton images[END_REF][START_REF] Kloster | Sherpa: an image segmentation and outline feature extraction tool for diatoms and other objects[END_REF][START_REF] Borges | A highly accurate level set approach for segmenting green microalgae images[END_REF][START_REF] Rojas Camacho | A tuning method for diatom segmentation techniques[END_REF]. At best, these methods are semi-automatic and require manual fine-tuning of parameters, which is impractical. For automatic bounding box detection, Convolutional Neural Networks (CNNs) can improve performance in terms of reliability and robustness [START_REF] Girshick | Fast r-cnn[END_REF][START_REF] Redmon | You only look once: Unified, realtime object detection[END_REF]. This has been demonstrated in microscopy image detection, mainly for planktonic organisms [START_REF] Shi | Study on holographic image recognition technology of zooplankton[END_REF][START_REF] Li | Plankton detection with adversarial learning and a densely connected deep learning model for class imbalanced distribution[END_REF], and also specifically for benthic diatoms [START_REF] Salido | A lowcost automated digital microscopy platform for automatic identification of diatoms[END_REF]. However, the approaches still suffer from overlapping bounding boxes when diatoms are close to each other. To overcome these difficulties, one can use rotated bounding boxes [START_REF] Li | A novel cnn-based method for accurate ship detection in hr optical remote sensing images via rotated bounding box[END_REF][START_REF] Zhong | Single-stage rotation-decoupled detector for oriented object[END_REF]. By estimating the orientations of objects, rotated bounding boxes are fitted around objects, resulting in minimal overlaps and thus cleaner detection.

Training supervised deep-learning networks requires thousands of labelled microscopy images.

Most of the time, labelling is performed manually. In addition to being a time-consuming and laborious process, human intervention is also a source of error and bias [START_REF] Sager | A survey of image labelling for computer vision applications[END_REF]. Thus, when available labelled data are limited, one can use synthetic datasets for pre-training the network before fine-tuning it with a real dataset [START_REF] Li | Plankton detection with adversarial learning and a densely connected deep learning model for class imbalanced distribution[END_REF]. Depending on the method adopted, images generated could be fully or partially synthetic. Learning-based approaches such as Variational Auto Encoders [START_REF] Kingma | Auto-encoding variational bayes[END_REF] and Generative Adversarial Networks [START_REF] Goodfellow | Generative adversarial nets[END_REF] generate fully synthetic data, where no part of the generated images contain real images. However, to generate high-quality images, these networks require thousands of images to train on, which is impractical for applications. In the present work, we adopted a method to generate partially synthetic data, (i.e., where parts of the generated images contain real images).

In the case of freshwater benthic diatoms, hundreds of digitized individual images are publicly available for various diatom species. In theory, they can also be retrieved in order to create thousands of synthetic images similar to real digital ones taken by a human expert. This offers a unique opportunity for the design of a relevant pre-labelled dataset for bounding boxes, as each individual is already boxed, allowing the training of automatic detection networks. Our method for synthetic data generation closely follows Copy-Paste augmentation as implemented for instance segmentation [START_REF] Dwibedi | Cut, paste and learn: Surprisingly easy synthesis for instance detection[END_REF][START_REF] Ghiasi | Simple copy-paste is a strong data augmentation method for instance segmentation[END_REF]. In Copy-Paste augmentation, the generated images use real images as background, then images of individual objects of interest are pasted at random locations on the real images. However, while pasting, the global consistency of the final images is not considered. This means that an object could be placed at locations where they usually do not occur. This could be detrimental in the case of single-stage architectures such as YOLO, which predicts from full images and hence, takes into account the global context of the images while making predictions. By contrast, we generate a synthetic background and place the individual diatoms and debris images, while considering the global consistency of the final image.

To generate realistic images, we consider different parameters while pasting individual objects, such as scaling, rotation, level of overlap of diatom and diatom-debris ratio. Finally, we perform sensitivity analysis to study the impact of these parameters on the performance of the network by evaluating using real datasets.

To summarize, we evaluated the usefulness of synthetic datasets for the automatic detection of diatoms in microscopy images using deep-learning. We first propose a method to generate thousands of synthetic datasets based on an off-the-shelf seamless blending approach. This dataset is used for pre-training a state-of-the-art detection network, YOLOv5 [START_REF] Jocher | ultralytics/yolov5: v3.1 -Bug Fixes and Performance Improvements[END_REF], in order to process real images with different levels of complexity (number and diversity of objects, including different diatom species and debris). Based on our experiments, fine-tuning the network previously trained on the synthetic dataset performed best in diatom detection, regardless of the type of real images. This confirms that synthetic datasets provide a useful alternative when the available labeled dataset is limited for training deep neural networks.

Materials and Methods

Initial Datasets

We used the following datasets for our analyses:

Individual diatom images

Individual diatom images were collected from several open-access taxonomic atlases [START_REF] Bey | Atlas des diatomées des cours d'eau de la région rhône-alpes. tome 1. Centriques, Monoraphidées. tome 2. Araphidées[END_REF][START_REF] Lalanne-Cassou | Atlas des diatomées d'ile de france[END_REF][START_REF] Peeters | Atlas des diatomées des cours d'eau du territoire bourguignon[END_REF]. From the atlases in PDF format, we automatically extracted diatom images using PDF scraping. Our final dataset consisted of 9230 grayscale images of individual diatoms belonging to 166 diatom species.

Real microscopy images

Real images were generated from two types of diatom samples: (1) natural samples collected in the field (Cleurie Stream, Vosges, France, hereafter referred to as CLEURIE) on hard mineral substrates and fixed by adding ethanol, showing natural communities consisting of various diatom species; (2) a sample collected from a lab culture of the diatom Nitzchia palea (hereafter referred to as NPAL). Each liquid sample was subjected to oxidation using hot hydrogen peroxide in order to remove organic material, rinsed with deionized water, then mounted on permanent microscope slides with Naphraxinto, a high refractive index resin in order to be visualized at 1000× magnification using light microscopy. CLEURIE and NPAL datasets finally consist of 120 and 175 images, respectively, each one representing a microscope view field containing multiple diatoms and debris.

Debris

To create the synthetic images of microscope slides, we needed to take into account the variable occurrence of debris potentially present on microscope images. From the real microscope images, we manually extracted 600 debris objects of different shapes and sizes.

Synthetic Microscope Image Generation

The synthetic dataset should closely mimic the real images obtained from light microscopy.

Let us consider the following procedure to create a seamless image I of width W and height H with images from the individual diatom dataset D:

1. Init I ′ , empty image of dimensions (W, H).

• Pick a subset S i ⊂ D composed of individual diatom and debris images.

• For each image T ij ∈ S i of width w ij and height h ij , -Apply data augmentation. Refer to Sec. 3.1 for the different augmentations and the parameters used.

-Find a random position P ij for T ij on I ′ such that it does not overlap any previously added diatom or debris images or the borders of I ′ by more than a certain threshold percentage.

2. Init I, empty image of dimensions (W, H). Let B be the set of pixels p I ′ of I ′ comprising the borders of T ij . Set the value of each pixel p I i of I as a weighted arithmetic mean such that:

p I i = p I ′ j ∈B val(p I ′ j )w(p I i , p I ′ j ) p j ∈B w(p I i , p I ′ j )
(1)

w(p, p ′ ) = e -||p ′ -p|| 2 (2)
where val(p) is the gray value of pixel p. This step generates the gray background, similar to those obtained from light microscopy.

3. Finally, place T ij at P ij on I using the Poisson Editing method for seamless blending [START_REF] Pérez | Poisson image editing[END_REF].

Object Detection

The The inputs to the network are images from microscopy, and the network predicts the coordinates of the bounding boxes. We used YOLOv5 for two types of detection: horizontal bounding boxes and rotated bounding boxes. While the standard YOLOv5 [START_REF] Jocher | ultralytics/yolov5: v3.1 -Bug Fixes and Performance Improvements[END_REF] regresses the values of the rectangular bounding boxes, YOLOv5 for rotated bounding boxes5 estimates the orientation in addition to the box coordinates.

Experiments

Baseline

Four augmentation techniques were used in order to generate the synthetic dataset: scaling, rotation, overlap between diatoms and debris, and diatom to debris ratio. To determine the sensitivity of each of the augmentation methods while evaluating using real microscopy images, and to choose the optimal parameter values, we performed a grid-search on the parameters without changing the neural network hyperparameters as summarized in Table 1. The definition and significance of each of these techniques and the parameter values are given below.

1. Scaling: Scaling refers to resizing the images of individual diatoms and debris while creating the synthetic dataset. This was carried out to represent the real microscopy images where diatoms belonging to a species can occur at varying scales. For an accurate scaling value, one could consider the actual size range in which the diatoms occur naturally. However, for our analysis, we randomly chose a scaling value from an exponential distribution to consider a wide range of scales for each diatom species. For the grid-search, we considered situations with and without scaling to generate the data.

Rotation:

The individual diatom images were randomly rotated at any angle between 0 and 360 • . Our grid-search considered two conditions: with and without rotating the images.

3. Overlap: For each individual diatom on the synthetic image, this refers to the level of overlap allowed between other diatoms or debris. We varied the level of overlap between the objects from 0 to 25%. The maximum threshold value was chosen as 25% to avoid hidden objects that would bias training and evaluation. For our parameter-tuning, we considered two cases: no overlap between the objects and overlap between objects.

Diatom-debris ratio:

This refers to the ratio between the number of diatoms and the number of debris objects while generating the images. We considered the following range of values:(1) diatom-debris ratio between 2 and 5;

(2) diatom-debris ratio between 0.1 and 0.5.

For the above two cases, the number of diatoms was between 6 and 10, and we varied the number of debris objects to achieve the desired ratio.

To study the impact of the synthetic dataset while training on the real datasets, we performed the following experiments: (1) in the 'Synthetic' experiment, we trained and evaluated the network on the synthetic images; (2) in the 'Zero-shot learning' experiment, we trained the network with the synthetic dataset and evaluated using the available real dataset; (3) in the 'Training from scratch' experiment, we trained the network with only the real dataset and evaluated using the real dataset; (4) in the 'Fine-tuning' experiment, we used the model trained on the synthetic

Evaluation Metrics

Detection is concluded when the intersection over union (IoU), which measures the extent of overlap between the predicted box and the ground truth box, is above a certain threshold. For a fixed threshold of IoU, precision gives the percentage of detections that are correct.

P recision = T rueP ositive T rueP ositive + F alseP ositive

(3)

For a fixed IoU threshold, recall gives the number of positive detections out of all the positives in the ground truth.

