Benjamin Lambert

Florence Forbes

Senan Doyle

Alan Tucholka

Michel Dojat

Beyond Voxel Prediction Uncertainty: Identifying brain lesions you can trust

Keywords: MS lesion, Detection, Deep Learning, Interpretabilty, Prediction

Deep neural networks have become the gold-standard approach for the automated segmentation of 3D medical images. Their full acceptance by clinicians remains however hampered by the lack of intelligible uncertainty assessment of the provided results. Most approaches to quantify their uncertainty, such as the popular Monte Carlo dropout, restrict to some measure of uncertainty in prediction at the voxel level. In addition not to be clearly related to genuine medical uncertainty, this is not clinically satisfying as most objects of interest (e.g. brain lesions) are made of groups of voxels whose overall relevance may not simply reduce to the sum or mean of their individual uncertainties. In this work, we propose to go beyond voxel-wise assessment using an innovative Graph Neural Network approach, trained from the outputs of a Monte Carlo dropout model. This network allows the fusion of three estimators of voxel uncertainty: entropy, variance, and model's confidence; and can be applied to any lesion, regardless of its shape or size. We demonstrate the superiority of our approach for uncertainty estimate on a task of Multiple Sclerosis lesions segmentation.

Introduction

Magnetic Resonance Imaging (MRI) is the standard imaging modality for the diagnosis and follow-up of Multiple Sclerosis (MS). It allows a direct observation of brain lesions produced by the disease and provides information about the pathology stage or treatment efficiency. Deep Learning (DL) approaches, based on a trained U-Net-like neural network, are invaluable tools to automatically delineate MS lesions [START_REF] Shoeibi | Applications of deep learning techniques for automated multiple sclerosis detection using magnetic resonance imaging: A review[END_REF]. Although powerful and versatile, these models provide segmentation maps that are typically opaque, with no indication regarding their certainty. This hinders full acceptance of DL models in clinical routine, for which uncertainty attached to the computerized results is essential for their interpretation and to avoid misleading predictions.

A variety of methods have been proposed to quantify the uncertainty attached to deep neural networks [START_REF] Abdar | A review of uncertainty quantification in deep learning: Techniques, applications and challenges[END_REF]. Among them, the Monte Carlo (MC) dropout stands out as one of the simplest approach, as it can be applied to any model trained with the dropout technique [START_REF] Srivastava | Dropout: a simple way to prevent neural networks from overfitting[END_REF]. Such a model can be interpreted as a Bayesian neural network, giving access to the interesting properties of these probabilistic models regarding quantification of their uncertainty [START_REF] Gal | Dropout as a Bayesian approximation: Representing model uncertainty in deep learning[END_REF]. More particularly at inference, for a given input, multiple stochastic forward passes are computed by keeping dropout activated, corresponding to empirical samples from the approximated predictive distribution. This produces a set of softmax probabilities that can further be used to compute uncertainty estimates. Applied to MRI segmentation, the MC dropout method produces uncertainty metrics for each voxel in the volume, resulting in so-called voxel-wise uncertainty maps [START_REF] Sander | Towards increased trustworthiness of deep learning segmentation methods on cardiac MRI[END_REF][START_REF] Jungo | Towards uncertainty-assisted brain tumor segmentation and survival prediction[END_REF][START_REF] Nair | Exploring uncertainty measures in deep networks for multiple sclerosis lesion detection and segmentation[END_REF]. The clinically-relevant information, however, is at a higher level, typically at the instance (lesion, tissue) level.

Natural ways to obtain such instance-wise uncertainties, meaning the uncertainties attached to each connected component within the output segmentation, are through a post hoc aggregation of voxel-wise uncertainty estimations. Existing approaches include computing the mean uncertainty of voxels belonging to the same class in the segmentation [START_REF] Roy | Bayesian QuickNAT: Model uncertainty in deep whole-brain segmentation for structure-wise quality control[END_REF] (thus producing one uncertainty estimate per class, rather than per connected component). In the context of MS, lesionwise uncertainty was also estimated using the logsum of the connected voxels uncertainties [START_REF] Nair | Exploring uncertainty measures in deep networks for multiple sclerosis lesion detection and segmentation[END_REF]. Using the mean implies that each component uncertainty contributes equally to the overall instance score, while the use of the logsum assumes that connected voxels are conditionally independent, given that they belong to the same instance. These highly simplified assumptions may degrade the quality of instance uncertainty computation. To go further, a side-learner called MetaSeg has been proposed to predict the Intersection Over Unions (IoU) of each individual segmented instance with the ground truth [START_REF] Rottmann | Prediction error meta classification in semantic segmentation: Detection via aggregated dispersion measures of softmax probabilities[END_REF]. For this task, a Linear Regression Model is trained based on a series of features derived from a standard segmentation model's output probabilities. The predicted score is then used as a marker of instance uncertainty. Yet, the input features of MetaSeg consist in averaged voxel-wise metrics, leading to the same restrictions than the previouslydescribed post hoc aggregation methods. Additionally, it has been proposed to train an auxiliary Graph (Convolutional) Neural Network (GCNN) using the outputs of a MC dropout U-Net (i.e. voxel-wise segmentation and uncertainty maps) to refine the predicted masks [START_REF] Soberanis-Mukul | Uncertainty-based graph convolutional networks for organ segmentation refinement[END_REF]. This approach, however, remains at the voxel level and focuses on 2D segmentation tasks.

In this work, we propose to build from the two last methods to overcome their respective limitations. Indeed, we implement a GCNN at the output of a trained MC dropout U-Net model. Using the predicted 3D segmentation outputs, each individual segmented lesion is modeled by a graph whose voxels are the interconnected nodes. Node features are determined by the input and output of the U-Net, comprising the voxel image intensities, the voxel predicted label, and voxel-wise uncertainty maps. We implement two alternative variants of the proposed GCNN, either classification or regression, to quantify lesions uncertainty. We test our framework on a task of 3D binary segmentation segmentation on MS data. Results demonstrate the superiority of our approach compared to known methods.

Our Framework: Graph modelization for lesion uncertainty quantification

Overview: Consider an input image X and a trained MC dropout segmentation model N with parameters W that produces a segmentation Y = N (X, W ) and a set of voxel-wise uncertainty maps U i (e.g. entropy, variance, PCS, etc.). Our objective is to quantify the uncertainty of each instance (i.e. lesion) in Y. To do so, we propose to train an auxiliary GCNN to predict this uncertainty directly from X, Y, and U i (see Figure 1).

Monte Carlo dropout model and voxel-wise uncertainty

We use a generic 3D U-Net [START_REF] Çiçek | 3D U-Net: learning dense volumetric segmentation from sparse annotation[END_REF] for its simplicity and popularity within the field, although our method can be employed with any segmentation model trained with dropout. We add 3D dropout [START_REF] Tompson | Efficient object localization using convolutional networks[END_REF] with a rate of p = 0.2 at the end of each encoding and decoding block. The model is trained on annotated datasets composed of pairs of images: (i) input T2-weighted FLAIR MRI sequences X and

(ii) associated ground truth MS lesions segmentation Y . At inference, dropout is kept activated and T forward passes are made for a new input volume x * . We chose T = 20, as it has been shown that it produces the best uncertainty estimates [START_REF] Orlando | U2-net: A bayesian u-net model with epistemic uncertainty feedback for photoreceptor layer segmentation in pathological oct scans[END_REF]. From this set of predictions, several well-known voxel-related uncertainty metrics are extracted: (see Figure 1, part A): the entropy [START_REF] Gal | Deep Bayesian active learning with image data[END_REF], the variance [START_REF] Kendall | Bayesian SegNet: Model Uncertainty in Deep Convolutional Encoder-Decoder Architectures for Scene Understanding[END_REF] and the Predicted Confidence Score (PCS) [START_REF] Zhang | Towards characterizing adversarial defects of deep learning software from the lens of uncertainty[END_REF].

