Emilie Morvant

Amaury Habrard
email: amaury.habrard@univ-st-etienne.fr

Stéphane Ayache

PAC-Bayesian Majority Vote for Late Classifier Fusion

Keywords: Machine Learning, Multimedia fusion, Multi-modality search, Ranking and re-ranking

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Introduction

Combining multimodal information is an important issue in Multimedia and a lot of research effort has been dedicated to this problem (see [START_REF] Atrey | Multimodal fusion for multimedia analysis: a survey[END_REF] for a survey). Indeed, the fusion of multimodal inputs can bring complementary information, from various sources, useful for improving the quality of any multimedia analysis method such as for semantic concept detection, audio-visual event detection, object tracking, etc.

The different modalities correspond generally to a relevant set of features that can be grouped into different views. For example, classical visual or textual features commonly used in multimedia are based on TF-IDF, bag of words, texture, color, SIFT, spatio-temporal descriptors, etc. Once these features have been extracted, another step consists in using machine learning methods in order to build classifiers able to discriminate a given concept.

Two main schemes are generally considered [START_REF] Snoek | Early versus late fusion in semantic video analysis[END_REF]. In the early fusion approach, all the available data/features are merged into one feature vector before the learning and classification steps. This can be seen as a unimodal classification. However, this kind of approach has to deal with heterogeneous data or features which are sometimes difficult to combine. The late fusion model works at the decision level by combining the prediction scores available for each modality. This is usually called multimodal classification or classifier fusion. Late fusion may not always outperform unimodal classification. Especially when one modality provides significantly better results than others or when one has to deal with imbalanced input features. However, late fusion scheme tends to give better results for learning semantic concepts in case of multimodal video [START_REF] Snoek | Early versus late fusion in semantic video analysis[END_REF]. Several methods based on a fixed decision rule have been proposed for combining classifiers such as max, min, product, sum, etc [START_REF] Kittler | On combining classifiers[END_REF]. Other approaches, often referred to as stacking [START_REF] Wolpert | Stacked generalization[END_REF], need of an extra learning step.

In this paper, we address the problem of late multimodal fusion at the decision level with stacking. Let h i be the classifier that gives the score associated with the i th modality for any instance x. A classical method consists in looking for a weighted linear combination of the different scores,

H(x) = n i=1 q i h i (x),
where q i represents the weight associated with h i . It is usually required that 0 ≤ q i ≤ 1 and n i=1 q i = 1. This linear weighting scheme can be seen as a majority vote. This approach is widely used because of its robustness, simplicity and scalability due to small computational costs [START_REF] Atrey | Multimodal fusion for multimedia analysis: a survey[END_REF]. It is also more appropriate when there exist dependencies between the views [START_REF] Wu | Optimal multimodal fusion for multimedia data analysis[END_REF]. An important issue is then to find an optimal way to combine the scores. One solution is to use machine learning methods to assess the weights [START_REF] Atrey | Multimodal fusion for multimedia analysis: a survey[END_REF]. From a machine learning standpoint considering a set of classifiers with a high diversity is generally a desirable property [START_REF] Dietterich | Ensemble methods in machine learning[END_REF]. One illustration is given by the algorithm AdaBoost [START_REF] Freund | Experiments with a new boosting algorithm[END_REF], frequently used as a multimodal fusion method. AdaBoost weights the classifiers according to different distributions of the training data, introducing some diversity, but requires at least weak classifiers to perform well. Another recent approach based on the portfolio theory [START_REF] Wang | Portfolio theory of multimedia fusion[END_REF] proposes a fusion procedure trying to minimize some risks over the different modalities and a correlation measure. While it is well-founded, it needs to define some appropriate functions and is not completely fully adapted to the classifier fusion problem since it does not directly take into account the diversity between the outputs of the classifiers.

We propose to study a new machine learning method, namely MinCq, introduced in [START_REF] Laviolette | From PAC-Bayes bounds to quadratic programs for majority votes[END_REF]. It proposes a quadratic program for learning a weighted majority vote over real-valued functions called voters (such as score functions of classifiers). The algorithm is based on the minimization of a generalization bound that takes into account both the risk of committing an error and the diversity of the voters, offering strong theoretical guarantees on the learned majority vote. In this article, our aim is to show the interest of this algorithm for classifier fusion. We provide evidence that MinCq is able to find good linear weightings but also very performing non-linear combination with an extra kernel layer over the scores. Since in multimedia retrieval, the performance measure is related to the rank of positive examples, we propose to extend MinCq to improve the Mean Average Precision. We base this extension on an additional order-preserving loss for verifying ranking pairwise constraints.

The paper is organized as follows. Section 2 deals with the theoretical framework of MinCq. We extend MinCq as a late fusion method in Section 3. Before concluding in Section 5, we evaluate empirically the MinCq late fusion in Section 4.

PAC-Bayesian MinCQ

In this section we present the algorithm MinCq of Laviolette et al. [START_REF] Laviolette | From PAC-Bayes bounds to quadratic programs for majority votes[END_REF] for learning a Q-weighted majority vote of real-valued functions (e.g. classifier scores). This method is based on the PAC-Bayes theory [START_REF] Mcallester | PAC-bayesian model averaging[END_REF]. We first recall the setting of MinCq.