Recall = T rueP ositive T rueP ositive + F alseN egative

(4)

In addition to these metrics, some Object Detection challenges (VOC, COCO, Open Images) define their own metrics that have become standard evaluation indicators. In this work, we also calculated mean Average Precision for IoU at 0.5 (mAP@0.5) and AP for IoU from 0.50 to 0.95 with a step size of 0.05 (mAP@0.5:0.95) as defined in [START_REF] Coco | Coco -evaluate -metrics[END_REF]. We used the same metrics for evaluating the horizontal and rotated bounding boxes.

Implementation Details

We used Adam optimizer [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF] and the learning rate was 0.0002. The batch size for training YOLOv5 (horizontal and rotated) was 16. For every category of synthetic dataset (D1-D16 in Table . 1), we generated 4000 images. While training the networks using synthetic and real datasets, we used 80% of the images for training, 10% for validation and 10% for testing. We trained our networks on GeForce GTX 2080 with 8 Gb RAM and GeForce GTX 1080 with 12 Gb RAM. The total training time for sensitivity analysis was 24 days.

Results

Horizontal bounding box detection

Figure 2 shows some examples of the horizontal bounding box detections from YOLOv5. Tables 2 and3 show the quantitative metrics for horizontal bounding box detection using YOLOv5 [START_REF] Jocher | ultralytics/yolov5: v3.1 -Bug Fixes and Performance Improvements[END_REF]. The following observations could be made based on the results:

Scaling: When evaluated on synthetic dataset, the precision, and recall (mIoU=0.50 and mIoU from 0.50 to 0.95) was higher when the images were not scaled (D1-D4; D9-D12 in Table . 2). This is because when there is no scaling, the diatoms are of uniform size, which leads to network overfitting. When evaluated on the real microscopy images, the CLEURIE dataset contained diatoms of varying size due to the presence of different species, and hence the presence of scaling in the synthetic dataset leads to better metric values on both zero-shot learning and fine-tuning.

NPAL images contained diatoms of the same species, and all diatoms were of rather uniform size.

Hence, the network generalized better when the synthetic dataset had no scaling.

Rotation: When the dataset contained no rotated diatoms (D1-D8), the mIoU 0.5:0.95 on the synthetic dataset was superior, but it did not generalize well to the CLEURIE and NPAL datasets. This is because networks detecting horizontal bounding boxes are designed to fit non-rotated objects better, hence the network generalizes well. The network draws tighter and more accurate bounding boxes around these objects, resulting in better mIoU values. However, real datasets contain diatoms present in random orientations. Thus, training datasets D9-D16 generalized better to CLEURIE and NPAL datasets compared with D1-D8.

Overlap: From the results, having overlaps had no significant impact on the precision and recall scores (mIoU=0.50 and mIoU from 0.50 to 0.95) of the network for synthetic and real datasets. However, the best metric values were obtained when a maximum overlap of 25% was allowed. This means that although overlap helps improve network performance, the network was able to generalize well even when the training datasets do not have overlap of diatoms.

Diatom-debris ratio: The network had better metrics when the diatom-debris ratio was in the range [2,5] for both the synthetic and real datasets. For the synthetic datasets, this could be explained by the reduced complexity of the images for the network to detect, since the debris objects were widely distributed. For both the CLEURIE and NPAL datasets, the diatom-debris ratio was greater than 1. Hence, when the synthetic dataset was characterized by a higher diatomdebris ratio, the network generalized better for these datasets.

Table 3 shows the results when the network was trained using only the real dataset. Compared with the performance when the network was trained only on real data, there was an improvement of 15% in precision (mIoU=0.50) and 8% in recall (mIoU=0.50) for the CLEURIE dataset and an improvement of 25% in precision (mIoU=0.50) and 11% in recall (mIoU=0.50) for the NPAL dataset when the network was trained on synthetic data and fine-tuned to real data.

Rotated Bounding Box Detection

Figure 3 shows some examples of the rotated bounding box detection from YOLOv5. Tables 4 and5 show the quantitative metrics for rotated bounding box detection. Since no rotation was included in D1-D8, we considered datasets D9-D16.

Scaling:

The evaluation metrics follow a similar trend to the horizontal box evaluation. Evaluation on the synthetic dataset achieved the best performance when there was no scaling, since the network overfits to the uniform diatom sizes. Scaling helps the network generalize better to the CLEURIE dataset due to the presence of different taxa of diatoms with varying scales. The NPAL dataset consists of diatoms of the same species with uniform sizes, hence the network generalizes best when trained with no scaling.

Overlap: When evaluated on the synthetic dataset, the network performed best when there was no overlap between the diatoms and debris. This is because, when there are no overlaps, there is no ambiguity due to occlusions for the network while learning the rotated bounding box parameters. For the CLEURIE and NPAL datasets, overlap helps the network generalize better.

Diatom-debris ratio: The network performed better when the diatom-debris ratio was in the range [2,5] for the synthetic and real datasets. This is because fewer debris objects means a sparser distribution compared with the diatoms, making them easier to discard.

Table 5 shows the results when the network was trained using only the real dataset. Compared with the performance when the network was trained only on real data, there was an improvement of 17% in precision and 22% in recall for the CLEURIE dataset and an improvement of 8% in precision and 23% in recall for the NPAL dataset, for an mIoU threshold of 0.5 when the network was trained on synthetic data and fine-tuned to real data.

Conclusion

In this work, we assessed the use of a synthetic dataset for diatom detection using deep-learning.

When the labelled dataset available for training the network is limited, one can use a synthetic dataset for training and fine-tune it to the real dataset. We proposed a method to generate thousands of synthetic datasets and performed a comprehensive analysis on the impact of the synthetic dataset when training horizontal and rotated object detection networks. Our results show that when the available labeled dataset is limited, fine-tuning from a network trained on synthetic dataset can significantly improve the performance of the network. This approach will thus be of interest for future work, which will focus on the implementation of automatic diatom segmentation and classification networks.

  Scaling factor=1; Diatom-debris ratio=[2,5]; Overlap=0; D2 No rotation; Scaling factor=1; Diatom-debris ratio=[2,5]; Overlap=0.25; D3 No rotation; Scaling factor=1; Diatom-debris ratio=[0.1,0.5Scaling factor=1; Diatom-debris ratio=[0.1,0.5]; Overlap=0; D12 With rotation; Scaling factor=1; Diatom-debris ratio=[0.1,0.5]; Overlap=0.25; D13 With rotation; Scaling factor=exponential; Diatom-debris ratio=[2allows to drastically reduce the number of real hand-labelized image.

Figure 1 :

 1 Figure 1: Pipelines for evaluating networks. First, individual diatoms and debris images are used to generate synthetic datasets for training the detection networks. There are four evaluation pipelines: Evaluation A -the network is trained and evaluated on synthetic datasets. Evaluation B -the network is trained on synthetic images and evaluated on real images (zero-shot detection). Evaluation C -the network is trained on a synthetic dataset, fine-tuned to real images, and evaluated on real images. Evaluation D -the network is trained and evaluated on real datasets

ZeroFigure 3 :

 3 Figure 3: Rotated bounding box detection from YOLOv5 using CLEURIE (top row) and NPAL (bottom row) real images. Green boxes indicate correct detection; blue indicates false negatives, and red boxes indicate false positives.

  deep-learning object detection networks can be distinguished into two types: one-stage, and two-stage. The one-stage family[START_REF] Liu | Ssd: Single shot multibox detector[END_REF][START_REF] Redmon | You only look once: Unified, realtime object detection[END_REF][START_REF] Lin | Focal loss for dense object detection[END_REF] generally

uses a unique network to extract and classify the bounding boxes, while two-stage networks

[START_REF] Girshick | Rich feature hierarchies for accurate object detection and semantic segmentation[END_REF][START_REF] Girshick | Fast r-cnn[END_REF]

first use a Region Proposal Network to propose multiple Regions of Interest (RoI) which are subsequently classified using a second generic backbone architecture.

One-stage detection networks are faster since they combine detection and classification in a single architecture, but are generally less accurate than two-stage networks. For our objective, we chose one stage YOLOv5

[START_REF] Jocher | ultralytics/yolov5: v3.1 -Bug Fixes and Performance Improvements[END_REF]

for detection due to its overall performance in terms of speed and accuracy. However, future work would incorporate Neural Architecture Search

[START_REF] O'neill | Evolutionary neural architecture search for high-dimensional skip-connection structures on densenet style networks[END_REF][START_REF] Xue | A self-adaptive mutation neural architecture search algorithm based on blocks[END_REF][START_REF] Xue | Partial connection based on channel attention for differentiable neural architecture search[END_REF]

to choose the optimal architecture. YOLOv5 is a real-time CNN for localizing objects on images. YOLO-based networks function by dividing the input into several grids responsible for the detection and localization of the objects they contain.

Table 1 :

 1 Augmentation methods and the parameters used to generate synthetic dataset

Table 2 :

