Baris Aksoy

Vincent Labatut

Murat Egi

Petar Denoble

Costantino Balestra

Richard Vann

Alessandro Marroni

ASSOCIATION RULES OF DCI PATIENT CLUSTERS AND RELIABILITY OF CLUSTERING ANALYSIS

Introduction

Decompression illness (DCI) is an adverse outcome of decompression and has a wide spectrum of signs and symptoms. The treatment plan often depends on the classification of DCI, which makes the correct classification of DCI crucial; however, there is no consensus on the classification of DCI. [START_REF] Benton | Dive Medicine[END_REF][START_REF] Buch | Cigarette Smoking and Decompression Illness Severity: A Retrospective Study in Recreational Divers[END_REF][START_REF] Vann | DAN's Annual Review on Decompression Illness, Diving Fatalities, and Project Dive Exploration[END_REF][START_REF] Golding | Decompression sickness during construction of the Dartford Tunnel[END_REF]. We have previously attempted DCI clustering with statistical methods [START_REF] Özyigit | Empirical Classification of DCS patients using Cluster Analysis[END_REF][START_REF] Ozyigit | Classification of DCI by Cluster Analysis[END_REF] and suggested that data mining techniques can be used as a decision support tool to determine the type of DCI.

Our recent study was based on a classification of DCI using multivariate statistics to assess naturally associated clusters of signs and symptoms based on 1929 cases reported by hyperbaric chambers to the Divers Alert Network from 1999 to 2003 [START_REF] Lloyd | Least Squares Quantization in PCM[END_REF]. The aim of this study is to validate the reliability of the previous work by applying three different alternative clustering methods, by comparing the results of two-step clustering analysis with the Perceived Severity Index (PSI) [START_REF] Golding | Decompression sickness during construction of the Dartford Tunnel[END_REF] and to validate the characteristics of patient clusters using association rules. Additionally, we will present the most interesting association rules detected by the A Priori algorithm on the same data.

Methods

Below are the details of the algorithms we used as an alternative to our previous work:.

K-Means Algorithm

The k-means method is a widely used geometric clustering algorithm based on the article proposed by Lloyd in 1982 [START_REF] Lloyd | Least Squares Quantization in PCM[END_REF]. Given a set of n instances, the algorithm uses a local search approach to partition the instances into k clusters. A set of k initial cluster centers is chosen arbitrarily. Each instance is then assigned to the cluster whose center is the closest, and the centers are recomputed as centers of mass of their assigned points. This is repeated until the clusters stabilize. It can be shown that no partition occurs twice during the course of the algorithm, and so the algorithm is guaranteed to terminate. The k-means method is still very popular today, and it has been applied in a wide variety of areas ranging from computational biology to computer graphics [START_REF] Chiu | A Robust and Scalable Clustering Algorithm for Mixed Type of Attributes in Large Database Environment[END_REF]. One of the shortcomings of the k-means algorithm is the necessity to specify the number of clusters.

COBWEB Algorithm

Unlike the k-means algorithm which iterates over the whole dataset, the COBWEB Algorithm [START_REF] Douglas | Knowledge acquisition via incremental conceptual clustering[END_REF] incrementally incorporates instances into a classification tree. The incorporation of an instance is a process of classifying the object by descending the tree along an appropriate path, updating counts along the way, and performing one of several operators at each level. These operators include: classifying the instance with respect to an existing class, creating a new class, combining two classes into a single class (merging), and dividing a class into several classes (splitting). There are two parameters: acuity (minimum value to avoid infinite values) and cutoff (a parameter to suppress growth to avoid overwhelmingly large hierarchy) [START_REF] Witten | Data Mining: Practical machine learning tools and techniques with Java implementations[END_REF].

Expectation-maximization Algorithm

The Expectation-Maximization (EM) algorithm [11] is a statistical approach that makes use of the finite Gaussian mixtures model. A mixture is a set of N Gaussian probability distributions, where each distribution represents a cluster. Additionally, a latent random variable models the probability for an instance to belong to a cluster. The algorithm has to estimate the parameters defining each cluster (i.e. mean and standard-deviation) and the latent variable distribution. The estimation process is iterative, with an expectation step consisting in using the current estimate of the parameters to process the expected value for the likelihood function, and a maximization step consisting in processing new estimates for the parameters by maximizing the previously processed likelihood. The algorithm terminates when cluster quality, i.e. expected likelihood, no longer shows significant improvement. The number of clusters can be either set by the user or estimated by the algorithm.