We consider binary classification tasks over a feature space

X ⊆ R d of dimension d. The label space is Y = {-1, 1}. The training sample is S = {(x i , y i )} m
i=1 where each example (x i , y i ) is drawn i.i.d. from a fixed -but unknown -probability distribution D defined over X × Y . We consider a space of real-valued voters H, such that ∀h i ∈ H, h i : X → R. Given a voter h i , the predicted label of x ∈ X is given by sign[h i (x)], where sign[a] = 1 if a ≥ 0 and -1 otherwise. Then, the learner aims at choosing a distribution Q over H -the weights q i -leading to the Q-weighted majority vote B Q with the lowest risk. B Q is defined by,

B Q (x) = sign [H Q (x)] , with H Q (x) = |H| i=1 q i h i (x).
The associated true risk R D (B Q ) is defined as the probability that the majority vote misclassifies an example drawn according to D,

R D (B Q ) = P (x,y)∼D (B Q (x) = y) .
In the case of MinCq, H has to be a finite auto-complemented family of 2n real-valued voters

H = {h 1 , . . . , h 2n } such that, ∀x ∈ X, ∀i ∈ {1, . . . , n}, h i+n (x) = -h i (x).
(1)

Moreover, the algorithm considers quasi-uniform distributions Q over H, i.e. the sum of the weight of a voter and its opposite is

1 n , ∀i ∈ {1, . . . , n}, Q(h i ) + Q(h i+n ) = q i + q i+1 = 1 n . ( 2
)
This constraint is not too restrictive since every distribution over H can be represented by a quasi-uniform distribution [START_REF] Laviolette | From PAC-Bayes bounds to quadratic programs for majority votes[END_REF]. The assumptions (1) and ( 2) are actually an elegant trick to avoid the use of a prior distribution over H which is often required by usual PAC-Bayesian method [START_REF] Mcallester | PAC-bayesian model averaging[END_REF], making the algorithm more easily applicable. We now present the principle of the algorithm MinCq. The core of MinCq is the minimization of the empirical version of a bound -the C-Bound -over the risk of the Q-weighted majority vote.

Theorem 1 (C-Bound [START_REF] Laviolette | From PAC-Bayes bounds to quadratic programs for majority votes[END_REF]) Given H = {h 1 , . . . , h 2n } a class of 2n functions, for any weights

{q i } 2n i=1 , i.e. distribution Q, on H and any distribution D over X ×Y , if E (x,y)∼D H Q (x) > 0 then R D (B Q ) ≤ C D Q
where,

C D Q = Var (x,y)∼D (yH Q (x)) E (x,y)∼D (yH Q (x)) 2 = 1 - (M D Q ) 2 M D Q 2 , with M D Q = E (x,y)∼D 2n i=1 yq i h i (x), and M D Q 2 = E (x,y)∼D 2n i=1 2n i ′ =1 q i q i ′ h i (x)h i ′ (x)
are respectively the first and the second moments of the Q-margin: yH Q (x).

Following some generalization bounds, MinCq proposes to minimize the empirical version of the

C-bound, C S Q = 1 - (M S Q ) M S Q 2
, over a sample S. The idea is to fix the empirical first moment M S Q to a margin µ > 0 and to minimize the empirical second moment M S Q 2 measuring the correlation of the voters. This leads to minimize the bound and thus the risk of the majority vote by taking into account the diversity between the voters.

Definition 1 (MinCq algorithm [START_REF] Laviolette | From PAC-Bayes bounds to quadratic programs for majority votes[END_REF]) Given a set H = {h 1 , . . . , h 2n } of voters, a training set S = {(x j , y j )} m j=1 , and a margin µ > 0, among all quasi-uniform distributions Q of empirical margin M S Q exactly equal to µ, the MinCq algorithm consists in finding one that minimizes the empirical M S Q 2 . Due to the auto-complemented (1) and quasi-uniformity (2) assumptions, the algorithm can be expressed as a quadratic program (M inCq) by only considering the first n voters h i ∈ H.

argmin Q Q t S M S Q -A t S Q, s.t. m t S Q = µ 2 + 1 2nm m j=1 n i=1 y j h i (x j ),
and ∀i ∈ {1, . . . , n}, 0

≤ q i ≤ 1 n , (M inCq)
where t denotes the transposed function, Q = (q 1 , . . . , q n ) t is the vector of the first n weights q i , M S is the n × n matrix formed by 1 m m j=1 h i (x j )h i ′ (x j ) for i and i ′ in {1, . . . , n}, and,

m S = 1 m m j=1 y j h 1 (x j ), . . . , 1 m m j=1 y j h n (x j ) t , A S = 1 nm n i=1 m j=1 h 1 (x j )h i (x j ), . . . , 1 nm n i=1 m j=1 h n (x j )h i (x j ) t .
Finally, the Q-weighted majority vote learned by MinCq is then

B Q (x) = sign[H Q (x)],
with

H Q (x) = n i=1 2q i - 1 n h i (x).
3 MinCq as a Late Fusion Method PAC-Bayesian MinCq has been proposed in the particular context of binary classification where the objective is to minimize the misclassification rate of the Q-weighted majority vote by taking into account the diversity of the voters. From a multimedia indexing standpoint, MinCq thus appears to be a natural way for late classifiers fusion to combine the predictions of classifiers separately trained from different modalities.

Concretely, given a training sample of size 2m we split it randomly into two subsets S ′ and S = {(x j , y j )} m j=1 of the same size. Let n be the number of modalities. For each modality i, we train a classifier

h i from S ′ . Let Technical Report V 1.0 H = {h 1 , . . . , h n , -h i , .
. . , -h n } be the set of the n associated prediction functions and their opposites. At this step, the fusion is achieved by MinCq: We learn from S the Q-weighted majority vote over H with the lowest risk. However, in many applications, such as multimedia document retrieval, people are interested in performance measures related to precision or recall. Since a low-error vote is not necessarily a good ranker, we propose an adaptation of MinCq to improve the popular Mean Averaged Precision (MAP). We first recall the definition of the MAP measured on S for a given real-valued function h. Let S + = {(x j , y j ) : (x j , y j ) ∈ S ∧ y j = 1} = {(x j + , 1)} m + j + =1 be the set of the m + positive examples from S and S -= {(x j , y j ) : (x j , y j ) ∈ S ∧ y j = -1} = {(x j -, -1)} m - j -=1 the set of the m -negative examples from S (m + + m -= m). For evaluating the MAP, one ranks the examples in descending order of the scores. The MAP of h is,