 2 Quantitative evaluation of horizontal bounding box detection. Higher values indicate better performance.

	M1-Precision; M2-Recall; M3-mAP@.5; M4-mAP@.5:.95.	Synthetic Zero-shot learning Fine-tuning	Datasets CLEURIE NPAL CLEURIE NPAL	M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4	D1 1.0001.000 0.995 0.990 0.422 0.461 0.406 0.174 0.848 0.549 0.694 0.413 0.930 0.917 0.974 0.797 0.816 0.958 0.918 0.756	D2 0.998 0.998 0.995 0.981 0.632 0.453 0.501 0.224 0.533 0.690 0.603 0.405 0.925 0.923 0.974 0.804 0.842 0.901 0.909 0.745	D3 0.999 1.000 0.995 0.973 0.573 0.458 0.512 0.222 0.543 0.620 0.567 0.310 0.939 0.920 0.975 0.802 0.936 0.829 0.9330.786	D4 0.998 0.999 0.995 0.967 0.605 0.605 0.614 0.254 0.639 0.775 0.683 0.257 0.921 0.934 0.976 0.802 0.837 0.944 0.924 0.775	D5 0.993 0.996 0.995 0.971 0.543 0.535 0.451 0.107 0.861 0.795 0.788 0.308 0.928 0.963 0.972 0.791 0.747 0.915 0.812 0.705	D6 0.995 0.991 0.995 0.968 0.579 0.404 0.444 0.195 0.475 0.423 0.464 0.275 0.933 0.923 0.979 0.791 0.805 0.928 0.903 0.747	D7 0.995 0.992 0.995 0.967 0.512 0.596 0.568 0.308 0.800 0.450 0.547 0.373 0.919 0.946 0.978 0.797 0.812 0.915 0.920 0.779	D8 0.995 0.991 0.995 0.965 0.802 0.513 0.645 0.353 0.766 0.507 0.531 0.284 0.907 0.948 0.971 0.786 0.840 0.887 0.892 0.770	D9 1.000 0.999 0.995 0.955 0.666 0.610 0.652 0.364 0.646 0.436 0.461 0.265 0.960 0.900 0.977 0.798 0.793 0.971 0.879 0.757	D10 0.998 0.991 0.995 0.915 0.791 0.642 0.675 0.173 0.805 0.761 0.741 0.253 0.910 0.954 0.977 0.791 0.767 0.972 0.850 0.723	D11 0.999 1.000 0.995 0.932 0.783 0.613 0.621 0.121 0.753 0.690 0.669 0.210 0.916 0.940 0.977 0.787 0.775 0.972 0.871 0.712	D12 0.999 1.0000.999 0.946 0.695 0.642 0.670 0.190 0.762 0.859 0.804 0.175 0.910 0.954 0.974 0.784 0.802 0.915 0.899 0.744	D13 0.988 0.990 0.995 0.954 0.681 0.484 0.546 0.194 0.844 0.789 0.861 0.199 0.931 0.928 0.977 0.788 0.841 0.901 0.890 0.728	D14 0.982 0.987 0.994 0.931 0.8000.7000.776 0.300 0.710 0.901 0.763 0.235 0.967 0.911 0.9810.806 0.805 0.930 0.886 0.749	D15 0.982 0.984 0.996 0.943 0.651 0.648 0.630 0.157 0.849 0.9580.913 0.244 0.917 0.948 0.975 0.771 0.779 0.943 0.873 0.753	D16 0.980 0.974 0.993 0.922 0.766 0.685 0.750 0.211 0.766 0.924 0.858 0.275 0.933 0.920 0.979 0.789 0.800 0.958 0.878 0.738

Table 3 :

 3 Quantitative evaluation of horizontal bounding box detection when the network is trained using only the real dataset.

	Dataset	Precision Recall mAP@0.5 mAP@0.5:0.95
	CLEURIE	0.840	0.894	0.864	0.611
	NPAL	0.744	0.864	0.847	0.594

Table 4 :

 4 Quantitative evaluation of rotated bounding box detection. Higher values indicate better performance.

	M1-Precision; M2-Recall; M3-mAP@.5; M4-mAP@.5:.95.	Synthetic Zero-shot learning Fine-tuning	Datasets CLEURIE NPAL CLEURIE NPAL	M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4	D9 0.983 0.992 0.9910.811 0.302 0.320 0.172 0.037 0.634 0.706 0.618 0.198 0.868 0.958 0.928 0.679 0.862 0.955 0.939 0.724	D10 0.975 0.987 0.988 0.793 0.536 0.414 0.346 0.027 0.8130.8060.8040.234 0.826 0.819 0.865 0.615 0.880 0.991 0.940 0.730	D11 0.968 0.958 0.985 0.750 0.520 0.475 0.372 0.027 0.755 0.838 0.781 0.176 0.868 0.957 0.925 0.699 0.879 0.982 0.942 0.737	D12 0.972 0.993 0.981 0.781 0.387 0.260 0.162 0.028 0.731 0.675 0.656 0.176 0.944 0.823 0.890 0.662 0.909 0.991 0.9680.769	D13 0.957 0.949 0.969 0.722 0.349 0.199 0.122 0.062 0.677 0.644 0.605 0.154 0.910 0.839 0.857 0.603 0.872 0.982 0.955 0.733	D14 0.953 0.941 0.958 0.721 0.393 0.398 0.268 0.049 0.640 0.665 0.605 0.154 0.927 0.823 0.857 0.626 0.880 1.000 0.937 0.715	D15 0.936 0.889 0.900 0.709 0.466 0.434 0.317 0.072 0.699 0.709 0.670 0.191 0.892 0.806 0.817 0.589 0.879 0.982 0.928 0.708	D16 0.936 0.876 0.903 0.680 0.5410.5310.4480.091 0.736 0.803 0.748 0.221 0.846 0.887 0.875 0.647 0.908 0.982 0.952 0.743

Table 5 :

 5 Quantitative evaluation of rotated bounding box detection when the network is trained using only the real dataset.

	Dataset	Precision Recall mAP@0.5 mAP@0.5:0.95
	CLEURIE	0.804	0.783	0.834	0.528
	NPAL	0.843	0.811	0.871	0.642

https://github.com/XinzeLee/RotateObjectDetection

Funding

The PhD scholarship for AV was funded by ANR [ANR-20-THIA-0010] and Région Grand-Est.

Additional financial support was provided by CNRS (ZAM LTSER Moselle, PNRIA) and Université de Lorraine (LUE).

Data availability

Image datasets and code used to generate synthetic dataset are available on the institutional repository DOREL ( https://doi.org/10.12763/UADENQ).

Fine-tuning

Ground Truth


Meyer Neldel

L Pichon

A Boukhenoufa

B Cretu

R Rogel

Improvement in the determination by 1/f noise measurements of the interface state distribution in polysilicon TFTs in relation with the compensation law of

Low-frequency (1/f ) noise is studied in N-channel furnace solid phase crystallized (FSPC) and in laser solid phase crystallized (LSPC) polysilicon TFTs biased from weak to strong inversion. Noise analysis is supported by the theory of charge carrier trapping/detrapping at the interface tunnelling into gate oxide traps.

The distribution of interface trap states (N T ) is deduced from the number of carriers trapped into the oxide. Noise measurements for devices biased from weak to moderate inversion allow the determination of the distribution of deep level trap states associated with dangling bonds type defects (N Tdb ); whereas measurements from moderate to strong inversion give the distribution of shallow level trap states (N Tts ) associated with strained bonds defects. The noise analysis clearly shows that the slope of both exponential distributions equals to the reverse of the Meyer Neldel energy E MN (0.035 eV and 0.055eV for FSPC and LSPC TFT respectively).

For LSPC devices the resulting distribution of interface states (N T =N Tdb +N Tts ) is one decade lower and it is attributed to the effects of the laser annealing on the active layer crystal quality.

I. INTRODUCTION

Electrical properties in polysilicon thin film transistors (TFTs) are strongly controlled by trapping effect at defects both located within the bulk of the polysilicon active layer [1] and at the gate insulator/active layer interface. Two main causes of bulk defects are usually invoked: dangling bonds and strained bonds corresponding to deep and shallow level trap states into the band gap respectively. The energy distributions of the corresponding trap states (N T ) are representative of both the gate oxide and the active layer qualities, and thus are related to the electrical properties of the devices. Two types of distributions are considered: Gaussian distribution with a maximum around the midgap (N Tdb ) and exponential band tailing (N Tts ) corresponding to dangling bonds and strained bonds type defects respectively [2,3]. Methods based on capacitance, resistivity, and conductance activation energy measurements exist to determine these distributions [4,5].

More recently two approaches based on low frequency (1/f) noise measurements in polysilicon

TFTs were proposed [6,7]. In this case, for N-channel polysilicon TFTs, 1/f noise is assumed to be due to carrier fluctuations usually modelled by carriers trapping/detrapping (T/D) from slow oxide traps located close to the interface. In the first study [6] a model taking into account additional fluctuations of the potential barriers height induced by the carriers trapping effect at the grain boundaries was suggested, and thus the average oxide and grain boundary trap densities were deduced. In the second study [7], based on the tunnelling theory of carriers into oxide traps, a method to determine the energy distribution of the interface states into the band gap was presented.

In these two studies it was shown that 1/f noise level is strongly dependent on both interface and active layer qualities, and thus on fabrication process parameters. Therefore 1/f noise measurements can be used as a diagnostic tool to qualify TFT technology. However, the separated contributions of N Tdb and N Tts on the 1/f noise level in relation with the interface quality were not clearly demonstrated.

We present a study of 1/f noise level on polysilicon TFTs biased from weak to strong inversion. Analysis is based on carriers T/D from oxide traps and the separated contributions of deep and shallow level trap states on the resulting energy distribution of interface states in relation with the crystal quality of the active layer is presented. In addition, a direct relation with the Meyer Neldel effect [8] of N Tdb and N Tts is demonstrated, and it constitutes an improvement in the determination of the interface states distribution. crystallized by a FSPC thermal annealing in vacuum at 600°C. For LSPC structures an additional SPC thermal annealing of the active layer is carried out by using a scan of an Ar laser beam. Laser annealing parameters, power beam (P=4.8W) and scan speed (v=70mm/s), were adjusted to limit contamination of the active layer from the glass substrate. Under these conditions the active layer remains in solid phase and does not present a high degree of crystal quality as usually provided by other types of laser crystallization processes. Moreover, a possible contamination from the glass substrate has been reported in such laser annealed polysilicon layer [9]. The gate insulator is made of a SiO 2 layer deposited by atmospheric pressure CVD (APCVD) technique at 390°C and annealed in nitrogen ambient for densification. The thickness of the gate oxide is 60nm and 50nm in SPC TFT and in LSPC TFT respectively. Electrodes are made of thermally evaporated aluminium. Finally, the devices were annealed into forming gas at 390°C. Noise measurements are carried out in a shielded environment by using a low noise transimpedance amplifier (EG&G 5182, 15fA/ Hz ) connected to the source electrode, followed by a low noise voltage amplifier (EG&G 5113, 4nV/ Hz ) and a HP 3562A dynamic signal analyzer. A HP 4156 B semiconductor parameter analyzer is used for static drain current measurements. All tested devices are biased in the linear mode (V DS =300mV) and operating from weak to strong inversion.