A Priori Algorithm

Agrawal et al. [START_REF] Lloyd | Least Squares Quantization in PCM[END_REF] introduced association rules by presenting an algorithm which generates significant logical rules involving items in a large database. An item is defined by a field and its value, for instance X=x. An association rule takes the form "if X=x then Y=y" where X=x and Y=y are items or item sets. Even for a small data set, many different association rules can be derived, so interest is restricted to those applying to a reasonably large number of instances, and having a reasonably high accuracy on the instances they apply to. The support, or coverage, of an association rule is the proportion of instances containing all the items used in the rule. Its confidence, or accuracy, is the ratio of the rule support to the support of its conditions (items located in the left part) alone. In other terms, the confidence corresponds to the proportion of instances for which the rule is correct, relatively to the total number of instances it applies to [START_REF] Witten | Data Mining: Practical machine learning tools and techniques with Java implementations[END_REF]. The association rule mining task is usually decomposed in two subtasks. One is to find item sets whose occurrences exceed a predefined threshold (minimum support) in the database, called frequent item sets. The second task is to generate association rules from those large item sets with a constraint of minimal confidence. The process starts with small frequent item sets, and then iteratively enlarge them until no frequent item set can be defined [START_REF] Kotsiantis | Association Rules Mining: A Recent Overview[END_REF].

Data and Tools

DCI data were retrieved from the Diving Injury Database maintained by DAN America and based on reports provided by participating hyperbaric facilities using a standardized Diving Incident Report Form (DIRF). We used 1929 DIRFs (1368 males -561 females) which were collected between 1999 and 2003. The average age of the patients is 37.94 with a range of 13 to 73. Twenty-five different signs and symptoms were listed: altered consciousness, confusion, dyspnea/chockes, cardiovascular signs (CV), pain, skin rash and mottling, localized skin swelling, abnormal sensations, hearing loss, vision problems, discoordination, muscular weakness, muscular problems, decreased skin sensitivity, bladder-bowel problems, headache, fatigue, nausea, dizziness, vertigo, paresthesia, tinnitus, numbness, paralysis and malaise. After a first exploratory analysis, an additional pain-only field was added, to improve performance for some algorithms.

The obtained clusters differ by their characteristic signs and symptoms. Our criterion was to look for clusters where each cluster have characteristic signs and symptoms that are distinct from other clusters. We favored those with features relevant with medical references. As a sample design, we used k-fold cross validation with 10 folds, which is the number of folds supposed to give best results in general [START_REF] Kohavi | A Study of Cross-Validation and Bootstrap for Accuracy Estimation and Model Selection[END_REF]. We used Weka 3.5.7 [START_REF] Witten | Data Mining: Practical machine learning tools and techniques with Java implementations[END_REF], which is an open source software issued under the GNU GPL, as a data mining tool. We also used SPSS 16.0.1 for calculating the associations among the classifications and data transformation.

Results

K-Means

We applied the K-means algorithm for 2 to 6 clusters, and found out 4 clusters was the best parameter value according to our criteria. The estimated clusters contain 708, 637, 300 and 204 patients, respectively. Table 1 shows the signs and symptoms distribution over the clusters. Numbness, paresthesia and decreased skin sensitivity are the most characteristic signs and symptoms for Cluster K1. Cluster K2 is characterized by fatigue, headache, skin, confusion, nausea, dyspnea/chokes, altered consciousness, vertigo, localized skin swelling, hearing loss and vision problems. Cluster K3 is the pain only cluster. Malaise, muscular weakness, paralysis and bladder bowel problems are the characteristics for Cluster K4.