M AP S (h) = 1 |m + | j:yj =1 P rec@j,
where P rec@j is the percentage of positive examples in the top j. The intuition behind this definition is that we prefer positive examples with a score higher than negative ones. To achieve this goal, we propose to learn with pairwise preference Fürnkranz and Hüllermeier (eds) (2010) on pairs of positive-negative instances. Indeed, pairwise methods are known to be a good compromise between accuracy and more complex performance measure like MAP. Especially, the notion of order-preserving pairwise loss was introduced in Zhang ( 2004) in the context of multiclass classification. Following this idea, Yue et al. [START_REF] Yue | A support vector method for optimizing average precision[END_REF] have proposed a SVM-based method with a hinge-loss relaxation of a MAP-loss. In our specific case of MinCq for multimedia fusion, we design an orderpreserving pairwise loss for correctly ranking the positive examples. Actually, for each pair (x j + , x j -) ∈ S + ×S -, we want:

H Q (x j + ) > H Q (x j -) ⇔ H Q (x j -) -H Q (x j + ) < 0.
This can be forced by minimizing (according to the weights q i ) the following hinge-loss relaxation of the previous equation,

1 m + m - m + j + =1 m - j -=1 H Q (x j -) -H Q (x j + ) + , (3)
where [a] + = max(a, 0) is the hinge-loss. In the setting of MinCq, with H auto-complemented (Eq.( 1)) and Q quasi-uniform (Eq.( 2)), we reduce the term (3) to,

1 m + m - m + j + =1 m - j -=1 n i=1 2q i - 1 n h i (x j -) -h i (x j + ) + . ( 4
)
To deal with the hinge-loss of (4), we consider m + ×m -additional slack variables ξ S + ×S -= (ξ j + j -) 1≤j + ≤m + ,1≤j -≤m - weighted by a parameter β > 0. We make a little abuse of notation to highlight the difference with (M inCq): Since ξ S + ×S -appear only in the linear term, we simply add (4) after the (M inCq) formulation. We obtain the quadratic program (M inCq P W ),

argmin Q,ξ S + ×S -Q t S M S Q -A t S Q + β Id t ξ S + ×S -, s.t. m t S Q = µ 2 + 1 2nm m j=1 n i=1 y j h i (x j ), ∀j + ∈ {1, . . . , m + }, ∀j -∈ {1, . . . , m -}, ξ j + j -≥ 0, ξ j + j -≥ 1 m + m - n i=1 2q i - 1 n h i (x j -) -h i (x j + ) ,
and ∀i ∈ {1, . . . , n}, 0

≤ q i ≤ 1 n , (M inCq P W )
where Id is the unit vector of size m + × m -. However, one drawback of this method is the incorporation of a quadratic number of additive variables (m + × m -) which makes the problem harder to solve. To overcome this problem, we propose to relax the constraints by considering the average score of the negative examples: We force the positive examples to be higher than the average negative scores. This leads us to the following alternative

argmin Q,ξ S + Q t S M S Q -A t S Q + β Id t ξ S + , s.t. m t S Q = µ 2 + 1 2nm m j=1 n i=1 y j h i (x j ), ∀j + ∈ {1, . . . , m + }, ξ j + ≥ 0, ξ j + ≥ 1 m + m - m - j -=1 n i=1 2q i - 1 n h i (x j -) -h i (x j + ) ,
and ∀i ∈ {1, . . . , n}, 0

≤ q i ≤ 1 n , (M inCq P W av )
where Id is the unit vector of size m + . Note that the two approaches still respect the framework of the original MinCq. We simply regularize the search of the weights for a Q-weighted majority vote leading to an higher MAP.

Finally, for tuning the hyperparameters (µ, β) we use a cross-validation process (CV). Instead of selecting the parameters leading to the lowest risk, we select the ones leading to the best MAP.

Experiments

In this section, we show empirically the interest of MinCq, and our extension, as a late fusion method with stacking (implemented with MOSEK solver). We experiment the MinCq-based approaches on the PascalVOC'07 benchmark [START_REF] Everingham | The PASCAL visual object classes challenge 2007[END_REF], where the goal is a list of 20 visual concepts to identify in images. The corpus is constituted of 5000 training and 5000 test images. In general, the ratio between positive and negative examples is less than 10%. For each concept, we generate a training sample constituted of all the training positive examples and negative examples independently drawn such that the positive ratio is 1/3. We keep the original test set.

Our objective is not to provide the best results on this benchmark but rather to evaluate if the MinCq-based methods could be helpful for the late fusion step in multimedia indexing. To do so, we split the training sample into two subsets, S ′ and S, of the same size. We consider 9 different visual features: 1 SIFT, 1 LBP, 1 Percepts, 2 HOG, 2 Local Color Histograms and 2 Color Moments. Then, we train from S ′ a SVM-classifier for each visual feature (with the LibSVM library [START_REF] Chang | LIBSVM: a library for support vector machines[END_REF] and a rbf kernel with parameters tuned by CV). The final classifier fusion is learned from S.

In a first series of experiments, the set of voters H is constituted by the 9 SVM-classifiers (MinCq also considers the opposites). We compare the 3 linear MinCq methods (M inCq), (M inCq P W ), (M inCq P W av ) to the following 4 baseline fusion approaches. • The one with the highest margin:

best(x) = argmax hi∈H |h i (x)|.

• The sum of the classifiers (unweighted vote):

Σ(x) = hi∈H h i (x).
• The MAP-weighted vote:

Σ MAP (x) = hi∈H M AP S (h i ) h i ′ ∈H M AP S (h i ′ ) h i (x).
In a second series, we propose to introduce non-linear information with a rbf kernel layer. We represent each example by the vector of its scores of the 9 SVM-classifiers, H being the set of kernels over the sample S: Each x ∈ S is seen as a voter k(•, x). We then compare our method to stacking with SVM tuned by CV (SVM rbf ). Note that we do not report the results of (M inCq P W ) in this context, because the computational cost is much higher and the performance is lower. The full pairwise version implies too many variables which may penalize the resolution of (M inCq P W ).