III-THEORETICAL ANALYSIS

It is widely admitted that in MOS transistors 1/f noise is due to channel conductivity fluctuation. Two mechanisms are proposed to explain it: i) carrier number fluctuation (∆N model) or ii) carrier mobility fluctuation (∆µ model). ∆N model is based on carriers Generation-Recombination (G-R) processes at defects. In this case 1/f noise is explained as a superposition of G-R spectra with a distribution of time constant associated with carriers trapping/detrapping processes into the gate oxide. This noise model was first suggested by Mc Worther [START_REF] Mc | Semiconductor Surface Physics[END_REF]. ∆µ model describes mobility fluctuation due to scattering mechanisms of carriers in homogeneous samples of high crystal quality, and it is proposed by Hooge [START_REF] Hooge | [END_REF]. There have been ample discussions concerning the origin of the 1/f noise but they have not been decisive. Some effects can be explained by either models, but none of them can be definitively excluded. where S IDS is the drain current power noise spectra density, S N the carrier number power noise spectra density, I DS the drain current, α the noise parameter and N the free carrier number. In addition, according to the G-R model the low frequency noise due to carrier fluctuations is described by [12]:

( ) ( ) [ ] f f f d f S N π τ π ∝ τ π + τ ∝ τ τ τ τ ∫ 2 2 arctan 2 1 1 0 1 0 2 (2)
with f the frequency and τ 0 (τ 1 ) the lower (upper) limit of time constants involved in the carriers T/D processes at defects. In this case 1/f noise spectrum is obtained for 1/τ 1 <<f<<1/τ 0 it means for long time constant τ 1 (see fig. 2. a). For carriers T/D at SiO 2 /poly-Si interface followed by tunnelling into oxide traps (see fig. 2. b) the time constant τ is expressed as τ=τ 0 exp(y/λ)exp((Ε C -Ε)/kT) with λ is the tunnel attenuation distance (≈0.1nm) and τ 0 ~10 -10 s [13].

In this case maximum time constant τ 1 can be very long. For example, considering that maximum value of E C -E can be half polysilicon band gap (0.56eV), and with y=2nm (typical oxide depth for simulated low frequency noise at f=1Hz) [13], τ 1 ~10 6 s. For carriers T/D at defects in the bulk of the polysilicon active layer τ =τ 0 exp((Ε C -Ε)/kT, and the maximum time constant is much shorter (τ 1 ~0.003s). In this case, bulk trap contribution cannot explain the 1/f noise spectra of S IDS /I 2 DS measured for 1Hz≤f≤300Hz (0.03s≤τ≤1s) on both FSPC and LSPC TFTs biased from weak to strong inversion (see fig. 2 in ref [14]). Therefore, our 1/f noise analysis is supported by the theory of carriers T/D processes at the interface tunnelling into gate oxide traps located close to the interface (Mc Worther model) related to long time constants.

Previous theoretical study [15] on the relevance of the Mc Worther model showed that theoretical behaviour of noise parameter α versus the gate voltage does not correspond to whose usually measured for crystalline MOS transistors. However, in the case of polysilicon TFTs, a previous experimental study [START_REF] Boukhenoufa | Proceedings of: Noise and Fluctuations: 18 th International Conference on Noise and Fluctuations-ICNF 2005[END_REF] showed that α measurements follow theoretical predictions, and thus our noise analysis is based on the corresponding theoretical calculations α reported in ref [15].

In this study, calculations of α are carried out by considering the effective number m* defined by 1/m * =1/m+1/(M-m) with m the number of trapped carriers into the oxide and M the number of traps uniformly distributed into the gate oxide. The number m is both controlled by the number of traps at the interface n t , the oxide depth y, and the inversion free carrier number N. So, m can be expressed as ( )

( ) ( ) ( ) N E n f f y y E m t t t - λ - = 1 / exp ,
, with f t =1/(1+exp((E-E F )/kT)) the Fermi factor, E F the Fermi level energy and E the trap energy [12]. Considering M(E)=m(E,y)×exp(y/λ) and

( ) ( ) ( ) ( ) λ - - = / exp 1 , , * y y E m y E m
, therefore the average value of m* is:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ox t t t 0x t 0 ox t t t λ/2t f 1 Nf E n dy y/λ exp - y/λ exp t N E n f 1 f E * m - ≈           - - - ≈ ∫ 2 1
(3) with t ox the thickness of the gate oxide (typically>>λ).

Furthermore it has been reported that [15]: A previous experimental study of 1/f noise on polysilicon TFTs reported the validity of these theoretical results [START_REF] Boukhenoufa | Proceedings of: Noise and Fluctuations: 18 th International Conference on Noise and Fluctuations-ICNF 2005[END_REF]. Our analysis is supported by relations ( 4) and ( 5) for the determination from n t (E) of the interface states energy distribution as described in the next section.

B. Interface trap states distribution

For the calculation of the interface trap states distribution in the TFTs, we assumed that the free carrier number N≈N C WLt si exp((E F -E C )/kT) with N c (~10 19 cm -3 ) the effective density of states into the conduction band, W(L) the width(length) of the channel and t si (=150nm for FSPC TFT and =100nm for FSPC TFT) the thickness of the active layer. Moreover, considering that when

E ≤ E F trap occupancy is high, in dynamical equilibrium E C -E F ≈E A -(E F -E) (see fig 2. b), thus Nf t (1-f t )=(N C WLt si )exp(-E A /kT)(1+exp((E-E F )/kT)) -2 .
Then the calculation of n t is possible and the corresponding interface trap states distribution N T (cm -2 eV -1 ) can be deduced considering traps within an energy band kT around the Fermi level (1+exp((E-E F )/kT)~2) by stating N T =n t /(WLkT).

Therefore, because thermal activation energy associated with T/D process is E A =E C -E, then for high values of E A (low gate voltages) according to (2), ( 3) and (4) we deduce:

( ) ( )       ≈ kT E S I kT t fN WL E N A DS I DS si C Tdb exp 32 2 2 (6)
In addition, for low values of E A (high gate voltages) according to (2), (3) and (5), we obtain: Previous experimental studies [START_REF] Pichon | [END_REF][START_REF] Pichon | EMRS 2008 Symposium E "Thin Film for large area electronics[END_REF] reported that the static drain current I DS and the 1/f noise drain current spectral S IDS , for TFTs biased from weak to strong inversion, (ie from sub-to abovethreshold region) are thermally activated both following the empirical Meyer Neldel (MN) rule [8] according to:

( )       -       λ ≈
( )( )      - β =       -         = 0 GS DS 0 A MN A DS V V V kT E exp E E exp I 0 DS 0 DS I I (8) and       -         ≈ kT E E E S S A MN A IDS IDS exp exp 0 (9)
with E MN the MN characteristic energy, V 0 the gate voltage corresponding to the minimum of the drain current on the transfer characteristics plotted in a semi-logarithmic scale, and β 0 (V DS ) the transconductance. Meyer Neldel effect is related to the multiple trapping processes of carriers [START_REF] Jackson | [END_REF], and in our case it is assumed to be the dominant carrier transport mechanism mainly occurring at the interface along the channel. It was also reported that the E MN could act as a factor of merit of the quality of the polysilicon active layer depending on the process fabrication parameters such as pressure deposition [START_REF] Pichon | [END_REF]. Moreover, it was shown that E NM is related to exponential distribution of tail states [20], and that its slope becomes steeper as E MN decreases [21]. In addition, study of numerical simulation of static drain current showed that MN effect is mainly controlled by the trap distribution associated with defects located at the gate insulator/active layer interface [22]. Values of E MN parameters are mentioned in the table I for FSPC and LSPC TFTs, and the plots of the corresponding activation energies versus the gate voltage are reported in the figure 3.

B. Results

Plots of I DS =f(V GS ) and S IDS /I 2 DS =f(I DS ) of the studied devices (operating in the linear mode) are displayed in the figures 4 and 5 respectively. The corresponding interface trap states distributions are then calculated according to ( 6) and ( 7) and reported in a semi logarithmic scale in the figure 6.

The resulting interface trap states distributions N T exhibits a linear increase for shallow level traps (strained bonds type defects) and a linear decrease for the deep levels into the band gap (dangling bonds type defects). The expected maximum of the deep states distribution usually depicted around the midgap is not observed and should be extrapolated. Indeed, the maximum of N T corresponds to the maximum value of E A measured around the minimum value of I DS (ie below 10 -9 A) but noise measurements for devices biased at such low level of the drain current is difficult. Furthermore, for LSPC TFTs plot of N T shows a deviation from exponential distribution versus activation energy at shallow trap levels (fig 6 . (b)). This can be explained by the (source/drain) access resistances 1/f noise contribution at high gate voltage (low activation energy). Indeed, this contribution is particularly pronounced for LSPC device due to the existing active layer/source(drain) region interface, that does not appear in FSPC TFTs, and acting as additional source of noise [23].

Moreover, previous work [24] reported that at high gate voltages the access resistances noise contribution (S Rac ) dominates channel noise, whereas the resistance of the channel (R c ) is still higher than the value of the total access resistance (R ac ). Under these conditions, in the linear mode, the total resistance (R) is R=R c +R ac ,≈R c and the resistance power spectra density (S R ) is S R =S Rc +S Rac ≈S Rac , with S Rac non dependant on the gate voltage. Thus, the normalized noise becomes S IDS /I 2 DS =S R /R 2 ∝exp(-2(E A /kT-E A /E MN )), whereas S IDS /I 2 DS ∝exp(E A /kT-E A /E MN ) without access resistances noise contribution. Therefore, determination of N T considering the access resistance noise contribution gives, according to (7), an exponential distribution at shallow trap levels with a lower slope for LSPC TFTs (fig. 6 (b)). Because our model is based on the domination of the channel noise, it is not applicable for determination of shallow level interface trap states distribution in the case of access resistances noise contribution within TFTs at high gate voltages.

In order to avoid this discrepancy in determination of N Tts , improved formulation can be given by injecting ( 8) and ( 9), into ( 6) and (7). Therefore, with E A =E C -E (see fig. 2. b) ( 6) and (7) respectively become: [START_REF] Mc | Semiconductor Surface Physics[END_REF] with N Tdb0 =32I 2 DS0 /(fN C (WL) 2 t si kTS IDS0 ) and [START_REF] Hooge | [END_REF] with N Tts0 =8(t ox /λ) 2 fN C t si S IDS0 /(I 2 DS0 kT). In this case the determination of the resulting interface states distribution given by ( 10)+( 11) can be deduced from measurements of the drain current activation energy at different gate bias, and from the saturated value of the normalized drain current noise spectra, S IDS0 /I 2 DS0 , measured at high gate bias (ie for E A <<kT) (see inset of the fig. 5). Furthermore, these two relations show the direct dependence of the interface trap states density on the MN characteristic energy, and valid the results of the numerical simulation of the static conduction in polysilicon TFTs reporting the strong dependence of the MN effect on the trap states distribution related to defects located at the interface [22]. In addition, equations [START_REF] Mc | Semiconductor Surface Physics[END_REF] and [START_REF] Hooge | [END_REF] agree with the two exponential distributions of trap state model used for the theoretical studies to correlate the Meyer Neldel effect with the statistical shift of the Fermi level within the density of states distribution [20,[START_REF] Overhof | Electronic Transport in hydrogenated Amorphous Semiconductors[END_REF].