Graph dataset generation

Inference on Validation Dataset and Connected Component Analysis. After training, the MC dropout U-Net is subsequently used to generate segmentation and uncertainty maps on the set-aside validation set of images. These predictions are used to generate training data for the auxiliary GCNN. We use a Connected Component Analysis (CCA) to identify each lesion in the segmentation masks using 26-connectivity -meaning that a lesion is defined by voxels that are interconnected by their faces, edges, or corner. For each lesion identified by CCA, we compute the Adjusted Intersection Over Union (IoU adj ) [START_REF] Rottmann | Prediction error meta classification in semantic segmentation: Detection via aggregated dispersion measures of softmax probabilities[END_REF] with the ground truth lesions (see Figure 1, part B). This variant of the IoU is suited for brain-abnormalities segmentation, where a connected component in the ground truth can be divided into several pieces in the predicted segmentation.

Identified lesions can exhibit a wide range of shape and size. To learn from these data, we must thus design a neural network that can be employed regardless of the shape and size of the input structure. GCNNs, which can be interpreted as a generalization of the classic convolutional networks to non-Euclidean and irregular data, are thus particularly suitable for this task.

From voxels to graphs We first slightly dilate each lesion mask to include surrounding voxels at the border between classes, which typically convey useful information about uncertainty. We then convert the dilated mask into a graph by representing its voxels by nodes and neighborhood relationships by edges. Each node is further defined by a set of n + 4 features: (i) the intensity of its corresponding voxel in each of the n input MRI sequences, (ii) its binarized label (1 for the observed lesion class and 0 for all other classes), and its 3 voxel-wise uncertainty estimates: (iii) entropy, (iv) variance and (v) PCS (see Figure 1, part C). In agreement with the aforementioned 26-connectivity CCA, each node (i.e. voxel) is connected in the graph to its 26 nearest neighbors.

GCNN architecture and training

Here, we use a lightweight GCNN architecture composed of 2 consecutive Graph Convolutional layers with a hidden dimension of h = 64, followed by a Linear layer (see Figure 1, part D). The model is trained using the graph dataset generated from the validation images, composed of graphs (transformed connected components obtained from the segmentation model) along with their associated ground truth (IoU adj ). As in [START_REF] Rottmann | Prediction error meta classification in semantic segmentation: Detection via aggregated dispersion measures of softmax probabilities[END_REF], we propose two versions of our model:

-In the classification approach (GCNN Classif ), the IoU adj labels are first binarized as follows: FP if IoU adj (graph) < ϵ, and TP if IoU adj (graph) ≥ ϵ. ϵ is a hyperparameter that we set to 0.1 in our experiments, so that lesions with an IoU adj very close to 0 are not wrongly considered as TP. The network is then trained using the Cross-Entropy Loss. At inference, structural uncertainty is quantified by the graph FP probability. -In the regression approach (GCNN Reg ), the model is directly trained to predict the graph IoU adj , using the MSE loss. At inference, we use 1-IoU adj as the structural uncertainty score.

3 Material and Method

Data

We combine two open-source MS datasets: from the University Hospital of Ljubljana (MSLUB) [START_REF] Lesjak | A novel public MR image dataset of multiple sclerosis patients with lesion segmentations based on multi-rater consensus[END_REF] and from the MICCAI 2016 MS segmentation challenge (MSSEG 2016) [START_REF] Commowick | Multiple sclerosis lesions segmentation from multiple experts: The MICCAI 2016 challenge dataset[END_REF]. We thus use 83 manually-annotated 3D T2-FLAIR sequences.

Images are resampled to a 1 mm isotropic resolution of 160 × 192 × 160 to focus on brain tissues, and intensities are normalized to zero mean and unit variance.

We opt for a 4-fold cross-validation scheme due to the limited number of images.

In each fold, we put aside 25% of the images for testing. From the remaining images, we use 20% for validation and 80% to train the model. During evaluation, results are averaged over the 4 folds. Due to the limited number of images, we extensively use Data Augmentation to train our models, comprising flipping, rotation, contrast alteration, gaussian noise and blurring.

Comparison with known approaches

To evaluate the relevance of our proposed GCNN Classif and GCNN Reg approaches, we implement in parallel known approaches to obtain instance uncertainty from the U-Net. We use the mean and logsum of the voxel-wise uncertainty of each lesions, with the 3 different types of uncertainty. We name these methods Entropy mean , Variance mean , PCS mean , Entropy logsum , Variance logsum , and PCS logsum . As pointed out in [START_REF] Nair | Exploring uncertainty measures in deep networks for multiple sclerosis lesion detection and segmentation[END_REF], using the logsum assigns a higher uncertainty to smallsize lesions. This appears sub-optimal as small lesions could be segmented with high confidence, especially in the case of MS lesions. To verify this point, we implement a naive approach, named Size, which attributes a lesion uncertainty inversely proportional to its size. The lesion size (number of voxels composing it) being S, its uncertainty is computed as 1/S.

Lastly, we implement an approach inspired from the MetaSeg framework [START_REF] Rottmann | Prediction error meta classification in semantic segmentation: Detection via aggregated dispersion measures of softmax probabilities[END_REF]. We extract a series of features from each connected component in the validation dataset, consisting in the mean entropy, variance and PCS, as well as the size of the lesion. We then train a Logistic Regression classifier from these 4 features to distinguish between True Positive (TP) and FP lesions (MetaSeg Classif ). Alternatively, we train a Linear Regression model to directly predict IoU adj (MetaSeg Reg ). We use the outputs of these models to obtain lesion uncertainty as described for the GCNN approach.

Evaluation Setting

For medical applications, the ideal uncertainty quantification should attribute a higher uncertainty to FP lesions than TP, to allow for proper interpretation and evaluation of the results. To evaluate this properly, we use Accuracy-Confidence curves [START_REF] Lakshminarayanan | Simple and scalable predictive uncertainty estimation using deep ensembles[END_REF]. Briefly, the principle is to set aside the τ % of the most uncertain predicted lesions among the test dataset, and measure the performance of the model on the remaining lesions by counting the number of FP and TP lesions. The threshold τ fluctuates between 0 (all lesions are kept) and 100 (all lesions are removed). By plotting the couples (FP, TP) at different thresholds, we obtain an Accuracy-Confidence curve and compute the AUC (Area Under the Curve) score reflecting the quality of the estimated lesion uncertainty. FP and TP counts are normalized in the range [0, 1] by dividing by the counts obtained without filtering (at τ = 0). This metric only depends on the ranking of uncertainties, thus is independent of the uncertainty ranges of each method ensuring a fair comparison. We additionally evaluate the segmentation performance of the U-Net on the test datasets using the Dice coefficient, as well as the total number of TP and FP lesions. Finally, for each method, we control the correlation between the estimated uncertainty and the lesion size using the Spearman's rank correlation coefficient (ρ).

Implementation Details

3D Segmentation U-Net Our segmentation framework was implemented using PyTorch [START_REF] Paszke | Pytorch: An imperative style, high-performance deep learning library[END_REF]. We opt for a patch approach to train the segmentation U-Net, meaning that the 160 × 192 × 160 MRI volumes are split into 3d patches of 160 × 192 × 32, decreasing the memory cost of training. We use a batch size of 5. The U-Net is trained with a combination of the Dice [START_REF] Milletari | V-net: Fully convolutional neural networks for volumetric medical image segmentation[END_REF] and Cross-Entropy loss, using the ADAM optimizer [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF] with a learning rate of 1e -4 until convergence. For the training of the segmentation models, a single NVIDIA T4 GPU was used.