Expectation-maximization

There is no need to specify the number of clusters for EM. The algorithm yielded four clusters containing 905, 253, 471 and 300 patients, respectively. We have set the maximum number of iterations to 100 but the iterations already repeated until convergence which did not exceed the maximum number of iterations. Table 1 shows the detailed results. Cluster E1 has numbness, paresthesia and decreased skin sensitivity as characteristic signs and symptoms. Weakness, muscular weakness, paralysis and bladder bowel problems are the characteristic signs and symptoms of Cluster E2. Cluster E3 is characterized by altered consciousness, confusion, dyspnea/chokes, localized skin swelling, skin rash and mottling, vertigo and hearing loss while Cluster E4 is the pain only cluster.

COBWEB

We applied the COBWEB algorithm with acuity and cutoff parameters ranging from 0.1 to 1 and from 0.2 to 0.35, respectively. The best results according to our criteria were obtained for an acuity of 1 and a cutoff of 0.33. Three clusters were estimated, with 264, 721 and 944 patients respectively. Table 1 shows the characteristic signs and symptoms for each cluster.

Weakness, paralysis, muscular weakness and bladder bowel problems are the characteristic signs and symptoms for Cluster C1. Numbness and paresthesia are the signs and symptoms for Cluster C2. While pain is a significant sign for Cluster C3, it also has other signs and symptoms: confusion, altered consciousness, dyspnea/chokes, skin rash and mottling, localized skin swelling, hearing loss, headache, fatigue, vertigo. Compared to the other classification methods results, we did not find a pain-only cluster, which may be due to the incremental nature of the algorithm.

Analysis

We analyzed our results by carrying out both whole classification and cluster-to-cluster comparisons. First, we used Goodman-Kruskal's lambda, which measures the association level between nominal random variables, to compare classifications from the tested algorithms and PSI diagnosis, as shown in Table 2. The association between the classifications from the different algorithms is generally very strong, varying between 0.343 and 0.827. The association level is higher between the algorithms (including Two-step) than between the algorithms and PSI (manual classification). This means firstly that all automatic classification methods led to very similar results, and secondly that the resulting clusters are close to those defined by medical experts. COBWEB is the most different, because it only found three clusters. Two-step clustering is the most similar to PSI classification. We performed a cluster-to-cluster comparison to check if these similarities were also present at this level. Table 3 shows the Pearson product-moment correlations between clusters from EM (E), k-means (K), COBWEB (C), Two-step (T), and PSI (P) classifications. Just by observing the characteristic signs and symptoms in Table 1, one may infer that correspondences exist between some clusters. This is confirmed by the strong correlations shown in Table 3, leading us to define four typical clusters. Typical cluster α correspond to clusters K1, E1, C2, T2 and P3 which are highly correlated (>0.9687), and all characterized by numbness and paresthesia. Typical cluster β corresponds to clusters K4, E2, C1, T3 and P1 which are highly correlated (>0.8808) and present muscular weakness, bladder bowel problems, paralysis and weakness. Typical cluster γ corresponds to clusters K2, E3 and T4, highly correlated (0.8120) and sharing altered consciousness, confusion, dyspnea/chokes, skin rash and mottling, localized skin swelling,, hearing loss and vertigo. Typical cluster δ is the pain only cluster, corresponds to clusters K3, E4, T1 and P4 (>0.9913). Cluster C3 is correlated to all K2,E3,T4 (>0,7441) and K3,E4,T1,P4 (0.9518), meaning it is likely a merge of typical clusters γ and δ. Cluster P2 is correlated with K2 (0.6230) and T4 (0.6065), and corresponds certainly to typical cluster γ, although it is not highly correlated with E3 (0.4928). Note that some PSI clusters are also non-trivially correlated to other clusters (E1:P1=0.6991, K1:P1=0.7387, E3:P4=0.7458), so the correspondence between PSI and the clustering methods is not straightforward.