( )         - ≈ MN C Tdb Tdb E E E N E N exp 0
( )         - - ≈ MN C Tts Tts E E E N E N exp 0
In the figure 7 plots of the resulting interface states densities deduced from (10)+( 11) are reported with those obtained according to (6)+ (7) for the two types of TFTs. By applying [START_REF] Mc | Semiconductor Surface Physics[END_REF] and [START_REF] Hooge | [END_REF] we can observe a good agreement with ( 6) and ( 7) except at shallow levels for LSPC TFTs (because of the access noise effect as previously mentioned). Differences in values of N T between the two methods of calculations is less than one decade and can be explained by induced estimation errors such as geometry (W, L, t ox ) and E MN parameters. In addition, because measurements from static drain current activation energy is possible at low current level corresponding to high activation energy values (see fig. 3 and4), determination of N T given by ( 10) and ( 11) allows a calculation over a larger range of trap levels, and more particularly close to the midgap. The results highlight that the density of the interface states is significantly lower for LSPC devices, and more particularly for deep states into the band gap corresponding to a lower density of dangling bonds type defects. This result is attributed to a better quality of the interface in relation with the improved crystal quality of the active layer due to the laser annealing.

In polysilicon films, the energy distribution of trap states can be modelled by the sum of a deep level Gaussian distribution N ds (E) with a maximum at energy E t near the midgap, and an exponential-like N ts (E) band tail near the conduction band edge following [2,3]:

( ) ( ) ( ) ( )         - - +         - - π = + = s c s d t d d ts ds T kT E E N s E E s N E N E N E N exp 2 exp 2 2 2 (12)
where N d is the total trap density per unit area, s d the standard deviation of the Gaussian distribution, N s and T s the characteristic parameters of the exponential trap distribution. These distributions can be related to distributions given by [START_REF] Mc | Semiconductor Surface Physics[END_REF] and [START_REF] Hooge | [END_REF]. Indeed, for shallow level ). As S IDS0 /I 2 DS0 and E MN parameter values depend on process fabrication parameters [START_REF] Pichon | [END_REF], it shows the link of the interface deep state distribution parameters with the gate insulator interface and the active layer qualities. For our studied devices, N Tts0 and N Tdbmax were estimated according to [START_REF] Mc | Semiconductor Surface Physics[END_REF] and [START_REF] Hooge | [END_REF] and are summarized in the table I. The values are convenient with previous published results [2,3].

These results show the dependence of the shape of the interface trap states distribution on the E MN parameter in relation with the quality of the interface. Our study validates the steeper slope of the exponential tail state distribution as E MN decreases as previously reported [2,21]. In addition, it confirms i) the change of E MN value in function of the polysilicon active layer quality [START_REF] Pichon | [END_REF], and ii) the strong influence of the interface trap states distribution on the Meyer Neldel effect for the drain current thermal activation [22].

  II. DEVICE AND EXPERIMENT DESCRIPTIONS 1/f noise is analysed on TFTs issued from two low-temperature (≤ 600°C) technologies: Furnace Solid Phase Crystallized (FSPC) polysilicon TFTs and Laser Solid Phase Crystallized (LSPC) polysilicon TFTs. Devices are fabricated on glass substrate. A thick APCVD (Atmospheric Pressure Chemical Vapour Deposition) SiO 2 layer is first deposited to prevent a possible contamination from the substrate because of severe thermal annealing during the fabrication process. FSPC TFTs (see fig.

  1. a) are elaborated with one poly-Si layer: the upper part is heavily in-situ n-type doped (source and drain regions), and the bottom part is undoped and is dedicated to the active layer. LSPC TFTs (see fig. 1. b) are fabricated with two poly-Si layers: the active layer is built in the undoped layer, whereas an in-situ n-type doped layer constitutes source and drain regions. For the two types of TFTs the polysilicon layers are deposited by LPCVD (Low Pressure-CVD) technique and are

  A. Carriers trapping/detrapping process at oxide traps It is widely admitted that low frequency noise in N-channel MOS transistors is due to carrier fluctuations. In this case, it can be described by the usual Hooge empirical relation of the normalized noise spectral density:

  i) for m*<<N, it means from weak to moderate inversion (at low gate voltages), trap occupancy is small and α increases as E F increases into the lower part of the band gap following the N/m * ratio and: m*>>N, it means from moderate strong inversion (at high gate voltages), trap occupancy is high and α decreases as E F increases into the upper part of the band gap following the m * /N ratio and: m * (≈M-m) acting as the trapping centre number.

  allow the calculation of the interface trap states distribution from E A and S IDS /I 2 DS both deduced from measurements of I DS and S IDS at various gate voltages. Significant values of N Tdb correspond to deep level trap states distribution (high values of E A ) assumed to be related to dangling bond type defects, whereas N Tts significant values correspond to shallow level ones (low values of E A ) associated with strained bond defects (band tailing). Thus the resulting interface trap states distribution (N T ) is N T = N Tdb + N Tts . IV. EXPERIMENTAL ANALYSIS A. Temperature dependence

  interface trap states distribution, parameters are clearly linked by standing N S =N Tts0 and kT s =E MN .For deep level interface trap states distribution the link is less evident, but considering the maximum of the distribution (N Tdbmax ) close to the midgap, it means with E A =E C -E≈E G /2 (≈0.56eV), thus N Tdbmax =N Tdb0 exp(E G /(2E MN) )=N d /(

  IV. CONCLUSIONIn this experimental analysis of the 1/f noise we emphasized the influence of the interface quality in polysilicon TFTs on specific description of noise. A new method is proposed to determine the energy distribution of the interface trap states in polysilicon TFTs from the 1/f noise measurements. It is based on the variations of the normalized drain current spectral density with the gate voltage. Two regimes of variations are considered corresponding respectively to the weak and the strong inversion regions. 1/f noise spectrum at low frequency is explained by the domination of the carriers trapping/detrapping process from defects located at the interface tunnelling into oxide traps. The contributions of related band tailing (strained bonds) and dangling bonds interface defects are separately estimated. Quantitative dependence, in relation with the interface quality, and thus with the TFT-technology, on the Meyer Neldel energy parameter of the two corresponding exponential distributions is shown. For LSPC TFT, both tails and dangling bonds states distributions are wider because of a higher Meyer Neldel energy, and the resulting interface trap states density is one decade lower than for FSPC TFT due to the effect of the laser annealing on the crystal quality of the active layer. The study confirms the validity of the tunnelling theory of trapped carriers into the gate oxide (Mc Worther theory) to explain 1/f noise in polysilicon TFTs.

Figure 1 :Figure 2 :

 12 Figure 1: Cross section of the tested devices: (a) FSPC TFT, (b) LSPC TFT

Figure 3 :

 3 Figure 3: Plots of E A versus V GS the FSPC and LSPC TFTs.

Figure 4 :

 4 Figure 4: Transfer characteristics plotted in the linear mode (V DS =300mV) of the tested FSPC TFTs (W/L=30µm/10µm) and the LSPC TFTs (W/L=80µm/60µm).

Figure 5 :

 5 Figure 5: Normalized drain current noise spectral density versus drain current for: (a) FSPC TFTs, (b) LSPC TFTs, (f=10 Hz, V DS =300mV). Inset: same plots in semi logarithmic scale.

Figure 6 :

 6 Figure 6: Plots of the interface trap state densities deduced from (5) and (6) for: (a) FSPC TFTs, (b) LSPC TFTs.

Figure 7 :

 7 Figure 7: Plots of the interface trap state distributions deduced from noise measurements given by (5)+(6) and by (9)+[START_REF] Mc | Semiconductor Surface Physics[END_REF] for FSPC and LSPC TFTs. Dotted plots: extrapolated interface trap state distributions deduced from static drain current activation energy given (9)+[START_REF] Mc | Semiconductor Surface Physics[END_REF].

Table caption Table I :

 captionI summary of interface state distribution parameters for the FSPC and the LSPC TFTs.

		E MN	N Tts0	N Tdbmax
		(eV)	(cm -2 eV -1 )	(cm -2 eV -1 )
	FSPC TFT	0.035	~10 15	~8×10 13
	LSPC TFT	0.055	~10 13	~3×10 11


B Gralak

Y Liu

S Guenneau

Magneto-electric coupling and magnetism in photonic crystals: A high-order homogenization approach in the constitutive equations

Keywords: Asymptotic expansion, effective medium, layered structure

on artificial magneto-electric coupling and magnetism in moderate contrast dielectric layered media via high-order homogenization to the three-dimensional setting. For this, we consider asymptotic expansions in the vector Maxwell system. We then illustrate effect of magneto-electric coupling with numerical simulations for a layered system alternating positively and negatively refracting index. We unveil hyperbolic behaviour and magneto-electric coupling and derive a sharper estimate for the anisotropic effective medium in [Ramakrishna et al.,

Introduction

There is currently renewed interest in artificial magnetism in high-contrast photonic crystals, and metamaterials, notably in the quasi-static regime. Artificial bianisotropy is another very active research area of effective medium theories in periodic structures, such as arrays of split ring resonators or swiss-rolls. However, it is not widely appreciated that there is even magnetic activity and magneto-electric coupling in low contrast layered media [START_REF] Liu | Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers[END_REF].

A readily accessible asymptotic regime through effective medium theories, much studied in the literature on multipole expansions, is for low frequency (quasi-static), or dilute composite, limits. Alternatively if one introduces a small positive parameter η, say a ratio of the array pitch by the wave wavelength, then so called two-scale asymptotic expansions can be used to identify leading order homogenized Maxwell's equations, and in this way, effective properties are obtained at fixed frequency [START_REF] Guenneau | Homogenization of finite photonic crystals[END_REF][START_REF] Wellander | Homogenization of the Maxwell equations: Case I. Linear theory[END_REF], that involve an artificially anisotropic effective permittivity. In case of high-contrast in the material parameters, say when a highly-conducting inclusion is surrounded by a dielectric matrix, two-scale expansions unveil artificial magnetism [5]. An alternative route to achieve not only artificial magnetism, but also magneto-electric coupling, is via high order homogenization as proposed in [START_REF] Liu | Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers[END_REF] with a homogenization algorithm using the transfer matrix formalism. This high-order homogenization algorithm has been used subsequently in [START_REF] Liu | Causality and passivity properties of effective parameters of electromagnetic multilayered structures[END_REF][START_REF] Popov | Operator approach to effective medium theory to overcome a breakdown of Maxwell Garnett approximation[END_REF][START_REF] Novitsky | Nonlocal homogenization of PT -symmetric multilayered structures[END_REF][START_REF] Lebbe | Stable GSTC formulation for Maxwell's equations[END_REF][START_REF] Cornaggia | An homogenized model accounting for dispersion, interfaces and source points for transient waves in 1D periodic media[END_REF]. However, the latter route becomes cumbersome with doubly and triply periodic structures. We thus revert to two-scale expansions, however with higher order corrections taken into account in the homogenized Maxwell system, as proposed for the conductivity equation in [START_REF] Bakhvalov | Homogenisation: averaging processes in periodic media[END_REF].