Graph Neural Networks We use the Deep Graph Library [START_REF] Wang | Deep graph library: A graph-centric, highly-performant package for graph neural networks[END_REF] to implement and train the GCNN models. The training procedure of our GCNN is standard: we use the ADAM optimizer with a learning rate of 1e -2 at the start of training, and progressively decreasing to 1e -5 . Graphs are presented to the network in the form of batches, composed of 10 graphs. Due to the small size of the GCNN models, they were trained on CPU, which took a couple of minutes in our experiments.

Results and Discussion

Accuracy-Confidence curves are presented in Figure 2 along with the corresponding AUC values. Segmentation performance and correlation coefficients are presented in Tables 1 and2. Experimental results show that both models of the proposed framework outperform the classical methods by a significant margin, and that their performances are similar with a very small advantage for the classification version. The naive Size approach achieves the lowest performance. Similarly, the logsum approaches, also strongly correlated with the lesion size, have poorer performance than the mean counterparts. Not surprisingly, in the context of MS lesions, the lesion size is not a satisfying proxy for uncertainty as small lesions can be segmented with high confidence. In our experimental setting, M etaSeg models do not outperform simpler methods. This is probably due to the overall simplicity of these models, failing to fully learn the relationships between the different input features.

Results show that our graph-based framework can be efficiently used to flag uncertain lesions, that are more likely to result in False Positives. The classification variant slightly outperforms regression. We hypothesize that this is due to the increased difficulty of predicting the exact IoU adj , compared to the binary classification setting. One drawback of our approach is that it requires an additional validation set containing enough lesions (typically a few hundreds) to allow GCNN training. However, as most DL pipelines rely on a set-aside validation set to control overfitting during training, these data can then be used for this purpose (as it was the case in this work). The requirement is thus not prohibitive and only necessitates a sufficiently large validation set.

Overall, our framework is computationally light as CCA is computed only once per MRI, followed by the graph generation step that can be parallelized among the lesions. Additionally, in the context of MS, most brain lesions are relatively small (less than 1000 voxels), which results in small graphs that are fast to generate. Finally we use 26-connectivity, meaning that a voxel is only connected to its closest neighbors, which reduces the computational burden.

Our approach enhances the binary voxel-wise predictions of the segmentation model with reliable and readable lesion-wise uncertainty estimates. In the classification setting, uncertainty is casted as the probability of a lesion being a false positive, which is a straightforward and intelligible definition. In a real world clinical application, this may help the clinician examine the automated segmentation in the light of the model's confidence, hence allowing a better interpretability of the results and a more trustable usage of the algorithm.

Future work will study the extension to multi-class segmentation, and inclusion of additional features such as the global location of the lesion within the MRI volume. Indeed, for brain disorders such as MS, the location of the lesion within the brain conveys information concerning uncertainty, as FP tend to be concentrated in specific brain regions.

Conclusion

This paper presents an innovative graph-based framework to quantify lesion-wise uncertainty. We demonstrate, with our approach, improvement of the predicted uncertainty, when compared to various known methods. The strength of our solution is its generic nature, making it compatible with any segmentation model trained with dropout.

Fig. 1 .

 1 Fig. 1. Illustration of the proposed framework for learning lesion uncertainty from the outputs of a Monte Carlo dropout model. See the text for details of each block.

Fig. 2 .

 2 Fig. 2. Accuracy-Confidence curves for the different methods. The associated AUC scores are indicated in brackets in the graphs legends.

Table 1 .

 1 U-Net segmentation performance on the MS dataset and number of TP and FP lesions for each fold.

	Fold	0	1	2	3
	Dice	0.672 0.645 0.705 0.693
	# TP lesions 829 597 715 871
	# FP lesions 525 294 353 454

Table 2 .

 2 Evaluation of uncertainty estimates (AUC values). ρ represents Spearman's rank correlation coefficient ρ.

		AUC (%) Spearman's ρ
	GCNN Classif	87.32	-0.78
	GCNNReg	87.10	-0.77
	Entropy mean	83.80	-0.42
	Entropy logsum	83.72	-0.97
	Variancemean	83.14	-0.44
	Variance logsum 82.99	-0.99
	PCSmean	83.79	-0.44
	PCS logsum	83.88	-0.98
	Size	80.30	-1.00
	MetaSeg Classif	83.10	-0.76
	MetaSeg Reg	83.42	-0.77


J Cotxet

F Gutty

G Baili

L Morvan

D Dolfi

S Guérandel

D Holleville

Laser Intensity and Frequency Stabilization Implemented on a Miniature CPT Clock Breadboard

Keywords: compact clock, noise reduction, cesium, CPT

We developed and implemented a miniature electro-optical bench for the stabilization of a dual-frequency laser beam for high-contrast CPT interrogation of cesium. Preliminary results of optical intensity and wavelength simultaneous stabilizations give respective noise reduction of 15 dB and 60 dB at low frequencies. These performances are in line with targeted clock stability of 5×10 -13 at 1 s.

I. INTRODUCTION

In order to further reduce the volume of atomic clocks while keeping high performances needed for on-board applications, recent studies focused their work on Coherent Population Trapping (CPT) of cesium, showing impressive frequency stability at both short and long time scales [START_REF] Holleville | Toward a High-Stability Coherent Population Trapping Cs Vapor-Cell Atomic Clock Using Autobalanced Ramsey Spectroscopy[END_REF].

We propose to implement a highly compact set-up for a CPT clock based on a miniature electro-optical bench and a disruptive dual-frequency dual-polarization Vertical External Cavity Surface Emitting Laser (DF-VECSEL) that generates directly two eigenmodes at 852 nm separated by 9.192 GHz [START_REF] Gredat | Optimization of laser dynamics for active stabilization of DF-VECSELs dedicated to cesium CPT clocks[END_REF] with perpendicular polarizations needed for Lin per Lin interrogation [START_REF] Zanon | High contrast Ramsey fringes with coherent population trapping pulses in a double lambda atomic system[END_REF]. The targeted relative frequency stability at 1 s of integration time is below 5×10 -13 . Extensive studies of the power stabilization function of the bench has been done for both short and long time scales and interactions with a dualfrequency laser beam were described [START_REF] Cotxet | Progress on a Highly Compact Cesium CPT Clock Based on a Dual-Frequency VECSEL[END_REF]. Here we present and discuss preliminary characterizations of simultaneous optical power stabilization and wavelength locking of the laser within the complete miniature electro-optical bench.

II. EXPERIMENTAL SET-UP

The miniature electro-optical bench, designed and developed with the company Kylia, provides all the functions needed for the laser beam stabilization within a low volume of 10 L (Fig. 1).

First, the laser is protected from back reflections with a single dual-polarization optical isolator. Second, the optical power stabilization is done by adjusting the transmission of an acousto-optic modulator (AOM) as described in [START_REF] Tricot | Power stabilization of a diode laser with an acousto-optic modulator[END_REF]. Third, saturated absorption spectroscopy using a shielded cesium gas cell generates an error signal [START_REF] Mccarron | Modulation transfer spectroscopy in atomic rubidium[END_REF] to lock the wavelength of the laser onto the D2 line. Forth, a fast photodetector converts down the optical beat note at 9.2 GHz between the two polarization modes into the radiofrequency domain for comparison to the local oscillator. Last, a third AOM pulses the laser intensity for Raman-Ramsey interrogation and the beam is extended to match the diameter of the Cs cell. These functions are implemented on a single silica board with free-space miniature optical components. Optical alignments and adjustments have been carefully optimized and fixed with robust bonding technics to guarantee a high mechanical stability.

As a first step and before integrating the dual-frequency laser, preliminary characterizations of the stabilization loops performances are conducted using a monomode distributed feedback (DFB) laser source providing few mW of optical power (Fig. 2).