Table 3. Correlations between clusters measured with Pearson's correlation

A priori algorithm

We used the A priori algorithm to find out association rules linking symptoms and signs to the estimated cluster, in order to have a better understanding of the characteristic signs and symptoms. We applied the algorithm with minimum support and minimum confidence ranging from 0.1 to 0.9 and from 0.4 to 0.9, respectively. Our goal was to get simple rules, appropriate for human interpretation, and not necessarily the most accurate ones. 4. Some of the association rules mined with the A priori algorithm Table 4 shows the main rules mined for typical cluster α, for each algorithm where signs FALSE (does not exist). Interestingly, rather than the presence of cluster α most prominent features, A priori used the absence of the other clusters' features to discriminate the instances. Numbness is required for K-means and Two-step, but decreased skin sensitivity and paresthesia are not considered. Instead, instances must not present malaise (a cluster β feature) or hearing loss, altered consciousness, dyspnea/chokes, skin rash and mottling, localized skin swelling (γ cluster)., The rules mined for the other clusters are also in accordance with our interpretation of the clustering results: they focus on the characteristic symptoms and signs we found to be characteristic, either by requiring their presence or by requiring their absence in the case of those characteristic of another cluster (details documented elsewhere [START_REF] Aksoy | [END_REF])

Signs and symptoms

Discussion

In this work, we applied three clustering algorithms to the DCI classification problem. Each one is based on different approaches: K-means is distance-based, Cobweb is categorical and EM is probabilistic. We compared the clusters estimated by these three algorithms, those manually defined by the PSI medical experts [START_REF] Vann | DAN's Annual Review on Decompression Illness, Diving Fatalities, and Project Dive Exploration[END_REF] and those previously estimated by Ozyigit et al. by Two-step clustering [START_REF] Ozyigit | Classification of DCI by Cluster Analysis[END_REF]. We used Goodman-Kruskal's lambda to measure the association level between the different approaches, revealing all classifications shared similarities. We consequently performed a detailed cluster-to-cluster comparison with Pearson's correlation coefficient. It appears some typical clusters characterized in terms of signs and symptoms appear whatever the considered classification method is. Typical cluster α is characterized by numbness and paresthesia. Typical cluster β is characterized by muscular weakness, bladder bowel problems, paralysis and malaise. Typical cluster γ is characterized by altered consciousness, confusion, dyspnea/chokes, skin rash and mottling, localized skin swelling, hearing loss and vertigo. Typical cluster δ is the pain only cluster, except for COBWEB, where this cluster was merged with typical cluster γ. To confirm these interpretations, we performed an association rule mining with the A priori algorithm, and focused only on rules with cluster as a conclusion. The results are consistent with our analysis, except the rules focus more on the absence of the signs and symptoms characteristic of other clusters, than on all the signs and symptoms we marked as characteristic.

The similarities observed between the estimated clusters and those defined manually show that different clustering methods can successfully classify DCI statistically. We can therefore conclude cluster analysis is a suitable technique to diagnose DCI type according to the patient's signs and symptoms. The long-term goal of this project is to optimize the treatment plan for a given DCI case, so the next step is to determine whether clustering is enough to define a treatment plan. Yet, we processed lambda between outcome and classifications and observed a close to zero association level (Documented elsewhere, [START_REF] Aksoy | [END_REF] ). This means diagnosis alone (be it manual or automatic) is not enough to predict the success or failure of the DCI treatment plan. Further studies will have to consider other data, such as patient age and sex, and treatment modalities to be able to perform a correct prediction. Such a prediction would then allow studying the relationships between symptoms, signs and other factors, and the outcome of the treatment plan, allowing to determination which data are relevant to select an adapted treatment plan.

Table 1 .