High-order homogenization of constitutive equations

We consider a finite photonic crystal (PC) that consists of a large number of periodic cells ηY = [0, η] 3 , filled with a dielectric medium with permittivity ε(x/η), which is periodic of period η in the bounded domain Ω (the PC) and which is equal to ε 0 in the unbounded domain outside Ω. As per usual one denotes by ε 0 µ 0 = c -2 , where c is the velocity of light in vacuum. Since we consider a dielectric, non-magnetic medium, illuminated by a time harmonic wave, the oscillatory electromagnetic field within the PC (E η , H η ) satisfies the first two Maxwell's equations

∇ x × E η (x) = iωµ 0 H η (x) , ∇ x × H η (x) = -iωε(x/η)E η (x) , (1)
where ω is the wave angular frequency (rad/s). Taking the divergence of (1), we can proceed with the following conditions

∇ x • µ 0 H η (x) = 0 and ∇ x • ε(x/η)E η (x) = 0 (2)
We assume the same asymptotic expansions of the electric and magnetic oscillatory fields E η and H η as in [START_REF] Guenneau | Homogenization of finite photonic crystals[END_REF]

E η (x) = E 0 (x, x/η) + N i=1 η i E i (x, x/η) + O(η N ) (3)
and

H η (x) = H 0 (x, x/η) + N i=1 η i H i (x, x/η) + O(η N ) (4)
One notes that E i and H i are periodic of period η in the second variable and besides from that E η and H η should be of finite energy on every compact set, and satisfy some outgoing wave condition, so that the diffraction problem of a plane wave illuminating the PC be well posed (existence and uniqueness of the solution) for each η. We refer to [START_REF] Guenneau | Homogenization of finite photonic crystals[END_REF][START_REF] Wellander | Homogenization of the Maxwell equations: Case I. Linear theory[END_REF] for such questions.

Let us adopt the following notations:

F η (x) = E η (x) H η (x) = F 0 (x, x/η) + N i=1 η i F i (x, x/η) + O(η N ) (5)
with

F i (x, x/η) = E i (x, x/η) H i (x, x/η) for i = 0, 1, . . . , N , (6)
and

M x = 0 i ε -1 0 ∇ x × -i µ -1 0 ∇ x × 0 , V (x/η) = -ω ε(x/η) -ε 0 ε 0 1 0 0 0 . (7)
Then, the Maxwell's equations ( 1) can be written as

ωF η (x) = M x F η (x) + V (x/η) F η (x) . (8)
Rescaling the differential operator M x as M x + η -1 M y and plugging the asymptotic expansion (5) in the Maxwell's equations ( 8) and further collecting terms of factors of same powers of η, we get a hierarchy of equations in powers of η with, at the leading order in η -1 ,

M y F 0 (x, y) = 0 (9)
and, at the next orders,

η 0 : ωF 0 (x, y) = M x F 0 (x, y) + M y F 1 (x, y) + V (y)F 0 (x, y) , η 1 : ωF 1 (x, y) = M x F 1 (x, y) + M y F 2 (x, y) + V (y)F 1 (x, y) , η 2 : ωF 2 (x, y) = M x F 2 (x, y) + M y F 3 (x, y) + V (y)F 2 (x, y) , . . . η i : ωF i (x, y) = M x F i (x, y) + M y F i+1 (x, y) + V (y)F i (x, y) . (10)
Applying the mean operator • over the periodic cell Y = [0, 1] 3 , and invoking the generalized Stokes' theorem applied over Y (a torus, so a manifold without boundary), this set of equations [START_REF] Cornaggia | An homogenized model accounting for dispersion, interfaces and source points for transient waves in 1D periodic media[END_REF] above imply

η 0 : ω F 0 (x, y) = M x F 0 (x, y) + V (y)F 0 (x, y) , η 1 : ω F 1 (x, y) = M x F 1 (x, y) + V (y)F 1 (x, y) , η 2 : ω F 2 (x, y) = M x F 2 (x, y) + V (y)F 2 (x, y) , . . . η i : ω F i (x, y) = M x F i (x, y) + V (y)F i (x, y) . (11)
These last equations provide the different orders of the homogenized Maxwell's equations.

Before applying this mean operator • over the periodic cell Y = [0, 1] 3 , the homogenization procedure requires to derive the expression of the components F i (x, y) of F η (x, x/η). For the leading order, this derivation starts with the equations ( 9) and [START_REF] Cornaggia | An homogenized model accounting for dispersion, interfaces and source points for transient waves in 1D periodic media[END_REF] for the transverse part and, for the longitudinal part, with the equations deduced from [START_REF] Liu | Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers[END_REF], see the sections 5.1 in the annex for details.

For the leading order, the equation ( 9) and the equation deduced from [START_REF] Liu | Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers[END_REF] show that the electric and magnetic components E 0 (x, y) and H 0 (x, y) are uncoupled. However, it is stressed that, for the next orders, the equations [START_REF] Cornaggia | An homogenized model accounting for dispersion, interfaces and source points for transient waves in 1D periodic media[END_REF] shows that the transverse parts of the electric and magnetic components E i (x, y) and H i (x, y) are generated by source-terms H i-1 (x, y) and E i-1 (x, y). Therefore, we have formally that

E i = W ee i E i-1 + W eh i H i-1 , H i = W he i E i-1 + W hh i H i-1 . (12)
According to the homogenized Maxwell's equations [START_REF] Bakhvalov | Homogenisation: averaging processes in periodic media[END_REF], the effective parameters describing the interaction between the electromagnetic field and the media are given at each order by

V (y)F i (x, y) , (13)
and the different orders of the homogenized electric displacement field and magnetic inductance are given by:

ω -V (y) F i (x, y) . (14)
Hence, from the formal expressions ( 12), it appears that, the equation above is reminiscent of the structure of a constitutive equation for an effective bianisotropic medium as in [START_REF] Liu | Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers[END_REF][START_REF] Kong | Electromagnetic Wave Theory[END_REF][START_REF] Serdyukov | Electromagnetics of bi-anisotropic materials : Theory and applications[END_REF]]

D hom (x) = ε hom E hom (x) + Ξ hom H hom (x) , B hom (x) = µ hom H hom (x) + t Ξ hom E hom (x) , (15)
where D hom and B hom are homogenized electric displacement field and magnetic inductance, Ξ hom and its symmetric t Ξ hom are rank-2 tensors of magnetic-electric coupling. Notice that, at the first order, the magnetic field H 1 (x, y) is the solution to this equations for its transverse and longitudinal parts [START_REF] Guerin | Singularity of the dyadic Green's function for heterogeneous dielectrics[END_REF] y) is thus generated by the solely source term E 0 (x, y), which means that the effective constitutive equation for B hom (x) reduces to a term driven by E hom (x). As a consequence, µ hom remains the vacuum permeability µ 0 at the order 1, and the effective permeability µ hom differs from µ 0 only from the order 2. This is in agreement with the results stated in [START_REF] Liu | Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers[END_REF], where the effective magneto-electric coupling is obtained from order 1 and then the effective magnetism from order 2.

∇ y × ∇ y × H 1 (x, y) = -iω∇ y × ε(y)E 0 (x, y) , ∇ y • H 1 (x, y) = 0 (16) since H 0 (x, y) = H 0 (x) = H 0 (x) (see section 5.2 in the annex). The component H 1 (x,

Illustrative application : the layered case

Derivation of the homogenized parameters in the layered case is quite straightforward, a detailed derivation can be found in [START_REF] Liu | Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers[END_REF][START_REF] Liu | Causality and passivity properties of effective parameters of electromagnetic multilayered structures[END_REF]. The interested reader can find details on how to compute V (0) in (31), a complete derivation of the leading order expression for the homogenized permittivity (34) in [START_REF] Pavliotis | Multiscale Methods: Averaging and Homogenization[END_REF], and for higher-order expressions we refer to [START_REF] Allaire | Optimization of dispersive coefficients in the homogenization of the wave equation in periodic structures[END_REF].

1 x 2 x 3 x     h Figure 1.
Schematic diagram of a layered Poor Man's lens, the permittivities of two dielectrics are ε + and ε -, the thicknesses of two layers are the same.

Hyperbolicity and optical activity in the Poor Man's lens

We apply the above HOH to a layered Poor Man's lens [START_REF] Pendry | Negative Refraction Makes a Perfect Lens[END_REF], shown in Fig. 1, which consists of an alternation of two dielectrics with permittivity equal to ε 1 /ε 0 (= +1) and ε 2 /ε 0 (= -1), and filling fraction are f 1 (= 1/2) and f 2 (= 1/2) respectively. The zeroth order approximation provides us with the same expressions as the classical homogenization

ε = ε ⊥ = ε 1 f 1 + ε 2 f 2 = 0 , ε -1 3 = ε -1 1 f 1 + ε -1 2 f 2 = 0 , µ = µ ⊥ = µ 3 = µ 0 . ( 17
)
If we move forward to the first order approximation, the permittivity and permeability are the same as in Eq.( 17), while we obtain the magneto-electric coupling as follows

K ⊥ = ωh 2 µ 0 (ε 1 -ε 2 )f 1 f 2 = ωh 4c 2 , K = ωh 2 µ 0 (ε 1 -ε 2 )f 1 f 2 1 - k 2 ω 2 ε 1 + ε 2 µ 0 ε 1 ε 2 = ωh 4c 2 . ( 18
)
where k 2 = k•k and k is the two-dimensional component of the three-dimensional wave vector k after projection in the plane (x 1 , x 2 ). Note that the magnetoelectric coupling parameter is isotropic at this order, since K ⊥ = K in this very specific sign-shifting configuration where ε 1 = -ε 2 . Further, considering the second order approximation, the effective permittivity and permeability expressions are [START_REF] Liu | Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers[END_REF]