III. RESULTS

The optical wavelength locking technique consists in applying a correction on the current supply of the laser. The frequency noise is measured through power spectral density of the error signal of the servo loop (Fig. 3). A drastic reduction of 60 dB is observed for low offset frequencies when the loop is closed, down to a noise floor of 40 dBHz²/Hz. For offset frequency above 500 Hz, the noise increases and an excess of noise appears at 30 kHz, the cut-off frequency of the servo loop.

The wavelength locking operates simultaneously with the optical power stabilization. This second function adjusts the optical transmission of an AOM to maintain a photodetected signal to a reference voltage level. In the loop, the relative intensity noise (RIN) of the laser naturally reproduces the reference voltage RIN across the full bandwidth of the servo loop. Fig. 4 presents the RIN measured out of the loop, just before the atomic resonator. When the optical power stabilization loop is closed, a reduction of the RIN of 15 dB is measured at an offset frequency of 100 Hz. The noise then reaches a floor of -156 dB/Hz between 1 kHz and 10 kHz, close to the noise of the reference voltage. For offset frequencies below 50 Hz, stray light and mechanical perturbations prevents the transfer of the reference voltage stability out of the loop. For offset frequencies above 20 kHz, the stabilization loop cut-off generates an excess of noise with a peak at 200 kHz. The simultaneous operation of the wavelength locking generates a 3 dB bump at its cut-off frequency (30 kHz), where the gain of the power stabilization loop lacks gain.

Contributions of the laser intensity noise and frequency noise to the clock frequency Allan deviation are estimated through AM to FM and FM to FM processes. The calculations are reported in TABLE I. When the laser is stabilized, these contributions are 10 times smaller than targeted stability of 5×10 -13 at 1 s. We developed a miniature electro-optical bench for the stabilization of a laser beam for CPT interrogation of Cs atoms. We demonstrated the simultaneous locking of both the optical power and the optical wavelength of a monomode DFB laser with performances comparable to noise reduction obtained in state-of-the-art clock laboratory set-ups.

These promising results paves the way to the implementation of a complete highly compact clock system that will be presented at the conference, using either two phase-locked DFB lasers or a single DF-VECSEL.

Fig. 1 .

 1 Fig. 1. Miniature electro-optical breadboard. The power consumption of each AOM is 1 W. The dimensions are 9.5 × 25 × 43 cm 3 .

Fig. 2 .

 2 Fig. 2. Intensity stabilization and frequency locking set-up. HWP: halfwave plate, BS: beam splitter, PBS: polarization beam splitter.

Fig. 3 .

 3 Fig. 3. Frequency noise of the laser when the wavelength locking is applied.

Fig. 4 .

 4 Fig. 4. Relative intensity noise of the DFB laser when both intensity and wavelength stabilizations are applied.

TABLE I .

 I CALCULATION OF LASER NOISES CONTRIBUTIONS TO ALLAN DEVIATION

Noise Allan deviation at 1 s (×10 13 )

	Intensity noise	0.57
	Frequency noise	0.10
	IV. CONCLUSIONS


J Cotxet

F Gutty

G Baili

L Morvan

D Dolfi

S Guérandel Affiliations

Laser power stabilization of a compact dual-frequency VECSEL for a cesium clock optical bench Authors

Introductions

Frequency standards are essential for numerous applications such as GNSS and telecommunication systems. Atomic clocks present the best frequency stability available on the market. Latest development aimed at demonstrating a high frequency stability clock in a portable prototype for onboard applications. One of the promising solutions for cesium compact clocks is based on the Coherent Population Trapping (CPT) phenomenon. Here atomic interrogation is carried out by two lasers modes with a frequency difference of 9.19 GHz, matching the separation of the hyperfine ground states of 133Cs. Laboratory clocks demonstrated impressive frequency stabilities of few 10 -13 at 1 s [START_REF] Tricot | Analyse et réduction des sources d'instabilité de fréquence dans une horloge CPT compacte[END_REF], averaging down below 10 -14 at 1000 s of integration time [START_REF] Hafiz | Toward a High-Stability Coherent Population Trapping Cs Vapor-Cell Atomic Clock Using Autobalanced Ramsey Spectroscopy[END_REF].

In order to reduce the size and complexity of these systems, we propose to implement a highly compact clock based on a dual-frequency vertical external cavity surface emitting laser (DF-VECSEL) at 852 nm associated with a miniature optical bench.

Discussion

The optically-pumped DF-VECSEL emits directly the two frequencies needed for CPT interrogation on two orthogonal polarizations. As these two eigenstates share the same cavity and pump noises, it has been shown that their relative intensity noises (RIN) will be correlated and in-phase [START_REF] Gredat | Fully-correlated multi-mode pumping for low-noise dual-frequency VECSELs[END_REF]. In order to further reduce the optical power fluctuations of the laser, we implemented and demonstrated a highly compact set-up for intensity stabilization, operating simultaneously on the two polarizations eigenstates, using miniature optical components and robust bonding technics (figure).

A partial reflection of the laser beam provides a photodetected signal is compared to a reference voltage and a feedback loop is applied to the transmission of an acousto-optic modulator to stabilize the optical power [START_REF] Tricot | Power stabilization of a diode laser with an acousto-optic modulator[END_REF]. Detection of the sum of both polarizations leads to the best overall reduction of the RIN of the two correlated eigenmodes. A reduction of the dual-polarization RIN of 25 dB is measured up to an offset frequency of 4 kHz, reaching a noise floor of -152 dB/Hz (figure). The RIN of the two orthogonal polarizations are similarly reduced by 13 dB, down to a noise floor of -140 dB/Hz, compatible with the targeted clock performances. This noise floor is limited by the amplitude of the correlation between the two intensities (close to 0.9 here). There is good agreement between the predictive model and the experimental results. The discrepancy between model and experiment of noise floors for the RIN of the sum of the polarization is attributed to the shot noise and detection noises that are not yet taken into account in the simulations.

Conclusion

We implemented and demonstrated the simultaneous optical power stabilization of two polarization modes emitted by a DF-VECSEL within a highly compact set-up. We also developed a predictive model for the RIN reduction that gives good agreement with experimental results, showing that in-phase highly correlated intensity noises could be similarly reduced with a single feedback loop.

Future work will now be conducted for optical frequency locking of the DF-VECSEL via a saturated absorption experiment and for beat note frequency stabilization onto the microwave atomic resonance frequency.

Figure :

 : Figure: (Left) Compact set-up for optical isolation and power stabilization of the laser. (Center) Predictive RIN reductions for each components when the sum of the polarizations is detected and stabilized. (Right) Experimental results of the RIN reduction.


Femto-St-Ensmm Besançon

Femto-St -Ensmm -Besançon

Pour répondre aux besoins grandissant de références de temps ultrastables, des études récentes ont permis le développement d'horloges atomiques sur cellules de hautes performances, présentant des stabilités relatives de fréquence de quelques 10 -13 à 1 s et descendant jusqu'à quelques 10 -15 à 10 000 s [START_REF] Femto-St -Ensmm -Besançon | Toward a High-Stability Coherent Population Trapping Cs Vapor-Cell Atomic Clock Using Autobalanced Ramsey Spectroscopy[END_REF]. Ces horloges se basent sur le phénomène de piégeage cohérent de population (Coherent Population Trapping ou CPT) qui apparait quand les atomes sont illuminés par deux fréquences optiques dont l'écart correspond à la fréquence de la transition hyperfine (9.2 GHz dans le cas du césium). Afin de rendre la partie physique plus compacte tout en maintenant des performances de stabilité de laboratoire, nous avons développé, en collaboration avec la société française Kylia, un banc optoélectronique miniature (i.e. de volume inférieur à 10 L) tout collé qui permet les asservissements et la mise en forme du laser avant l'interrogation CPT du césium (figure 1).