 1 Signs and symptoms for K-Means, EM and COBWEB clusters

	Signs and symptoms	K1	K-means K2 K3	K4	E1	E2	EM	E3	E4	C1	COBWEB C2 C3
	Altered consciousness	5	28	0	10	6	10	27	0	11	8	20
	Confusion	32	78	0	32	35	36	71	0	35	28	77
	Dyspnea/chokes	27	45	0	20	19	26	47	0	27	10	54
	CV	2	1	0	1	1	2	1	0	2	1	1
	Pain	431 159 300	45 370 102 163 300	96 179 652
	Skin Rash and Mottling	20	89	0	13	27	15	80	0	15	30	76
	Localized Skin Swelling	4	17	0	1	4	1	17	0	1	5	16
	Abnormal Sensations	12	17	0	8	10	10	17	0	10	10	17
	Hearing Loss	2	13	0	1	1	2	13	0	1	4	11
	Vision Problems	16	26	0	11	12	16	25	0	15	13	25
	Discoordination	19	24	0	17	18	21	21	0	21	18	21
	Muscular Weakness	63	7	2 186	9 243	4	2 237	10	11
	Muscular Problems	22	25	0	5	23	9	20	0	7	19	26
	Decreased Skin Sensitivity	140	48	0	60 152	72	24	0	73	95	76
	Bladder Bowel Problems	7	2	0	27	1	34	1	0	33	2	1
	Headache	40	96	0	21	59	22	76	0	24	49	81
	Fatigue	59 110	0	26	77	29	89	0	32	60 100
	Nausea	29	57	0	10	41	11	44	0	12	35	46
	Dizziness	46	95	0	27	58	31	79	0	31	55	78
	Vertigo	11	28	0	8	9	10	28	0	9	13	25
	Paresthesia	615	51	0	74 593 132	15	0 128 565	26
	Tinnitus	1	3	0	0	1	0	3	0	0	1	2
	Numbness	753 135	0 115 846 148	9	0 163 649 167
	Paralysis	44	8	0	71	8 111	4	0 107	5	11
	Malaise	74	28	0 204	43 237	26	0 255	20	31

Table 2 .

 2 Association between clusters measured with Goodman-Kruskal's lambda

		EM	COBWEB	K-Means Two-Step PSI
	EM	-	0.621	0.759	0.827	0.405
	COBWEB	0.621	-	0.449	0.549	0.350
	K-Means	0.759	0.449	-	0.802	0.343
	Two-Step	0.827	0.549	0.802	-	0.444
	PSI	0.405	0.350	0.343	0.444	-


T V Shubina

M M Glasov

Nikolay A Gippius

A A Toropov

David Lagarde

Pierre Disseix

Joël Leymarie

Bernard Gil

G Pozina

J P Bergman

M M Glazov

B Monemar

Delay and distortion of slow light pulses by excitons in ZnO

Keywords: number(s): 42, 50, Ct

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

I. INTRODUCTION

The phenomenon of light retardation in a dispersive medium has been studied from the beginning of the last century, [START_REF] Brillouin | Wave Propagation and Group Velocity[END_REF] for both local dielectric 2 and exciton-polariton systems. 3 Garrett and McCumber 4 examined the propagation of a Gaussian light pulse near a resonant absorption line. In addition to exceptional variation of the group velocity, they predicted unchanged width and shape of the pulse for many exponential absorption depths. Early experiments showed no significant changes in the temporal shape of the pulse at the excitonic resonances in GaAs (Ref. 5) and GaP. 6 The distortion of the pulse spectrum was not considered, while the optical dispersion implies a different group velocity and attenuation for the higher-and lower-energy edges of the pulse. 7,8 That can result in pulse curvature in the time-energy plane and a shift of its maximum. The narrow absorption line can produce a hole in the pulse spectrum, causing temporal beats. 9 More recent time-resolved (TR) measurements have found such features of pulse distortion in various semiconductors. [10][11][12][13][14] Resumption of the interest in slow light was stimulated by possible applications in information processing. 15 That requires a delay significantly exceeding the pulse duration, while their ratio is limited when quantum coherent effects are exploited. The resonant dispersion allows one, in principle, to realize a high ratio, but at the cost of severe signal attenuation. Wide-gap semiconductors, like ZnO and GaN, possess strong excitonic resonances that makes them suitable for such light retardation. Light velocity as low as 6000 km/s in ZnO (Ref. 13) and 2100 km/s in GaN (Ref. 14) was measured below the bound exciton (BX) lines. In view of these findings, the question about the distortion of a light pulse has been revived. 16 Here, we report on delay and distortion of light pulses by excitons in ZnO. The results are interesting for two reasons: First, ZnO possesses numerous lines related to different states of donor-bound excitons. The fine spectrum of these excitons is probed by time-of-flight spectroscopy via the local distortion of the pulse shape at different energies. Second, simulation of the general shape of the transmitted pulse permits us to determine basic parameters of exciton-polariton resonances, inherent to bulk ZnO. They turn out to be somewhat different from those given by surface-probing techniques.