ε = ε 1 f 1 + ε 2 f 2 + ω 2 h 2 6 µ 0 f 1 f 2 (ε 1 -ε 2 )(ε 1 f 1 -ε 2 f 2 ) 1 - k 2 ω 2 ε 1 + ε 2 µ 0 ε 1 ε 2 = ω 2 h 2 12c 2 ε 0 , ε ⊥ = ε 1 f 1 + ε 2 f 2 + ω 2 h 2 6 µ 0 f 1 f 2 (ε 1 -ε 2 )(ε 1 f 1 -ε 2 f 2 ) = ω 2 h 2 12c 2 ε 0 , ε 3 -1 = ε -1 1 f 1 + ε -1 2 f 2 - ω 2 h 2 6 µ 0 f 1 f 2 (ε 1 -ε 2 )( f 1 ε 1 - f 2 ε 2 ) 1 - k 2 ω 2 ε 1 + ε 2 µ 0 ε 1 ε 2 = - ω 2 h 2 12c 2 ε 0 , µ = µ 0 - ω 2 h 2 6 µ 2 0 f 1 f 2 (ε 1 -ε 2 )(f 1 -f 2 ) = µ 0 , µ ⊥ = µ 0 - ω 2 h 2 6 µ 2 0 f 1 f 2 (ε 1 -ε 2 )(f 1 -f 2 ) 1 - k 2 ω 2 ε 1 + ε 2 µ 0 ε 1 ε 2 = µ 0 , µ 3 -1 = µ -1 0 + ω 2 h 2 6 f 1 f 2 (ε 1 -ε 2 )(f 1 -f 2 ) = µ -1 0 . ( 19
)
We note that our effective medium is highly anisotropic when ω tends to 0 but, unlike for the static limit studied in [START_REF] Ramakrishna | Imaging the near-field[END_REF], it is also hyperbolic since the ε 3 component takes negative values. This hyperbolic behaviour can be supported by the following argument. Let

ε 1 = ε 0 and ε 2 = -ε 0 (1 -δ), with 0 < δ 1.
Then it is straightforward to check that, at the zeroth order,

ε = ε ⊥ = ε 1 f 1 + ε 2 f 2 = ε 0 f 1 -ε 0 (1 -δ)f 2 = ε 0 δf 2 , ε -1 3 = ε -1 0 f 1 -ε -1 0 (1 -δ) -1 f 2 ≈ -ε -1 0 δf 2 , (20)
which shows that both perturbative corrections, with respect to small frequencies [START_REF] Pierre | Appropriate truncation for photonic crystals[END_REF] and to small deviation from the sign shifting [START_REF] Śmigaj | Validity of the effective-medium approximation of photonic crystals[END_REF], lead to a hyperbolic behaviour of the multilayered Poor Man's lens. Moreover, our modelling is different at the first-order correction [START_REF] Ramakrishna | Imaging the near-field[END_REF] since it brings some magnetoelectric coupling that is not accounted for in the effective medium model derived in [START_REF] Ramakrishna | Imaging the near-field[END_REF]. This first order correction is not surprising since it has been known for over a decade that truncation of photonic crystals matters in their effective properties [START_REF] Pierre | Appropriate truncation for photonic crystals[END_REF][START_REF] Śmigaj | Validity of the effective-medium approximation of photonic crystals[END_REF][START_REF] Simovski | On electromagnetic characterization and homogenization of nanostructured metamaterials[END_REF][START_REF] Yang | Retrieving the effective parameters of metamaterials from the single interface scattering problem[END_REF][START_REF] Vinogradov | Additional effective medium parameters for composite materials (excess surface currents)[END_REF]. Finally, we also notice that there is no effective magnetism, just like in [START_REF] Ramakrishna | Imaging the near-field[END_REF]. As a numerical illustration performed with Matlab, let us consider again the multilayered Poor Man's len, the total number of the unit layers is N = 20; the parameters of the two layers constituting each unit layer are

ε 1 /ε 0 = 1, µ 1 /µ 0 = 1, ε 2 /ε 0 = -1 + 0.0001 * i, µ 2 /µ 0 = 1,
a small absorption being introduced to ensure the convergence in Matlab. Note that the p-and s-polarized incident waves coincide under a normal incidence (k = 0), i.e. t p = t s = t. Now we would like to compare the transmission properties of the stack and its effective medium obtained by the above homogenization method. The transfer matrix method in [START_REF] Liu | Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers[END_REF] is introduced to compute the transmission coefficients as a function of the normalized frequency ωh/(2πc) ≡ η, Fig. 2 shows both the transmission curve of the stack (solid line) and that of the effective medium (dashed line) at the 2nd order approximation with parameters in Eqs.( 17)- [START_REF] Pierre | Appropriate truncation for photonic crystals[END_REF]. The ordinate in the figures is the real part of the transmission coefficient, while the abscissa is the normalized frequency η. We stress that, in this section, the non-dimensional parameter η is the normalized frequency ωh/(2πc), and it is thus different than the parameter η used in the section 2 (that was related to a non-dimensional length); however, in both cases, the homogenization is studied for vanishing parameter η.

It can be observed in Fig. 2 that the transmission curve of the effective homogeneous medium at the second order approximation (dashed line) fits well with the curve of the stack (solid line) at the normalized frequencies η lower than 0.4. With a higher order approximation, an improved agreement between the transmission curves of the stack and its effective medium can be achieved, see [START_REF] Liu | Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers[END_REF].

real( ) t

 

0 i    Figure 2.
Transmission curves of the multilayered stack (solid line) and effective medium at 2nd order (dashed line) approximation.The permittivities of the two dielectric layers are

ε 1 /ε 0 = 1, ε 2 /ε 0 = -1 + 0.0001 * i, respectively.
Furthermore, considering an oblique incidence with θ i = 30 • in p-polarization, as well as s-polarization, the transmission curves of the stack and its effective medium are depicted in Fig. 3. The asymptotic approximation at the second order is in good agreement up to the normalized frequency 0.5 in s-polarization and 0.3 in p-polarisation. Notice that this numerical example confirms the hyperbolic feature of the Poor Man's lens at low frequencies.

/ε 0 = 1, ε 2 /ε 0 = -1 + 0.0001 * i, respectively. L  G  nh 2 h 2 h 2 x 1 x 3 x Figure 4.
Geometry of a finite P-T symmetric multilayered structure, the unit cell consists of two dielectrics with balanced loss and gain, i.e. ε L and ε G , the thicknesses of two layers are the same, the total number of layers is 2N.

The P-T symmetric layered case

We turn to a second application of the high order homogenization in the configuration the P-T symmetric system studied in [START_REF] Novitsky | Nonlocal homogenization of PT -symmetric multilayered structures[END_REF]. Fig. 4 shows the considered P-T symmetric multilayered structure, which is a periodic structure with balanced loss and gain, i.e. the permittivities of the two component layers are

ε L = ε + iε , ε G = ε -iε
with the same value of loss and gain coefficients. Same HOH method is applied to such a P-T multilayered structure, the expressions of effective parameters for classical approximation are

ε = ε ⊥ = ε 1 f 1 + ε 2 f 2 = ε , ε -1 3 = ε -1 1 f 1 + ε -1 2 f 2 = ε (ε 2 + ε 2 ) -1 , µ = µ ⊥ = µ 3 = µ 0 . ( 21
)
while for the homogeneous medium at 1st order approximation, the entries of the magneto-electric coupling tensor are (in normal incidence)

K ⊥ = K = ωh 2 µ 0 (ε 1 -ε 2 )f 1 f 2 = i ωh 4 µ 0 ε . ( 22
)
When we consider the second order approximation, we obtain

ε = ε ⊥ = ε 1 f 1 + ε 2 f 2 + ω 2 h 2 6 µ 0 f 1 f 2 (ε 1 -ε 2 )(ε 1 f 1 -ε 2 f 2 ) = ε - ω 2 h 2 12 µ 0 ε 2 , ε 3 -1 = ε -1 1 f 1 + ε -1 2 f 2 - ω 2 h 2 6 µ 0 f 1 f 2 (ε 1 -ε 2 )( f 1 ε 1 - f 2 ε 2 ) = ε - ω 2 h 2 12 µ 0 ε 2 1 ε 2 + ε 2 , µ = µ ⊥ = µ 0 - ω 2 h 2 6 µ 2 0 f 1 f 2 (ε 1 -ε 2 )(f 1 -f 2 ) = µ 0 , µ 3 -1 = µ -1 0 + ω 2 h 2 6 f 1 f 2 (ε 1 -ε 2 )(f 1 -f 2 ) = µ -1 0 . ( 23
)
For the next orders, we refer to [START_REF] Liu | Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers[END_REF]. Fig. 5 shows the comparison of transmission for the multilayer and for the effective medium up to the seventh order in the P-T symmetry configuration proposed in [START_REF] Novitsky | Nonlocal homogenization of PT -symmetric multilayered structures[END_REF]. . Transmission curves of PT-symmetric multilayer and the effective medium at 0th, 1st, 2nd, 3rd and 7th orders approximation as a function of ε /ε 0 . The permittivities of the two dielectric layers are ε 1 /ε 0 = 2 + iε /ε 0 , ε 2 /ε 0 = 2 -iε /ε 0 , respectively. The period thickness (a) d=100nm and (b) d=200nm. A normal incident wave with λ = 1.55µm is considered, the multilayered structure consists of N=20 slabs.

The results are in good agreement and demonstrate the validity of the present high order homogenization in this P-T symmetry configuration. In particular, the seventh order bring a significative improvement of the approximation for values of ε /ε 0 below 20 for d = 100nm and below 6 for d = 200nm.

Conclusion

In this paper, we introduced a high-order homogenization for the Maxwell's equations, devoted to constitutive equations of periodic dielectric media. We have shown that at leading order, one simply achieves some anisotropic permittivity. However, at the first order some magneto-electric coupling occurs as the homogenized constitutive equation has the structure of that for a bianisotropic medium. At the second order some artificial magnetism occurs. Interestingly, if one would consider some periodic medium with spatially varying permittivity and permeability, the proposed algorithm would be slightly modified as in [START_REF] Guenneau | Homogenization of 3D finite photonic crystals with heterogeneous permittivity and permeability[END_REF] the equations at order η -1 would lead to a leading order asymptotic expansion of the form H 0 (x, y) = H 0 (x) + ∇ y [V(y) • H 0 (x)], and the same annex problem as (31) and some anisotropic permeability would be achieved at leading order as in (34). The homogenized Maxwell system at leading order was already derived for spatially varying permittivity and permeability using asymptotic expansions of the electric and magnetic fields (3)-(4) at first order in all four Maxwell's equations ( 1)-( 2) [START_REF] Wellander | Homogenization of the Maxwell equations: Case I. Linear theory[END_REF] and in the curlcurl Helmholtz operator by substituting the second equation into the first one in (1) thus working with the magnetic field unknown [START_REF] Guenneau | Homogenization of 3D finite photonic crystals with heterogeneous permittivity and permeability[END_REF]. However, our derivation is based on higher-order terms in the asymptotic expansions of the Maxwell's equations ( 1)-( 2), and a series of ansatz at higher orders for the electric and magnetic field unknowns allows to identify higher order homogenized effective parameters. If one proceeds with the high-order homogenization in the constitutive laws when both permittivity and permeability are heterogeneous, magneto-electric coupling occurs at first order as in the present case. We then recall the effective parameters for a layered structure at second order approximation already derived in [START_REF] Liu | Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers[END_REF], which has been reproduced with the exact same derivation in [START_REF] Popov | Operator approach to effective medium theory to overcome a breakdown of Maxwell Garnett approximation[END_REF]. We retrieve the numerical results of [START_REF] Novitsky | Nonlocal homogenization of PT -symmetric multilayered structures[END_REF] for a P-T symmetric layered structure at second order approximation, and show improved results at seventh order. We also apply the high-order homogenization to a layered medium alternating layers of identical thickness with positive and negative permittivity (known as the layered Poor Man's lens) and at leading order we retrieve the effective medium permittivity in [START_REF] Ramakrishna | Imaging the near-field[END_REF]. However, we find that some magneto-electric coupling occurs at the first order and, at second order, there is no magnetic activity. Thus the layered Poor Man's lens behaves effectively as a chiral medium with extreme effective permittivity, and moreover it is shown to be hyperbolic.