Les fonctions implémentées sur le banc sont (i) l'isolation optique ; (ii) la stabilisation de l'intensité optique ; (iii) l'asservissement de la fréquence d'émission laser sur une transition optique du césium ; (iv) la détection du battement microonde et (v) la mise en forme des pulses pour l'interrogation Ramsey des atomes du résonateur. La conception du banc a été pensée pour permettre d'atteindre une stabilité de fréquence inférieure à 5×10 -13 à 1 s et pour être compatible avec l'utilisation d'un faisceau laser bifréquence bipolarisation à 852 nm (transition D2 du césium). Les caractérisations préliminaires présentées ci-dessous ont cependant été obtenues avec une source laser DFB commerciale monomode. L'étude complète de la stabilisation optique d'un laser émis par une cavité VECSEL bifréquence ont été présentés précédemment [START_REF] Cotxet | Développement d'un banc miniature de stabilisation de puissance optique pour horloge atomique compacte[END_REF].

Pour l'asservissement de la fréquence optique émise un montage de spectroscopie par absorption saturée dans une cellule de césium blindée permet de générer un signal d'erreur par transfert de modulation [START_REF] Mccarron | Modulation transfer spectroscopy in atomic rubidium[END_REF]. Le signal correction issu d'un contrôleur PI est ensuite appliqué à l'entrée de modulation de l'alimentation en courant du laser. Les résultats de la réduction du bruit de fréquence du laser sont présentés en figure 2.a. Une forte réduction de 50 dB est observée aux basses fréquences de Fourier jusqu'à un plancher de 40 dBHz²/Hz. Au-delà de 500 Hz le bruit remonte et un excès de bruit de 6 dB apparait à 30 kHz avant de revenir au bruit du laser libre.

La stabilisation de l'intensité optique est réalisée en ajustant la transmission d'un modulateur acousto-optique pour maintenir la puissance détectée par une photodiode égale à une tension de référence [START_REF] Tricot | Power stabilization of a diode laser with an acousto-optic modulator[END_REF]. Les résultats de la stabilisation sont présentés en figure 2 Hors de la boucle, juste avant le résonateur atomique, le RIN du laser est dégradé par rapport au signal mesuré dans la boucle. Sous 100 Hz, le passage par l'AOM de mise en forme des pulses (ici en fonctionnement continu) perturbe fortement le RIN de plusieurs dizaines de dB. L'asservissement de la fréquence étant en fonctionnement simultané, une bosse de bruit de 3 dB apparait à la fréquence de coupure de l'électronique (30 kHz) où la stabilisation de puissance manque de gain pour la corriger. Audelà de 200 kHz, le RIN revient au niveau du bruit du laser en fonctionnement libre. Les contributions des bruits laser à la stabilité de fréquence relative de l'horloge sont estimées par le calcul des processus de transfert AM-FM et FM-FM (tableau 1). En considérant une séquence d'interrogation pulsée, ces effets sont particulièrement significatifs sur la plage de fréquence entre 50 Hz et 10 kHz. Quand le laser est asservi, les contributions calculées sont dix fois inférieures au niveau de stabilité d'horloge visé.

Bruits du laser

Ecart-type d'Allan à 1 s (×10 -13 ) Bruit d'intensité 0.57 Bruit de fréquence 0.10 Tableau 1. Estimations des contributions des bruits du laser à la stabilité de fréquence relative de l'horloge.

Cette caractérisation complète du banc miniature valide la conception réalisée et est prometteuse pour la mise en place du système d'horloge complet. En particulier, les travaux sont en cours pour vérifier la qualité du signal CPT issu du résonateur avec deux sources DFB asservies en phase à par injection optique et ainsi obtenir une première mesure de stabilité de fréquence.

En parallèle de cette étude, le développement d'une cavité laser unique émettant directement deux modes de polarisation orthogonaux espacés en fréquence de 9.2 GHz [START_REF] Gredat | Optimization of laser dynamics for active stabilization of DF-VECSELs dedicated to cesium CPT clocks[END_REF] a récemment montré la faisabilité des asservissements simultanés des deux fréquences optiques émises. Une fois finalisés, ces deux axes permettront la mise en place d'une horloge aux dimensions inédites pour ces niveaux de performance.

Références :

  Stabilisations d'intensité et de fréquence laser implémentées sur un banc miniature d'horloge CPT Jérémie Cotxet 1, 2 , François Gutty 1 , Ghaya Baili 1 , David Holleville 2 ,Loïc Morvan 1 , Stéphane Guérandel 2 et Daniel Dolfi 1 1 Thale Research & Technology 1 avenue Augustin Fresnel, 91767 Palaiseau, France 2 LNE-SYRTE, Observatoire de Paris, Université PSL, CNRS-UMR 8630, 61 avenue de l'Observatoire, 75014 Paris, France

  .b. Dans la boucle d'asservissement, le bruit d'intensité relatif (RIN) du laser est fortement réduit jusqu'au niveau de la tension de référence utilisée et atteint un plancher de -158 dB/Hz entre 100 Hz et 10 kHz. Au-delà de la bande passante de l'électronique, un excès de bruit de 4 dB apparait à 200 kHz.

Figure 1 :

 1 Figure 1 : Banc miniature et schéma d'asservissement et de mise en forme du laser pour l'interrogation atomique.

Figure 2 :

 2 Figure 2 : a. Bruit de fréquence du laser quand l'émission est asservie sur une transition optique du césium. b. Bruit d'intensité relatif du laser en fonctionnement libre et quand les asservissements sont en marche.


Femto-St-Ensmm Besançon

Femto-St -Ensmm -Besançon

La stabilisation de la puissance optique est réalisée en ajustant la transmission d'un modulateur acousto-optique pour maintenir le signal détecté par une photodiode égal à une tension de référence à bas bruit [4]. Grâce à l'architecture de pompage du VECSEL, dite double spot, les intensités optiques des deux états propres sont fortement corrélées et en phase. Cette propriété de la cavité permet de réduire simultanément les fluctuations des deux modes avec un unique signal de détection. Deux configurations ont ainsi été étudiées : (i) détection des fluctuations d'une seule polarisation ou (ii) détection des fluctuations des deux polarisations croisées. La Figure 1.a illustre les spectres de RIN de chacune des polarisations et de leur somme dans le cas (ii). Une réduction de RIN de 13 dB identique sur chacune des polarisations est mesurée tandis qu'une une réduction plus importante, de 25 dB, jusqu'au niveau de bruit de la tension de référence, est relevée sur la somme des polarisations. Une modélisation analytique du système en boucle fermée permet de déterminer les limites théoriques d'un tel asservissement et de prédire le fort impact des corrélations et de l'équilibre entre les deux intensités. En particulier, la partialité des corrélations entre les deux intensités (amplitude de 0.9) limite ici la réduction de RIN de chacun des modes laser quand c'est leur somme qui est asservie La stabilisation de la puissance optique ainsi que les asservissements simultanés des fréquences optiques émises par un prototype de DF-VECSEL à 852 nm ont été réalisés. Il a ainsi été montré que la réduction du RIN est limitée par l'amplitude des corrélations entre les bruits d'intensité des deux modes. Le contrôle simultané des deux fréquences optiques grâce à une architecture optique modifiée et un circuit électronique adapté a aussi été validé pour la première fois et permet une réduction simultanée des bruits de fréquence et de phase du laser. L'étude d'un montage d'horloge CPT complet est actuellement limitée par l'offset de la longueur d'onde centrale de la raie D2 du Cs. Les travaux sont en cours pour réduire les pertes générées par l'introduction des différents éléments optiques intra-cavité et ainsi obtenir une émission laser intense et résonante avec le césium.