In either case, the hyperparameters of MinCq-based methods are tuned with a grid search by a 5-folds CV. The MAP-performances are reported on Table 1, we can make the following remarks.

• On the right, for the first experiments, we clearly see that the linear MinCq-based algorithms outperform on average the linear baselines. At least one MinCq-based method produces the highest MAP, except for "boat" for which h best is the best. We note that the order-preserving hinge-loss is not really helpful: The classical (M inCq) shows the best MAP. In fact, this can be explained by the limited number of voters.

• On the left, with a kernel layer, at least one MinCq-based method achieves the highest MAP and for 17/20 both are better than SVM. Moreover, M inCq rbf P W av with the averaged pairwise preference is the best for 17 concepts, showing the order-preserving loss is a good compromise between improving the MAP and keeping a reasonable computational cost.

• Globally, kernel-based MinCq methods outperform the other methods. Moreover, at least one MinCq-based approach is the best for each concept showing PAC-Bayesian MinCq is a good alternative for late classifiers fusion.

Conclusion

We propose in this paper to make use of a well-founded learning quadratic program called MinCq as a novel multimedia late fusion method. PAC-Bayesian MinCq was originally developed for binary classification and aims at minimizing the error rate of the weighted majority vote by considering the diversity of the voters [START_REF] Laviolette | From PAC-Bayes bounds to quadratic programs for majority votes[END_REF]. In the context of multimedia indexing, we claim that MinCq thus appears naturally appropriate for late classifier fusion in order to combine the predictions of classifiers trained from different modalities. Our experiments show that MinCq is a very competitive alternative for classifier fusion. Moreover, the incorporation of average order-preserving constraints is sometimes able to improve the MAP-performance measure. Beyond these results, such PAC-Bayesian methods open the door to define other theoretically well-founded frameworks to design new algorithms in many multimedia tasks such as multi-modality indexing, multi-label classification, ranking, etc.

Table 1 :

 1 MAP obtained on the PascalVOC'07 test sample. On the left, experiments with rbf kernel layer. On the right, without.

	concept	MinCq rbf P W av	MinCq rbf	SVM rbf	MinCqP W av MinCqP W	MinCq	Σ	ΣMAP	best	h best
	aeroplane	0 .513	0 .513	0.497	0.487	0.486	0.526	0.460	0.241	0.287	0.382
	bicycle	0.273	0.219	0.232	0.195	0.204	0 .221	0.077	0.086	0.051	0.121
	bird	0.2659	0.264	0.196	0.169	0.137	0 .204	0.110	0.093	0.113	0.123
	boat	0.267	0.242	0.240	0.1593	0.154	0.159	0.206	0.132	0.079	0 .258
	bottle	0 .103	0.099	0.042	0.112	0.126	0.118	0.023	0.025	0.017	0.066
	bus	0.261	0.261	0.212	0 .167	0.166	0.168	0.161	0.098	0.089	0.116
	car	0.530	0.530	0.399	0 .521	0.465	0.495	0.227	0.161	0.208	0.214
	cat	0.253	0.245	0.160	0 .230	0.219	0.220	0.074	0.075	0.065	0.116
	chair	0.397	0.397	0.312	0 .257	0.193	0.230	0.242	0.129	0.178	0.227
	cow	0.158	0.177	0.117	0.102	0.101	0 .118	0.078	0.068	0.06	0.101
	diningtable	0.263	0.227	0.245	0.118	0.131	0.149	0 .153	0.091	0.093	0.124
	dog	0.261	0.179	0.152	0 .260	0.259	0.253	0.004	0.064	0.028	0.126
	horse	0.495	0.4504	0.437	0.3011	0.259	0.303	0 .364	0.195	0.141	0.221
	motorbike	0.295	0.284	0.207	0.1412	0.113	0.162	0 .193	0.115	0.076	0.130
	person	0.630	0.614	0.237	0 .624	0.617	0.604	0.001	0.053	0.037	0.246
	pottedplant	0.102	0.116	0.065	0 .067	0.061	0.061	0.057	0.04	0.046	0.073
	sheep	0.184	0.175	0.144	0.0666	0 .096	0.0695	0.128	0.062	0.064	0.083
	sofa	0.246	0.211	0.162	0.204	0 .208	0.201	0.137	0.087	0.108	0.147
	train	0.399	0.385	0.397	0.331	0.332	0 .335	0.314	0.164	0.197	0.248
	tvmonitor	0 .272	0.257	0.230	0.281	0.281	0.256	0.015	0.102	0.069	0.171
	Average	0.301	0.292	0.234	0.240	0.231	0 .243	0.151	0.104	0.100	0.165

problem (M inCq P W av ) with only m + additional variables.

  E. Morvant, A. Habrard, S. Ayache PAC-Bayesian Majority Vote for Late Classifier Fusion • The best classifier of H: h best = argmax hi∈H M AP S (h i ).

* This work was supported in part by the french project VideoSense ANR-09-CORD-026 of the ANR in part by the IST Programme of the European Community, under the PASCAL2 Network of Excellence, IST-2007-216886. This publication only reflects the authors' views.


S Azaizia

K Saleh

O Llopis

A Rissons

Evaluation of low cost solutions for the transmission through optical fiber of low phase noise OCXO signals

Various fiber optics links are evaluated for the transmission of OCXO signals on relatively short distances (1 m to 1 km). The case of VCSEL lasers is compared to the case of the more classical DFB laser. The DFB is still the best device, particularly in terms of signal to noise ratio, but single-mode VCSELs are becoming interesting as a low cost alternative to this device. I.