Our approach is different from that in [START_REF] Maurel | Sensitivity of a dielectric layered structure on a scale below the periodicity: A fully local homogenized model[END_REF] which considers some high-order homogenization for the electromagnetic problem based on a dynamic correction encapsulated in the boundary conditions at the interface between the homogenized stack and surrounding medium. Other works on improved effective medium approximation of layered media include [START_REF] Popov | Operator approach to effective medium theory to overcome a breakdown of Maxwell Garnett approximation[END_REF][START_REF] Gralak | Negative index materials and time-harmonic electromagnetic field[END_REF][START_REF] Gorlach | Boundary conditions for the effective-medium description of subwavelength multilayered structures[END_REF][START_REF] Gralak | Negative index materials: at the frontier of macroscopic electromagnetism[END_REF][START_REF] Lannebere | First principles homogenization of periodic metamaterials and application to wire media[END_REF].

The first two lines of the equation above imply:

∇ y × ∇ y × E 1 (x, y) = -∇ y × ∇ x × E 0 (x, y) , ∇ y × ∇ y × H 1 (x, y) = -iω∇ y × ε(y)E 0 (x, y) . (26)
and the last two lines imply:

∇ y • ε(y)E 1 (x, y) = -∇ x • ε(y)E 0 (x, y) , ∇ y • H 1 (x, y) = -∇ x • H 0 (x, y) . ( 27
)
Applying the mean operator • over the periodic cell Y = [0, 1] 3 , and invoking the generalized Stokes' theorem applied over Y (a torus, so a manifold without boundary), the first two lines in the set of equations ( 25) of order η 0 above imply

∇ x × E 0 (x, y) = iωµ 0 H 0 (x, y) , ∇ x × H 0 (x, y) = -iω ε(y)E 0 (x, y) . ( 28
)
The third line in equation ( 24) provides us with the homogenized permittivity simply by assuming the leading order term of the asymptotic expansion (3) has the following form [START_REF] Guenneau | Homogenization of finite photonic crystals[END_REF]

E 0 (x, y) = E 0 (x) + ∇ y V (0) (y) • E 0 (x) (29)
This form obviously fulfills the partial differential equation given by the first line in [START_REF] Guenneau | Homogenization of 3D finite photonic crystals with heterogeneous permittivity and permeability[END_REF], since the composition of the curl operator ∇ y × with ∇ y vanishes and plugging it in the third eqaution of (24) leads to the annex problem [START_REF] Guenneau | Homogenization of finite photonic crystals[END_REF] ∇

y • ε(y) E 0 (x) + ∇ y V (0) (y) • E 0 (x) = 0 (30)
or equivalently

∇ y • ε(y) e i + ∇ y V (0) i (y) = 0 , i = 1, 2, 3 , (31)
where • denotes the mean operator over the periodic cell Y = [0, 1] 3 and V (0) is a rank-1 tensor such that V (0) = 0 and and {e 1 , e 2 , e 3 } is the canonical basis in Euclidean space R 3 . We now apply the mean operator to the third line in the equation ( 25) of order η 0 :

∇ x • ε(y)E 0 (x, y) + ∇ y • ε(y)E 1 (x, y) = 0 , (32)
where, thanks to the divergence theorem, the second term reduces to the integral over the boundary ∂Y of the unit cell Y , ∂Y εE 1 • nds. Furthermore, noting that E 1 (x, y) is periodic over Y and the normal n is anti-periodic, this term vanishes. Using the ansatz (29), we get

∇ x • ε(y) I + ∇ y V (0) • E 0 (x) = 0 (33)
which provides the homogenized equation for the electric field at order 0, with effective rank-2 anisotropic permittivity tensor with coefficients

ε (0) hom,ij = ε(y) δ ij + ∂V (0) i ∂y j , i = 1, 2, 3 , (34)
where δ ij is the Kronecker symbol.

As for the leading order term of the asymptotic expansion (4), it is given by

H 0 (x, y) = H 0 (x) , (35)
since, from [START_REF] Guenneau | Homogenization of 3D finite photonic crystals with heterogeneous permittivity and permeability[END_REF], ∇ y × H 0 (x, y) = 0 and ∇ y • H 0 (x, y) = 0. We note that from ( 28) and (33) we get the homogenized Maxwell's equations at the leading order:

∇ x × E (0) hom = iωµ 0 H (0) hom (x) , ∇ x × H (0) hom (x) = -iωε (0) hom E (0) hom , (36) with E
(0) hom (x) = E 0 (x) , H (0) hom (x) = H 0 (x) .
(37)

Effective parameters at higher orders

Let us now proceed with the next terms E 1 (x, y) and H 1 (x, y) in the expansion of the fields (3) and (4). The equation [START_REF] Maurel | Sensitivity of a dielectric layered structure on a scale below the periodicity: A fully local homogenized model[END_REF] shows that each of these first-order term is the solution to a linear equation with two forcing terms generated by the zeroorder components E 0 (x, y) and H 0 (x, y). Hence the first order introduces a coupling between the electric and magnetic fields.

Let us now proceed with the terms of order η 1 following the equation ( 25) in a hierarchy of equations in powers of η. We can see that all terms H i are divergence free, so all the effective properties will come from ∇ • (ε -1 E) = 0. More precisely, ∇ x × E 1 (x, y) + ∇ y × E 2 (x, y) = iωµ 0 H 1 (x, y) , ∇ x × H 1 (x, y) + ∇ y × H 2 (x, y) = -iωε(y)E 1 (x, y) , ∇ x • ε(y)E 1 (x, y) + ∇ y • ε(y)E 2 (x, y) = 0 ,

∇ x • H 1 (x, y) + ∇ y • H 2 (x, y) = 0 . (38)
Applying the mean operator • over the periodic cell Y = [0, 1] 3 , and invoking the generalized Stokes' theorem applied over Y (a torus, so a manifold without boundary), the first two lines in the set of equations (38) of order η 1 above imply ∇ x × E 1 (x, y) = iωµ 0 H 1 (x, y) , ∇ x × H 1 (x, y) = -iω ε(y)E 1 (x, y) .

(39)

Applying the mean operator • over the periodic cell Y = [0, 1] 3 , and making use of the divergence theorem applied over Y , the last two lines in the set of equations (38) of order η 1 imply

∇ x • ε(y)E 1 (x, y) = 0 , ∇ x • H 1 (x, y) = 0 , (40)
and, in addition with the leading order, lead to ∇ x • ε(y) E 0 (x, y) + ηE 1 (x, y) = 0 ,

∇ x • µ 0 H 0 (x, y) + ηH 1 (x, y) = 0 . ( 41
)
These two equations are the starting point to derive the effective constitutive equations at the order η 1 . Indeed, using that H 0 (x, y) = H 0 (x) is independent of the variable y, we have ∇ x • H 0 (x, y) = ∇ x • H 0 (x) = 0 and ∇ y × ∇ x × H 0 (x, y) = ∇ y × ∇ x × H 0 (x) = 0. Therefore, the second and fourth lines of equation ( 25

We can write

H 1 (x, y) = H 1 (x) + W 1 (y) × E 0 (x) (44)
where W 1 (y) is a rank-1 tensor solution of

∇ y × ∇ y × W 1 (y) = -iω∇ y ε(y) (45)
We note that since ∇ y • H 1 (x, y) = 0, one has that ∇ y • (W 1 (y) × E 0 (x)) = 0. Similarly, the first and third lines of equation [START_REF] Maurel | Sensitivity of a dielectric layered structure on a scale below the periodicity: A fully local homogenized model[END_REF] imply

∇ y × ∇ y × E 1 (x, y) = -∇ y × ∇ x × E 0 (x, y) , ∇ y • ε(y)E 1 (x, y) = -ε(y)∇ x • E 0 (x, y) , (46)
since ∇ y × H 0 (x, y) = ∇ y × H 0 (x) = 0 and the expression for E 1 (x, y) ensues.

Figure 3 .

 3 Figure 3. Transmission curves of the multilayered stack (solid line) and effective medium at second order approximation (dashed line) as a function of the normalized frequency η = ωh/(2πc) under an oblique incidence with θ i = 30 • : (a) s-polarization; (b) p-polarization. The permittivities of the two layers are ε 1 /ε 0 = 1, ε 2 /ε 0 = -1 + 0.0001 * i, respectively.

Figure 5

 5 Figure 5. Transmission curves of PT-symmetric multilayer and the effective medium at 0th, 1st, 2nd, 3rd and 7th orders approximation as a function of ε /ε 0 . The permittivities of the two dielectric layers are ε 1 /ε 0 = 2 + iε /ε 0 , ε 2 /ε 0 = 2 -iε /ε 0 , respectively. The period thickness (a) d=100nm and (b) d=200nm. A normal incident wave with λ = 1.55µm is considered, the multilayered structure consists of N=20 slabs.

  ) imply∇ y × ∇ y × H 1 (x, y) = -iω∇ y × [ε(y)E 0 (x, y)] , ∇ y • H 1 (x, y) = 0 . (42)From the first line in (43) and (29)∇ y × ∇ y × H 1 (x, y) = -iω∇ y × [ε(y) E 0 (x) + ∇ y V (0) (y) • E 0 (x) ]= -iω∇ y ε(y) × E 0 (x)