II. EXPERIMENTAL DETAILS

The time-of-flight experiments were performed using freestanding c-plane ZnO samples with thickness L = (0.3-2) mm supplied by Tokyo Denpa Co. Pulses of a picosecond tunable laser (Mira-HP, second harmonic) hit the sample placed in a He cryostat. Their detected temporal width (15-30 ps) was determined by instrumental accuracy of the system as a whole (dependent on the jitter of the laser pulses and spectral resolution of the monochromator, etc.). A Hamamatsu streak camera with a 2 ps temporal resolution (for the fastest time range) was used to record the TR images of the collected light.

Two schemes of measurements were exploited. In the forward (transmission) geometry, excitation and detection are carried out from different sides of the sample. In the backward geometry, this is done from the same side [Fig. 1(a), inset]. In both cases, we were able to register a series of replicas, arising due to the light reflection from the crystal boundaries. Propagating like polaritons, 14 they cover a distance of nL, where n is an odd (even) number for the forward (backward) geometry. Thus, their study is the same as the investigation of a set of samples with identical parameters but different lengths. The initial time moment when light impinges on the sample surface was determined in the forward geometry by means of measurements of the light pulse without the sample. In the backward geometry, we used for this purpose a light streak directly reflected from the surface (0L in our notation), which precedes the series of replicas in a time-resolved image. This way is more precise because it excludes variation of the delay time due to readjustment between two different measurements.

Supplementary continuous-wave (cw) photoluminescence (PL), transmission, and reflection measurements were performed at different temperatures using a He-Cd laser (for PL) or a tungsten lamp (for the others). In the PL spectra, the number (2-4) and intensity of the dominant BX lines vary among the samples, while their energies are almost identical.

III. RESULTS AND DISCUSSION

Typical images of the light pulses propagating at different energies are shown in Fig. 1(a). Below the BX lines, a series of replicas can be seen. The higher-energy edge of the curved light stripe is observed in the close vicinity of the A exciton at 3.374 eV with a delay of about 1.6 ns. This light stripe gradually narrows due to increasing absorption. Further shift of the pulse toward higher energies results in its full quenching accompanied by an increase of the BX PL. These images apparently show that the general curvature and delay of the pulses follow the slowdown of the group velocity due to the optical dispersion controlled by the exciton-polariton resonances.

In contrast, the influence of the BX lines is spectrally local. They provide dips cutting the pulses into several parts and induce extra retardation of light nearby. Because even a very weak line causes noticeable absorption if a sample is thick enough, the number of BX lines resolved by the time-of-flight spectroscopy (similar to transmission) is higher than in the PL spectra [Fig. 1(b)]. The spectral cross sections of the pulses (Fig. 2) allow us to reconstruct the fine structure of the BX spectrum, which contains at least 18 lines in the 3.356-3.374 eV range (Table I). These lines can be divided into three groups: (i) Two sets of A and B excitons bound to neutral donors D 0 X, separated by a gap of ∼4.5 meV. They produce the strongest local changes. (ii) Excitons bound to ionized donors D + X, at higher energies. iii) Weak lines related to the vibrational-rotational (vr) states of the bound excitons. Besides, a few additional lines are discovered by this time-of-flight study.

To model the pulse delay and distortion, the spectrum of the impinging laser pulse E 0 (ω) is approximated as a Gaussian with central frequency ω 0 and the envelope maximum in the time domain at t = 0. The amplitude and phase of the pulse in the frequency domain are given by the Fourier transform at the boundary of the medium, z = 0. 8 In the linear regime, the electric field at the spatial point z = L can be described as