  Gutty 1 , Ghaya Baili 1 , David Holleville 2 , Stéphane Guérandel 2 , Loïc Morvan 1 , Daniel Dolfi 1 1Thales Research & Technology, 1 Avenue Augustin Fresnel, 91767 Palaiseau Cedex, France 2 LNE-SYRTE, Observatoire de Paris, UMR 8630, 61 Avenue de l'Observatoire, 75014 Paris, France Pour répondre aux besoins grandissant de références de temps ultrastables, des études récentes ont permis le développement d'horloges atomiques sur cellules de hautes performances, présentant des stabilités relatives de fréquence de quelques 10 -13 à 1 s et descendant jusqu'à quelques 10 -15 à 10 000 s [1]. Ces horloges se basent sur le phénomène de piégeage cohérent de population (Coherent Population Trapping ou CPT) qui apparait quand les atomes sont illuminés par deux fréquences optiques dont l'écart correspond à la fréquence de la transition hyperfine (9.2 GHz dans le cas du césium). Afin de simplifier et rendre plus compacte la génération de ce champ bifréquence, nous proposons ici une approche originale basée sur un Dual Frequency Vertical External Cavity Surface Emitting Laser (DF-VECSEL) à 852 nm (raie d'absorption D2 du césium) dont l'architecture permet l'oscillation simultanée de deux polarisations croisées et séparées, de manière contrôlée, de 9.2 GHz [2]. La dynamique de classe A du laser, associée au couplage entre les états propres du VECSEL et aux propriétés du pompage, confèrent aux deux fréquences optiques des corrélations d'intensité. Ces dernières ont fait l'objet d'études approfondies, en particulier portant sur leur optimisation pour réduire le bruit d'intensité relatif (Relative Intensity Noise ou RIN) et le bruit de phase du battement [3]. Dans le présent papier, nous proposons une architecture modifiée du VECSEL bifréquence bipolarisation associée à un circuit électronique permettant, pour la première fois, d'asservir simultanément et indépendamment les deux fréquences optiques. Sur cette même architecture, nous évaluons la stabilisation simultanée des deux intensités optiques sur une référence commune. Enfin, nous discutons les résultats obtenus et leur applicabilité à l'horloge Cs CPT en cours de construction. L'architecture du DF-VECSE est présentée ci-dessous. Il s'agit d'un ½-VCSEL en cavité externe.

Figure 1 :

 1 Figure 1 : Gauche : Architecture du VECSEL bifréquence bipolarisation. ½-VCSEl : milieu à gain, PBS : Polarisation beam Splitter YVO4, EO : Cristal MgOSLT, Etalon : filtre biréfringent YVO4. Droite : Photo du coeur du prototype du VECSEL. L'oscillation simultanée des deux états propres de polarisation est obtenue en diminuant le couplage non linéaire via une séparation spatiale des deux modes sur le ½-VCSEL. Ici, la séparation spatiale est choisie égale à 190µm, combinée avec un pompage optique double-spot. Quand les fluctuations des deux faisceaux de pompe sont corrélées, nous montrons que les fluctuations d'intensité des deux états propres de polarisation sont fortement corrélées (proche de 1) et en phase. Deux cristaux électro-optiques (EO) en MgO:SLT sont insérés tête-bêche dans la cavité laser. En plus de réduire l'effet thermo-optique inhérent aux cristaux utilisés, il est ainsi possible de contrôler la fréquence d'émission de chacun des modes laser indépendamment grâce à une combinaison linéaire de

Figure 1 Figure 2

 12 Figure 1Figure2: a. Spectres de RIN du VESCEL asservi en puissance. b. Spectres de bruit de fréquence du mode de polarisation vertical asservi. c. Spectres de bruit de phase du battement entre les modes laser asservi.

  M. Abdel Hafiz et al., "Toward a High-Stability Coherent Population Trapping Cs Vapor-Cell Atomic Clock Using Autobalanced Ramsey Spectroscopy", Physical Review Applied 9, 064002, 2018. [2] P. Dumont et al., "Low-noise dual-frequency laser for compact Cs atomic clocks", Journal of Lightwave Technology, 32.20 : 3817-3823, 2014. [3] G. Gredat et al., "Optimization of laser dynamics for active stabilization of DF-VECSELs dedicated to cesium CPT clocks", J. Opt. Soc. Am. B 37, 1196-1207, 2020. [4] F. Tricot et al., "Power stabilization of a diode laser with an acousto-optic modulator", Review of Scientific Instruments, 89(12):113112, 2018.


Réductions des bruits d'un prototype de VECSEL bifréquence à 852 nm pour des applications d'horloge atomique CPT

RESUME

Nous proposons de mettre en oeuvre une horloge atomique compacte exploitant une source laser bifréquence innovante à 852 nm. Les asservissements nécessaires de la puissance optique, de la fréquence optique d'un mode ainsi que de la fréquence du battement entre les deux modes ont été réalisés sur un prototype de VECSEL bipolarisation bifréquence.

MOTS-CLEFS :

VECSEL, RIN, bruit de phase, horloges atomique.

INTRODUCTION

Pour répondre aux besoins grandissant de références de temps ultrastables, des études récentes ont permis le développement d'horloges atomiques sur cellules de très hautes performances, présentant des stabilités relatives de fréquence de quelques 10 -13 à 1 s et descendant jusqu'à quelques 10 -15 à 10 000 s [ 1 ]. De telles horloges se basent sur le phénomène de piégeage cohérent de population (Coherent Population Trapping ou CPT) qui apparait quand les atomes sont illuminés par deux fréquences optiques dont l'écart correspond à la fréquence de la transition hyperfine (9.2 GHz dans le cas du césium). Afin de simplifier et compacter la génération de ce champ bifréquence, nous proposons ici une approche originale basée sur un Dual Frequency Vertical External Cavity Surface Emitting Laser (DF-VECSEL) à 852 nm (raie d'absorption D2 du césium) dont l'architecture permet l'oscillation simultanée de deux polarisations croisées et directement séparées de 9.2 GHz.

La dynamique de classe A du laser, associée au couplage entre les états propres du VECSEL et aux propriétés du pompage, confèrent aux deux fréquences optiques des corrélations d'intensité. Ces dernières ont fait l'objet de nombreuses études, en particulier portant sur leur optimisation pour réduire le bruit d'intensité relatif (Relative Intensity Noise ou RIN) et le bruit de phase du battement [ 2 ]. Pour réduire d'avantage les bruits du laser à des niveaux compatibles avec un fonctionnement d'horloge, les asservissements externes de la puissance optique et des fréquences émises ont été mis en place sur un montage prototype.

STABILISATION DE LA PUISSANCE OPTIQUE

La stabilisation de la puissance optique est réalisée en ajustant la transmission d'un modulateur acousto-optique pour maintenir le signal détecté par une photodiode égal à une tension de référence bas bruit [ 3 ]. Grâce à l'architecture de pompage du VECSEL, dite double spot, les intensités optiques des deux états propres sont fortement corrélées et en phase. Cette propriété de la cavité permet de réduire simultanément les fluctuations des deux modes avec un unique signal de détection. Deux configurations ont ainsi été étudiées : (i) détection des fluctuations d'une seule polarisation ou (ii) détection des fluctuations des deux polarisations croisées. La Figure 1-gauche illustre les spectres de RIN de chacune des polarisations et de leur somme dans le cas (ii). Une réduction de RIN de 13 dB identique sur chacune des polarisations est mesurée tandis qu'une une réduction plus importante, de 25 dB, jusqu'au niveau de bruit de la référence de tension, est relevée sur la somme des polarisations.