INTRODUCTION

Optical fiber is today an efficient technique for clock distribution, and numerous systems already use this technique for medium or short distances transmissions (ex : antennas networks), or even on long distance applications (ex : clock comparison). Most of these systems are based on costly laser modules, either amplitude modulated DFB lasers, which includes an optical isolator and temperature control, or high spectral purity lasers (coherent transmission). Vertical Cavity Surface Emitting Lasers (VCSELs) are low cost lasers, which are today available at long wavelengths, 1300 nm and 1500 nm, and which may be used for communications. It is thus interesting to compare their performances for this application, and particularly for an operation on short or medium distances (1 m to 1 km).

Another interest is for the power splitting required for a clock distribution on various receivers (ex : 10 MHz reference signal distribution towards a set of 10 or more HF synthesizers). If this power split is realized in the optical domain, using a medium or high power laser and an optical power splitter, the noise floor in the RF domain increases rapidly because of the optical losses. Indeed, the quadratic detection at the photodiode level results in a 2 dB RF losses for each 1 dB optical losses (and, as an example, an optical splitting by 10 results in, at least, 20 dB RF losses). If the power splitting is performed in the RF domain, then the resulting signals can be used to modulate a set of VCSEL for the optical transmission. The relatively low power and low cost VCSEL laser is well fitted to this solution, in which a lower RF power is used to modulate the laser current.

Among these studies, the case of the transmission of a low phase noise 10 MHz signal is particularly interesting. This signal is generally provided in a system by an oven controlled crystal oscillator (OCXO). The transmission of high spectral purity signals using fiber optics links has been described in many papers [START_REF] Logan | High stability microwave fiber optic systems[END_REF][START_REF] Bibey | Very low phase noise optical links : experiments and theory[END_REF][START_REF] Quadri | A Low Phase Noise Optical Link for Reference Oscillator Signal Distribution[END_REF][START_REF] Santarelli | High performance frequency dissemination for metrology applications with optical fibers[END_REF][START_REF] Holman | Precise frequency transfer through fiber network using 1.5-μm mode-locked sources[END_REF], and particularly the classical approach in which the reference signal directly modulates the current of a medium power DFB laser. The phase noise floor, for this type of link, is mainly related to the laser relative intensity noise (RIN) and the laser modulation index m, at least for a point to point link (m = P P Δ , with P being the average optical power and ΔP the maximum deviation from P due to the modulation; m ≤ 1). In case of a division on a large number of receivers, the noise floor is increased due to the optical losses and it may degrade even further if the shot noise limit is reached. Moreover, on long fiber links (typically above 1 km), some optical nonlinear effects may limit the phase noise performance [START_REF] Okusaga | Optical scattering induced noise in RF-photonic systems[END_REF] at high optical power. For these reasons, and because of the cost argument, the replacement of the DFB laser by a VCSEL device, which operates at lower power, must be investigated.

II. DEVICES UNDER TEST AND MAIN EXPERIMENT

We have studied four types of optical links at 10 MHz using four different lasers :

1) A telecommunications grade laser module (Mitsubishi), based on a 20 mW DFB laser emitting at 1550 nm, which includes an optical isolator, a temperature control and a photodiode.

2) A single mode VCSEL laser (Raycan) emitting about 0.5 mW at a wavelength of 1310 nm.

3) A single mode VCSEL laser (Raycan) emitting about 0.5 mW at a wavelength of 1550 nm.

4) A multimode VCSEL laser (Ulm Photonics) emitting about 10 mW at a wavelength of 850 nm.

The DFB laser module is ready to use : it includes an RF input which feeds an internal bias Tee. A dedicated test fixture has been designed for the three other devices, which includes a 10 MHz bias Tee.

On the receiver side, an InGaAs photodiode (Thorlabs FGA04) is used to recover the RF signal. The photodiode delivers the RF current to a 50 Ω load, through a bias Tee (Mini-Circuits). The same photodiode is used in the three experiments at large wavelength (1310 to 1550 nm), and is replaced by a silicon photodiode for the 850 nm case (Thorlabs FDS02). The RF signal is measured at the photodiode output directly, or is amplified prior measurement if its level is close to (or below) -20 dBm.

Figure 1 represents the schematics of the main experiment. A low phase noise 10 MHz OCXO (Wenzel Premium SC-50104609) is used to modulate the laser. The signal goes through a fiber spool and is detected by a photodiode and then, reaches the phase noise measurement system (Agilent E5052B cross-correlation phase noise analyser). As it is not a residual phase noise measurement set-up, the phase noise floor is imposed by the OCXO phase noise (which is much lower than the optical links noise at carrier offsets above 100 Hz). This noise floor is depicted in Table 1, with the typical values from Wenzel and the measured values on our oscillator.

For the single mode lasers (1310 nm and 1550 nm), three different length of single mode fiber are used in this experiment : 1 m (no spool), 100 m, 1 km. In the case of the multimode laser (850 nm), a 300 m multimode spool is used in place of the 100 m and 1 km spools.

III. DFB LASER LINKS

The DFB laser is biased at 80 mA and delivers an optical CW power of 17.8 mW. The modulation index is set at 0.52, in order to preserve the laser for long term operation. The RIN performance at 10 MHz has been measured for this laser at -162 dB/Hz. Under those conditions, the SSB phase noise floor is given by the noise to signal ratio divided two. The equation below [START_REF]Fiber-Optic Communication Systems[END_REF] allows to compute this phase noise floor.

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + + = 2 2 2 2 10 4 2 10 log 10 ) ( PD PD RIN I m R T k I m q m f L (1)
with L(f) being the RF phase noise in dBc/Hz, RIN the laser relative intensity noise performance in dB/Hz at the frequency of interest (10 MHz), m the RF modulation index, I PD the mean current at the photodiode output, T the absolute temperature, R the photodiode load, q = 1.6 10 -19 C and k = 1.38 10 -23 m 2 kg s -1 . In equation ( 1), the main contributor is the first term in case of a point to point link. The computed phase noise floor using ( 1) is very close to the one computed using only this term, and is about -155 dBc/Hz.