I * 0 (D + X A ) 3.3733 1 × 10 -7 X 3.3635 5 × 10 -7 I 0 (D + X A ) 3.3721 1 × 10 -6 I vr 6 (D 0 X A ) 3.3623 1 × 10 -6 I 0a (D + X A ) 3.
E(L,t) = ∞ -∞ E 0 (ω)T (L,ω)e -iωt dω 2π . (1)
Here, the amplitude transmission coefficient T (L,ω) takes into account the multiple reflections and attenuation of light impinging normally on a slab with parallel boundaries. [START_REF] Landau | Electrodinamics of Continuous Media[END_REF] It depends on the complex wave vector of light in the sample, k(ω) = (ω/c) √ ε(ω), with ε(ω) being the dielectric function of the medium. The pulse shape in the time-frequency representation is calculated by means of the Gabor transform with the temporal window of ∼15 ps. [START_REF] Chui | Wavelet Analysis and Its Applications[END_REF] For a medium with several exciton-polariton resonances, ε(ω) is written as ε(ω) = ε b + j X j , where ε b is the background dielectric constant (taken as 8.1) and X j are the contributions of these resonances. When the resonances are assumed to be homogeneous, an expression taking into account the spatial dispersion can readily be used. 20 With inhomogeneous broadening induced, e.g., by structural imperfections, each X j term can be represented by the convolution of the line with a Gaussian centered at the same frequency: 14

X j = f j ω 0,j ω 0,j + βk 2 + ξ -i j -ω 1 √ π j exp -ξ 2 2 j
dξ.

(

Here, each j resonance is characterized by a frequency ω 0,j , an oscillator strength f j , and a damping term j . j describes the inhomogeneous width. Spatial dispersion is taken into account by the term βk 2 = (h 2 /2M j )k 2 , where the effective masses M j are assumed to be infinite for the BX and equal to 0.9m e for each free-exciton resonance. Equations ( 1) and ( 2) account for the spatial dispersion in a simplified way. They are valid for ω not too close to ω 0,j , which is sufficient for our purposes. Variation of the group velocity v g (ω) = dω/dk, controlled mostly by f j , determines the different delay T = L/v g of the pulse constituents and, hence, the pulse curvature.

In our simulations, all noticeable BX lines were taken into account. h = 1 μeV and h = 75 μeV characterize these narrow lines. The f BX values corresponding to the observed local distortion are given in Table I. For the sake of illustration, we present the modeling of the intricate pattern appearing when the central energy of the pulse falls into the gap between the closely situated I 6 and I 7 lines. The light propagates there as an extremely narrow ∼0.3 meV stripe [Fig. 1(a)]. Figures 3(a)-3(c) show that a twice higher f value taken for the D 0 X B transitions (4 × 10 -6 vs 2 × 10 -6 ) provides an unacceptably long delay and too strong distortion of the pulse.

To determine the parameters of the exciton-polariton resonances, we consider exclusively the data on the processes taking place inside the crystal, namely, cw transmission and pulse propagation. We abstain from using the data from reflection and PL spectroscopies, because they probe mostly a region close to the surface. The modeling is simplified owing to the fact that each excitonic parameter is responsible predominantly for some particular characteristics. The derived exciton-polariton parameters are collected in Table II (the distant C exciton is neglected). The modeling reveals two severe limitations:

(1) The damping term h must be as small as ∼3 μeV at low temperature. Higher values would result in full opaqueness in the transmission spectra in the range where the narrow BX lines are clearly resolved [Fig. 1(b)]. This is independent of the model used to describe ε(ω), because the absorption depends on the imaginary part of the dielectric function, which is determined by this term.

(2) The inhomogeneous width h cannot exceed 0.5 meV for the A exciton resonances. This restriction arises from the observation of the transmitted light at 3.374 eV. With stronger broadening, which may occur in crystals of worse quality or with a temperature rise, the pulse propagating at this energy would be fully attenuated.

These constraints based on the signal attenuation are more rigorous than a requirement imposed on the effective damping parameter, which should be less than the longitudinaltransverse splitting ω LT to maintain the polaritonic modes. 21 Moreover, our data do not exclude the possibility that the excitonic resonances inside the crystal are homogeneous with h of a few μeV.

As a result, we were able to simulate TR images using the same exciton-polariton parameters for all samples (the BX parameters were slightly varied). Close agreement is observed between the simulated [Fig. 3(d)] and experimental [Fig. 1(a)] images of the pulses of different energies. Figure 4 presents the light replicas measured in the different samples. Their shapes are also well consistent with the calculated delay dependencies.