Une modélisation analytique du système en boucle fermée permet de déterminer les limites théoriques d'un tel asservissement et de prédire le fort impact des corrélations et de l'équilibre entre les deux intensités. En particulier, la partialité des corrélations entre les deux intensités (amplitude de 0.9) limite ici la réduction de RIN de chacun des modes laser quand c'est leur somme qui est asservie.

ASSERVISSEMENT DES FREQUENCES LASER EMISES

Deux cristaux électro-optiques (EO) sont insérés tête-bêche dans la cavité laser. En plus de réduire l'impact de l'effet thermo-optique, il est ainsi possible de contrôler la fréquence d'émission de chacun des modes laser indépendamment grâce à une combinaison linéaire de tension appliquée aux cristaux EO. Pour stabiliser l'émission laser, deux asservissements sont mis en place simultanément. D'une part, la fréquence optique d'un mode de polarisation est asservie sur le flanc d'une frange d'interféromètre dont le discriminateur de fréquence est représentatif de celui d'une transition atomique (~1 V/MHz). D'autre part, la fréquence du second mode est asservie par une boucle à verrouillage de phase de la fréquence du battement entre les modes laser sur un oscillateur local à quelques gigahertz. Les résultats des réductions de bruits associées sont présentés en Figure 1 (au centre et à droite). Une forte réduction des bruits est mesurée pour des fréquences de Fourier inférieures à 100 kHz. Cette limite de correction est donnée par la bande passante des amplificateurs de tensions utilisés pour contrôler les EO.

CONCLUSION

La stabilisation de la puissance optique ainsi que les asservissements simultanés des fréquences des modes laser émis par un prototype de DF-VECSEL à 852 nm ont été réalisés. Il a ainsi été montré que la réduction du RIN est limitée par l'amplitude des corrélations entre les bruits d'intensité des deux modes. Le contrôle de chacune des fréquences de manière indépendante grâce à une paire de cristal EO a aussi été validé et permet une réduction simultanée des bruits de fréquence et de phase du laser.

L'étude d'un montage d'horloge complet est actuellement limitée par les réglages et l'accordabilité du DF-VECSEL, inhérents à la conception mécanique du prototype. Les travaux sont en cours pour réduire les pertes générées par l'introduction des différents éléments optiques intracavité et ainsi obtenir une émission laser intense et résonante avec le césium.

Cotxet 1 , 2 , François Gutty 1 , Ghaya Baili 1 , David Holleville 2 ,Stéphane Guérandel 2 , Loïc Morvan 1 , Daniel Dolfi 1 1

 12112211 Thales Research & Technology, 1 Avenue Augustin Fresnel, 91767 Palaiseau Cedex, France 2 LNE-SYRTE, Observatoire de Paris, UMR 8630, 61 Avenue de l'Observatoire, 75014 Paris, France jeremie.cotxet@thalesgroup.com

Figure 1 :

 1 Figure1: Gauche : Spectres de RIN du VESCEL asservi en puissance. Centre : Spectres de bruit de fréquence du mode de polarisation vertical asservi. Droite : Spectres de bruit de phase du battement entre les modes laser asservi.


Jeremie Cotxet

François Gutty

Ghaya Baili

Loic Morvan

Daniel Dolfi

David Holleville

Stephane Guerandel

An innovative laser bench for a high performance compact cesium CPT clock

Keywords: Cesium CPT clock, intensity noise, frequency noise

We developed and implemented a miniature electro-optical bench for the stabilization of a dual-frequency laser beam for high-contrast CPT interrogation of cesium. Preliminary results of optical intensity and wavelength simultaneous stabilizations give respective noise reduction of 15 dB and 60 dB at low frequencies. These performances are in line with targeted clock stability of 5×10 -13 at 1 s. For longer time scales, a study of the optical power stability shows a drastic reduction of power fluctuations and highlights the temperature sensitivity of the optical components.

INTRODUCTION

High performance and compact atomic clocks are needed for on-board applications such as Galileo next generation systems. Recent studies focused their work on Coherent Population Trapping (CPT) of cesium, showing impressive frequency stability at both short and long time scales [START_REF] Hafiz | Toward a high-stability coherent population trapping Cs vapor-cell atomic clock using autobalanced Ramsey spectroscopy[END_REF] .

We propose to implement a highly compact set-up for a CPT clock based on a miniature electro-optical bench and a disruptive dual-frequency dual-polarization Vertical External Cavity Surface Emitting Laser (DF-VECSEL) that generates directly two eigenmodes at 852 nm separated by 9.192 GHz 2 with perpendicular polarizations needed for Lin per Lin interrogation [START_REF] Zanon | High contrast Ramsey fringes with coherent-population-trapping pulses in a double lambda atomic system[END_REF] . The targeted relative frequency stability at 1 s of integration time is below 5×10 -13 , averaging down below 10 -14 at 1 h. Extensive studies of the power stabilization function of the bench has been done for both short and long time scales and interactions with a dual-frequency laser beam were described [START_REF] Gredat | Optimization of laser dynamics for active stabilization of DF-VECSELs dedicated to cesium CPT clocks[END_REF][START_REF] Cotxet | Progress on a Highly Compact Cesium CPT Clock Based on a Dual-Frequency VECSEL[END_REF] . Here we present and discuss preliminary characterizations of simultaneous optical power stabilization and optical frequency locking of the laser within the complete miniature electro-optical bench.

MINIATURE ELECTRO-OPTICAL BENCH

The miniature electro-optical bench, designed and developed with the French company Kylia, provides all the functions needed for the laser beam stabilization within a low volume. The whole set-up is protected from external perturbations by a rigid box of dimensions 9.5×25×43 for a total volume of just above 10 L (Figure 1).

First, the laser is protected from back reflections with a single dual-polarization optical isolator. Second, the optical power stabilization is done by adjusting the transmission of an acousto-optic modulator (AOM) as described in reference [START_REF] Tricot | Power stabilization of a diode laser with an acousto-optic modulator[END_REF] . Third, saturated absorption spectroscopy using a shielded cesium gas cell generates an error signal [START_REF] Mccarron | Modulation transfer spectroscopy in atomic rubidium[END_REF] to lock the wavelength of the laser onto the D2 line. Forth, a fast photodetector converts down the optical beat note at 9.2 GHz between the two polarization modes into the radiofrequency domain for comparison to the local oscillator. Last, a third AOM pulses the laser intensity for Raman-Ramsey interrogation and the beam is extended to match the diameter of the Cs cell.

These functions are implemented on a single silica board with free-space miniature optical components. Optical alignments and adjustments have been carefully optimized and fixed with robust and space-qualified bonding technics to guarantee a high mechanical stability. *jeremie.cotxet@thalesgroup.com As a first step and before integrating the dual-frequency laser, preliminary characterizations of the stabilization loops performances are conducted using a monomode distributed feedback (DFB) laser source providing few milliwatts of optical power (Figure 2).

REDUCTION OF FREQUENCY AND INTENSITY NOISES

Optical frequency-lock loop

The saturated absorption spectroscopy signal is modulated by the AOM 2 at a frequency 𝑓 𝑚 = 600 kHz. The photodetected signal generated by PD 2 is then sent to a lock-in amplifier that demodulates the error signal used for the locking loop. The optical frequency of the laser is stabilized on one of the saturated peaks of Cs D2 line by applying a correction signal to the laser current supply.

The frequency noise is measured through the power spectral density of the error signal of the servo loop (Figure 3). A drastic reduction of 60 dB is observed for low offset frequencies when the loop is closed, down to a noise floor of 40 dBHz²/Hz. For offset frequency above 500 Hz, the noise increases and an excess of noise appears at 30 kHz, the cutoff frequency of the servo loop. Peaks between 50 Hz and 1 kHz are attributed to the laser current supply and can be filtered.

Figure 3. Frequency noise of the laser when the wavelength locking is applied: in free running regime (green) and in locked regime (red).