Concerning the 1/f phase noise, there is no simple analytical formula. This phase noise contribution comes from various nonlinear conversion phenomena, and particularly from the conversion of the laser 1/f RIN. The computation of the conversion coefficient would be beyond the scope of this paper, and this noise will be only qualitatively discussed. The case of the 100 m and 1 km links is a bit different. If the close to carrier phase noise has not changed compared to the 1 m case (OCXO noise below 10 Hz and 1/f noise between 10 Hz and 100 Hz), the phase noise floor above 1 kHz offset is much higher than the theoretically computed phase noise floor (about 10 dB higher). Actually, it is not exactly a noise floor, because in the 100 m case the phase noise drops again above 20 kHz offset. This increase of the noise far from the carrier cannot be explained by a change in the signal to noise ratio. Indeed, the losses in 100 m or even 1 km of SMF28 fiber are almost negligible (0.2 dB/km), even if the connector losses are included in the computation.

Therefore, a specific phenomenon occurs in long fibers. Two hypothesis may arise to explain this noise contribution. The first one is a conversion of the laser AM or, more probably, of its FM noise due to some specific phenomenon inside the long fiber (ex : frequency dispersion). The second one is the direct generation of a noise component inside the fiber.

The FM and AM noise of this laser have been measured, and a close to 1/f shape has been found on the 1 Hz to 100 kHz frequency range (see Figure 5 for AM noise). Thus, such a noise contribution should be converted with a similar spectral shape, meanwhile a large flat is observed on the phase noise measurement. Therefore, the second hypothesis has been examined, and we have measured the noise due to the Rayleigh scattering in the fiber using a measurement approach described in details in [START_REF] Okusaga | Optical scattering induced noise in RF-photonic systems[END_REF]. The signal backscattered from the fiber is separated from the input signal using a circulator and is then mixed on a photodiode with the input signal. Figure 3 shows the result of this experiment performed with a 1 km fiber spool. It is clear that the same spectral shape is observed than in the phase noise measurement. Thus, this increased phase noise floor may be due to the Rayleigh scattering inside the fiber, and can be limited by reducing the optical power in the fiber (but in this case, a compromise has to be found with the signal to noise ratio).

IV. SINGLE MODE VCSELS LASER LINKS

The link is then realized using a VCSEL (Raycan) in place of the DFB laser module. The laser is biased between 7 mA and 10 mA, and delivers an optical power between 0.5 mW and 1 mW (maximum level, close to saturation). An RF attenuator is added between the OCXO and the laser, in order to adjust the laser modulation index, which is set between 0.55 and 0.7 (RF power in the range of 0 dBm to 3 dBm). An optical isolator is also added at the laser output (1550 nm laser). In the case of the 1310 nm laser, and because no isolator was available in the laboratory at this wavelength, the isolator has been replaced by a 3 dB optical attenuator.

The RF output power of such a link can be easily computed from the modulation index m, the received optical power P opt , the photodiode sensitivity S PD and the photodiode RF load R using equation ( 2) :

2 2 2 2 1 opt PD RFout P m S R P = (2)
with m = 0.7, P opt = 1 mW, R = 50 Ω and S PD = 0.8 A/W, the maximum RF power at the output of this optical link is found to be -21 dBm. This level is of course much lower than the one obtained for the DFB link, in which the output RF power on the photodiode load is about +2 dBm, under the conditions described in paragraph III. Because the phase noise measurement system requires a signal higher than -20 dBm at its input, an amplifier has been added between the photodiode and the analyser. The first measurements were performed with a commercial amplifier, and finally a specific amplifier has been designed for these links. This amplifier uses a low 1/f noise silicon bipolar transistor BFR92, and has been optimized for 10 MHz operation using Agilent ADS software. Particularly, an unbypassed emitter resistance R e of 8 Ω has been used, both to stabilize the device and to reduce the amplifier phase noise. Finally, the amplifier has been realized and has demonstrated a gain of 19 dB at 10 MHz, together with a low residual phase noise, and particularly a noise floor of about -165 dBc/Hz. Figure 4 depicts the phase noise measured at the output of the optical link which uses the 1310 nm single mode VCSEL device. A noise floor in the range of -135 dBc/Hz is observed. Between 3 Hz and 200 Hz, the noise is clearly of 1/f type and, very close to the carrier (1 Hz), the OCXO phase noise is found again. The phase noise floor is close to what could be expected using equation [START_REF] Logan | High stability microwave fiber optic systems[END_REF] for this laser, although for the 1 km spool case, the noise is increased of about 4 dB compared to the two other cases. The high 1/f noise contribution is not too much surprising, because these lasers are small devices (with high current density) and thus feature a relatively high level of 1/f AM (and probably FM) noise. As an example, the VCSEL low frequency RIN is depicted in Figure 5, and compared to the same data measured on the DFB laser. The phase noise measured with the 1550 nm single mode VCSEL is depicted in Figure 6. This time, a relatively high phase noise floor can be observed, of about -120 dBc/Hz, which is much higher than the noise floor which can be computed from the 10 MHz RIN performance of this laser (about -140 dB/Hz). Up to now, we have no clear explanation of this behavior and further experiments will be performed with this device. The multimode 850 nm VCSEL (Ulm Photonics) delivers a much higher optical power than the single mode ones. In our experiment, this device was biased at 35 mA, delivering about 4 mW optical power, and the RF modulation index m was fixed at 0.7.

The laser has been directly connected to a silicon photodiode (Thorlabs FDS02), using a SMA -FC/PC cable. In another configuration, a 300 m multimode fiber spool is placed between the laser and the photodiode.