Simultaneous locking of optical power and frequency

To stabilize the optical power, a locking loop adjust the RF power driving the AOM 1 to keep the photodetected signal generated by PD 1 at the level of a reference voltage (Vref). The optical power stabilization and frequency locking were operated simultaneously. Figure 4 presents the relative intensity noise (RIN) measured in the loop with PD 1 and out of the loop, at the position of the atomic resonator, with PD 3. Figure 4. Relative intensity noise of the DFB laser when both intensity and wavelength stabilizations are applied: in free running regime (green), in locked regime in the loop (blue) and out of the loop (red). The reference voltage noise is given in dark grey. The frequencylock loop noise contribution is represented with a red arrow.

In the loop, the (RIN) of the laser naturally reproduces the reference voltage RIN across the full bandwidth of the servo loop, down to a noise floor of -158 dB/Hz. For offset frequencies above 20 kHz, the stabilization loop cut-off generates an excess of noise with a peak at 200 kHz. Spurs appear due to electronics perturbations when measuring the signal used for the locking loop. These parasitic peaks only appears when measurement is done in the loop and shall not degrade the optical signal sent through the atomic resonator, as described in the next paragraph.

Out of the loop, when the optical power stabilization loop is closed, a reduction of the RIN of 15 dB is measured at an offset frequency of 100 Hz. The noise then reaches a floor of -156 dB/Hz between 1 kHz and 10 kHz, close to the noise of the reference voltage. The outloop signal is measured after AOM 3, on the diffracted beam of order -1. This leads to a high degradation of the RIN of the laser beam below Fourier frequencies of 100 Hz, preventing the transfer of the reference voltage stability out of the loop. The simultaneous operation of the wavelength locking generates a 3 dB bump at its cutoff frequency (30 kHz), where the gain of the power stabilization loop lacks gain. Few parasitic peaks appears and are due to the current supply of the laser. Filtering these peaks will be part of future works.

Estimation of clock instabilities

Contributions of the laser intensity noise and frequency noise to the clock frequency Allan deviation 𝜎 𝑦 (𝜏) can be estimated through AM to FM (amplitude modulation to frequency modulation) and FM to FM (frequency modulation to frequency modulation) processes. The expression of RIN contribution to clock instability is:

𝜎 𝑦,𝑅𝐼𝑁 (𝜏) = 2 𝜋 Δ𝜈 𝜈 0 𝜎 𝑅𝐼𝑁 𝐶 √ 𝑇 𝑐 𝜏 , ( 1
)
where of the atomic signal, 𝜈 0 is the clock frequency (9.192 GHz for cesium), 𝐶 and Δ𝜈 are respectively the linewidth and the contrast of the atomic signal, 𝑇 𝑐 is the cycle time of the clock and 𝜎 𝑅𝐼𝑁 is the deviation of the RIN averaged over the clock pulsed sequence:

𝜎 𝑅𝐼𝑁 2 = 2 ∫ sinc 2 (𝜋𝜏 𝑚 𝑓)[1 -cos(2𝜋𝑓𝑇 𝐶 ) ∞ 0 RIN(𝑓)𝑑𝑓 , ( 2
)
where 𝜏 𝑚 is the measurement time during the sequence. The atomic interrogation pulsed sequence is especially sensitive to noises at Fourier frequencies between 100 Hz and 10 kHz.In the same way, we can define the deviation of the frequency noise of the laser 𝜎 𝑓 . Knowing the clock frequency sensitivity to optical frequency variations of the laser (4 Hz/GHz), the contribution of this noise is defined as:

𝜎 𝑦,𝑓 (𝜏) = 4 × 10 -9 𝜎 𝑓 𝜈 0 √ 𝑇 𝑐 𝜏 . ( 1
)
The calculations of both contributions are reported in Table 1 for integration time of 1 s.

Table 1. Estimation of the laser noises contributions to Allan deviation of the clock.

Laser noise

Allan deviation at 1 s (×10 13 ) Intensity noise 0.70

Frequency noise 0.13

The intensity noise of the laser contributes to 0.7×10 -13 . For comparison, contribution in free running regime is twice higher, reaching a level of 1.4×10 -13 . The optical frequency noise only contributes to 0.13×10 -13 . When the laser is stabilized, these contributions are one order of magnitude smaller than targeted stability of 5×10 -13 at 1 s.

LONG-TERM OPTICAL POWER STABILITY

Due to light shift effects [START_REF] Arditi | Pressure, light, and temperature shifts in optical detection of 0-0 hyperfine resonance of alkali metals[END_REF] , the power instability tends to be the limiting parameter for long-term CPT clock frequency stability. We measured the optical power of the laser over long periods of time (tens of hours), out of the loop as well as in the loop. The results, presented as relative fluctuations 𝜎 𝑃 /𝑃, are given in Figure 5.

For short integration times (below 30 s), the resolution of the data logger used for the measurement generates a white noise, the quantification noise. This noise is on the order of few 10 -6 and averages down as 𝜏 -1/2 . It limits the measure of low optical power fluctuations.

In the loop, the fluctuations are close to the voltage fluctuations of the reference of few 10 -7 . We find a high correlation between the fluctuations of the electronics temperature and the power in the loop (Figure 6a). These fluctuations however only limit the stability after 1 000 s of integration time and are not detrimental yet for the aimed clock performances. Out of the loop, the optical power fluctuations are greatly reduced in comparison to the free running regime. A reduction of more than two orders of magnitude is measured at 10 000 s. However the level of the inloop fluctuations is not reached. By monitoring the temperature of the optical components of the set-up, we found a high correlation between the temperatures fluctuation of the PBS used for the separation of inloop and outloop paths (Figure 6b). We attribute this effect a. b.

to the temperature sensitivity of the PBS transmission and reflection coefficients. A much lower temperature sensitivity of the photodiode has also been observed.

These levels of optical power stability are just enough to stay below 10 -14 for the clock relative frequency Allan deviation at 10 000 s. Methods to further mitigates light shift effect, such as autobalanced Ramsey spectroscopy, have also been successfully implemented on CPT clocks [START_REF] Hafiz | Toward a high-stability coherent population trapping Cs vapor-cell atomic clock using autobalanced Ramsey spectroscopy[END_REF] and could be used on our system to reach even lower level of clock stability.

CONCLUSION

We developed a miniature electro-optical bench for the stabilization of a laser beam for CPT interrogation of Cs atoms. We demonstrated the simultaneous locking of both the optical power and the optical wavelength of a monomode DFB laser with performances comparable to noise reduction obtained in state-of-the-art clock laboratory set-ups. Optical power stability has also been studied over long periods of time. Sensitivity to temperature fluctuations of optical components were highlighted, especially of the PBS.

These promising results are all in line with the aimed clock frequency stability and paves the way to the implementation of a complete highly compact clock system. Further works have now to be done to implement the DF-VECSEL source for the generation of the dual-frequency electromagnetic field need for the CPT interrogation.

Figure 1 .

 1 Figure 1. Miniature electro-optical breadboard. The power consumption of each AOM is 1 W. The dimensions are 9.5×25×43 cm 3 .

Figure 2 .

 2 Figure 2. Intensity stabilization and frequency locking set-up. AOM: acousto-optic modulator; HWP: half-wave plate; PD: photodiode; BS: beam splitter; PBS: polarization beam splitter.

Figure 5 .

 5 Figure 5. Relative power fluctuations of the laser: in free running regime (green), in locked regime in the loop (blue) and out of the loop (red). The reference voltage fluctuations are given in dark grey.

Figure 6 .

 6 Figure 6. a. Evolution of the optical power fluctuation in the loop (blue) and of the reference voltage electronics temperature (dark grey). b. Evolution of the optical power fluctuation out of the loop (red) and of the temperature of optical components (light green).