The measured phase noise at the photodiode output is depicted in Figure 7. Although the phase noise shape looks very similar to the one measured with the single mode VCSELs devices, the value of the phase noise floor has been found to be strongly dependent on the bias conditions (it has been optimised in this case down to -130 dBc/Hz). The RIN spectrum shape (not shown here) is also rapidly changing with the bias conditions, probably because of the multimode operation. Therefore, even if the link phase noise performance looks fairly good, we believe that this device is not well fitted for high quality time and frequency applications. Finally, the measured main phase noise data are depicted in Table 2. At 1 Hz offset, all the lasers are able to transfer the OCXO performance. At 10 Hz offset, only the DFB laser can keep the OCXO performance, and a degradation of about 10 dB is observed with all the VCSELs devices. At 100 kHz, which corresponds to the noise floor, the best performance is reached with the DFB laser (-153 dBc/Hz), but this performance is not maintained on long links. A relatively good performance has also been measured with the single mode 1310 nm VCSEL at this frequency offset (-135 dBc/Hz).

VI. CONCLUSION

The investigations described in this paper demonstrate that the medium power DFB laser is still the best device for time&frequency applications. However, the single mode VCSEL, today available at telecommunications wavelengths, is becoming interesting. It is still limited in signal to noise ratio, but it is reliable enough for high integration time (close to carrier performance) and could be used in clock distribution as an alternative to optical power splitting. Finally, the multimode VCSEL looks interesting at first glance, but the noise parameters are too much sensitive to the bias conditions to recommend this device for high quality analog applications.

Figure 1 .

 1 Photodiode

Figure 2 .

 2 Figure 2. Phase noise at the output of the 10 MHz optical links realized with the DFB laser module (and comparison to the OCXO phase noise) Figure 2 depicts the measurements performed on the optical links realized with the DFB laser. The measurement noise floor (OCXO) is also depicted on the same graph. The measurement performed with 1 m fiber shows above 10 kHz offset a white noise floor at about -153 dBc/Hz, which is close to the theoretical noise floor computed using equation (1).Between 10 Hz and a few hundred Hz, a 1/f phase noise dominates this noise floor, and between 1 Hz and 10 Hz, the measured phase noise at the output is the one of the OCXO.

Figure 3 .

 3 Figure 3. Measurement of the noise backscattered in the fiber (same method as in[START_REF]Fiber-Optic Communication Systems[END_REF]) in the case of the 1 km spool and the DFB laser. Each spectrum corresponds to a different input inside the

Figure 4 .

 4 Figure 4. Phase noise at the output of the 10 MHz optical links realized with the 1310 nm single mode VCSEL

Figure 5 .

 5 Figure 5. RIN performance measured at low frequency for the 1550 nm DFB laser and the 1310 nm VCSEL laser

Figure 6 .

 6 Figure 6. Phase noise at the output of the 10 MHz optical links realized with the 1550 nm single mode VCSEL

Figure 7 .

 7 Figure 7. Phase noise at the output of the 10 MHz optical links realized with the 850 nm multimode VCSEL

Table 1 :

 1 OCXO phase noise (measurement noise floor)

Table 2 :

 2 Phase noise data measured at 10 Hz and 100 kHz offset

ACKNOWLEDGMENT

This work has been performed in the frame of VERTIGE contract, sponsored by DIRECCTE Midi-Pyrénées, and in relation with Microsystèmes company (Toulouse).


Abdellah Ogbi
email: abdellah.ogbi@ec-lyon.fr

Laurent Nicolas

Ronan Perrussel

Damien Voyer

Calculation of DEP force on spherical particle in non-uniform electric fields

I. INTRODUCTION

When a dielectric particle is placed in a medium of different electrical properties and is subjected to a spatially nonuniform electric field, a force is exerted on the particle. This force called the dielectrophoretic (DEP) force, may be used to manipulate and characterize many types of particles, such as biological cells under special conditions [START_REF] Pethig | Review Article Dielectrophoresis: Status of the theory, technology, and applications[END_REF].

Here we calculate the DEP force on a homogeneous spherical particle in non-uniform fields, which are generated by two practical microsystems beyond the most commonly used to perform DEP experiments. The point-plate electrode structure (Fig. 2) enables to generate highly non-uniform field suitable for particle manipulations such as feedbackcontrolled DEP levitation [START_REF] Kaler | Dielectrophoretic spectra of single cells determined by feedback-controlled levitation[END_REF]. The four electrode chamber (Fig. 1) is another microsystem configuration used in dielectrophoretic experiments such as cell sorting and trapping [START_REF] Pethig | Review Article Dielectrophoresis: Status of the theory, technology, and applications[END_REF]. First we introduce the force calculation approaches and describe the electrode systems, where the particle of interest is placed. Then the results for the force obtained from the dipole approximation are presented and compared with the corresponding results obtained from the rigorous Maxwell stress tensor method both for verification and for determination of higher order force terms contribution to the total dielectrophoretic force.

II. DIELECTROPHORETIC FORCE CALCULATION

In the particle electromechanics theory, the simplest and most direct approach to calculate the force and the torque is a two-step process known as the effective dipole moment method, whereby first one calculates the dipole moment induced in the particle by the electric field and then the force on the particle follows from standard formulas of classical electrodynamics [START_REF] Jones | Electromechanics of Particles[END_REF]. Consider a homogeneous spherical particle with relative permittivity " 2 and conductivity 2 immersed in a dielectric fluid with relative permittivity " 1 and conductivity 1 . According to the effective dipole moment method, the DEP force expression is given by, F = (p (1) • r)E, where p (1) = 4⇡" 1 K (1) E, [START_REF] Jones | Electromechanics of Particles[END_REF] where p (1) is the effective dipole moment, E the imposed electric field and K (1) = (" 2 " 1 )/(" 2 + 2" 1 ) the complex frequency-dependent Clausius-Mossotti factor. The complex permittivity of different materials in the system is given by " i = " 0 " i j i 2⇡f where j 2 = 1, " 0 the vacuum permittivity and f the electric field frequency.

Equation [START_REF] Jones | Electromechanics of Particles[END_REF] for the force on a dipole is not exact for any situation of particle in a non-uniform electric field. This approximation breaks down in some situations described by Jones [START_REF] Jones | Electromechanics of Particles[END_REF]. The particle shape effect was partly treated in [START_REF] Ogbi | Numerical Identification of Effective Multipole Moments of Polarizable Particles[END_REF].

Here electric field non-uniformity effect is examined. The multipole moment expansion enables to extend the dipole moment approach, introducing higher-order terms [START_REF] Jones | Electromechanics of Particles[END_REF]:

F = N X n F (n) = p (1) • rE + p (2) • rrE + . . . (2)
where p (1) , p (2) are the first multipole moments [START_REF] Jones | Electromechanics of Particles[END_REF]. A practical method for the multipole moments factors estimation for any particle shape has been introduced in [START_REF] Ogbi | Numerical Identification of Effective Multipole Moments of Polarizable Particles[END_REF].

From (2) a force expression may be derived for spherical particles using linear multipoles, given by [START_REF] Jones | Electromechanics of Particles[END_REF]

F z = N X n 2⇡" 1 R 2n+1 n!(n 1)! Re(K (n) ) @ @z ✓ @ n 1 E z @z n 1 ◆ 2 (3)
where

K (n) = (" 2 " 1 )/(n" 1 + (n + 1)" 2 )
is the generalized Clausius-Mossotti factor. Another method for calculating the DEP force used by Sauer and Schlögl and studied in [START_REF] Rivette | A note on the electrostatic force and torque acting on an isolated body in an electric field[END_REF] is based on integration the Maxwell stress tensor over the surface of the particle.

A. Particle in a slightly non-uniform electric field

A variety of four-electrode configurations and correction factors for the field strength are available in the literature, and it is well established that the most homogeneous fields are generated by the electrodes of polynomial geometry where the polynomials defining the electric potential are derived from Laplace's equation analysis [START_REF] Pethig | Review Article Dielectrophoresis: Status of the theory, technology, and applications[END_REF]. A spherical particle with relative permittivity " 2 = 2.55F/m and conductivity 2 = 10 2 S/m, suspended in a fluid medium with relative permittivity " 1 = 77.8F/m and conductivity 1 = 10 3 S/m, is placed on the vertical symmetry axis of the four electrode chamber with the particle center at heights H = 80µm and H = 160µm above the electrode plane. The distance distance between opposing electrode tips is d = 400µm. The 3d electric field distribution is first calculated solving the Laplace equation in the system (without the cell) using the finite element analysis software COMSOL Multiphysics. The force is then estimated according to the dipole moment and Maxwell stress methods and the results are shown in Fig. 3.

B. Particle in a highly non-uniform electric field

The same particle and surrounding fluid medium properties as in subsection II.A. are considered. The particle is subjected to a highly non-uniform field generated by a pointplate electrode, with a distance d between the point and the plane electrode of 100µm, as shown in Fig. 2. The particle is positioned near the point (region of high field gradients), with its center at h = 80µm above the plane electrode. To calculate the electric field distribution to which the cell will be exposed in this chamber, we first solved the Laplace equation (with the suspending medium but without the cell) using the finite element method. Introducing a polar coordinates (r, , z) and taking advantage of the electrode structure symmetry, only a 2d calculation in the (r, z) plane is needed. The spherical particle of radius R = 15µm is positioned on the central axis.

III. NUMERICAL RESULTS

The force experienced by the particle placed in the slightly non-uniform electric field of the four-electrode chamber is calculated using the dipole approximation and the Maxwell stress tensor integrated over the surface of the particle and the results are shown in Fig. 3. As suggested by the plots, the effective dipole method produces a result globally consistent with the Maxwell stress tensor approach. When the particle is closer to the electrodes (H = 80µm), the dipole moment method slightly overestimates the force at low frequencies with respect to the corresponding values for the Maxwell stress tensor analysis, while the approximation appear to be more accurate when the particle is far from the electrodes (H = 160µm).

The force experienced by the particle placed in the highly non-uniform electric field generated by the point-plate electrode system is calculated using the dipole approximation, the higher-order (quadrupole, octopole,...) moments and the Maxwell stress tensor. The results are shown in Fig. 4. It

F ( 1) +F (2) F ( 1) +F ( 2) +F (3) F(MT) f c Fig. 4. Force calculation results using the multipole moment and Maxwell stress tensor methods.

may be seen from the data that, the force value in the more accurate Maxwell stress tensor spectra are shifted with respect to the corresponding values for the dipole analysis. It may be seen from the data that the higher-order term contribution corrects the dipole approximation witch underestimates the total force in particular at low and high frequencies. However, the crossover frequency f c value matches well in different calculations. According to the dipole mo-

ment model f c = 1 2 ⇣ ( 1
2 )( 2 +2 1 ) ("2 "1)("2+2"1) ⌘ 1/2 . This frequency point at which the DEP force changes sign is of particular significance for the dielectric parameters identification from DEP spectra [START_REF] Huang | Membrane changes associated with the temperature-sensitive P85gag-mos-dependent transformation of rat kidney cells as determined by dielectrophoresis and electrorotation[END_REF].

IV. CONCLUSION

We have calculated the dielectrophoretic force acting on a spherical particle in slightly and highly non-uniform electric fields generated by four polynomial electrodes and pointplate electrodes, using the dipole moment method, higherorder moments (quadrupole, octopole,..), comparing the results to the accurate Maxwell stress tensor analysis, showing that higher-order terms enables to correct the force estimation related to the electric field non-uniformity effect.

Fig. 1 .

 1 Fig. 1. A spherical particle placed in microsystem of polynomial electrodes.

Fig. 2 .

 2 Fig.2. Point-plate electrode system and a particle.

Fig. 3 .

 3 Fig. 3. Force calculation results using the effective dipole and the Maxwell stress tensor methods, for H = 80µm (left) and H = 160µm (right).