In addition, our modeling has revealed an obvious contradiction between the results of volume-and surface-probing experiments (Table II). The successful fitting of reflectivity spectra [Fig. 1(b), inset] requires the empirical damping term h * A = 0.75 meV in the model of homogeneous resonances. 20 A similar result can be obtained assuming the resonances as inhomogeneous with h A ∼ 0.9 meV. However, both variants would provide too strong attenuation of the propagating light. Further, the splitting between the PL peaks, ascribed to transverse and longitudinal A exciton emission, is 1.6-2 meV, whereas the rough estimation using the bulk parameters gives ω LT ≈ 3 meV. This dependence on the measurement technique is suggestive of a certain deterioration of the excitonic resonances at the surface, which can be induced by modified structural properties. 22,23 IV. CONCLUSION In conclusion, a strong resonant delay of light pulses in ZnO is accompanied by their severe attenuation and shape distortion. Simulation of these effects enables us to determine the excitonic parameters of bulk ZnO, as well as to establish that the resonances must have a low damping and limited broadening to allow the light propagation. Time-of-flight spectroscopy gives an opportunity to investigate the fine structure of a spectrum.

FIG. 1 .

 1 FIG. 1. (Color online) (a) Combined images of four pulses propagating through the 0.3 mm sample (logarithmic intensity scale). The boundaries between them are marked by white lines. The delay time dependence (dashed line) is calculated neglecting the BX lines to show the dominant contribution of the free-exciton resonances. The inset presents the scheme of the signal detections in the forward and backward geometries; labels nL mark the light replicas. (b) Spectra of cw PL, transmission, and reflection (the inset) measured in the same sample. A T and A L denote the transverse and longitudinal A exciton emission peaks. Dashed lines show the transmission spectra simulated neglecting the BX lines with h j (1) 3 μeV and (2) 30 μeV.

FIG. 2 .

 2 FIG. 2. Selected spectral cross sections of the light pulses passing the 0.3 mm sample, made at different delay times from 110 ps (bottom) up to 1200 ps (top). The dips corresponding to several important lines are labeled. The shape of the initial light pulse is shown by the dotted curve.

  3717 1 × 10 -6 I vr 7 (D 0 X A ) 3.3620 1 × 10 -6 X 3.3705 3 × 10 -7 I vr 8 (D 0 X A ) 3.3616 1 × 10 -6 X 3.3695 3 × 10 -7 I 5 (D 0 X A ) 3.3614 2 × 10 -6 X 3.3685 1 × 10 -6 I 6 (D 0 X A ) 3.3608 5 × 10 -6 I B 5 (D 0 X B ) 3.3669 1 × 10 -6 I 7 (D 0 X A ) 3.3600 5 × 10 -6 I B 6 (D 0 X B ) 3.3653 2 × 10 -6 I 8 (D 0 X A ) 3.3593 5 × 10 -9 I B 7 (D 0 X B ) 3.3647 2 × 10 -6 I 9 (D 0 X A ) 3.3566 1 × 10 -9

FIG. 3 .

 3 FIG. 3. (Color online) Simulations of the shapes of Gaussian light pulses propagating through the 0.3 mm sample. (a)-(c) A pulse centered at 3.365 eV modeled (a) neglecting BX lines, and taking into account the BX lines with (b) f = 2 × 10 -6 and (c) f = 4 × 10 -6 . Arrows mark the experimental delay of the pulse center. (d) Pulses centered at 3.354, 3.360, 3.366, and 3.373 eV modeled using the data of TablesI and II. The black line presents the calculated delay dependency for relative intensity >10 -7 . In each particular image with boundaries marked by white lines, the intensity (logarithmic scale) is normalized to the maximum value.

FIG. 4 .

 4 FIG. 4. (Color online) Images of nL replicas recorded in the 0.3 mm (a) and 1 mm (b) samples in the backward and forward geometry, respectively. 0L marks the pulse directly reflected from the surface. The black lines present the calculated delay dependencies shown for intensity of the transmitted light larger than 10 -7 of the incident light intensity.

TABLE I .

 I Parameters of bound excitons in bulk ZnO obtained through the pulse distortion analysis. The notation follows Ref. 19; X denotes unidentified lines.

	Transition	E (eV)	f BX	Transition E (eV)	f BX

TABLE II .

 II Low-temperature exciton-polariton parameters derived from volume-and surface-probing measurements.

	Parameter	A exciton	B exciton
		TR and cw transmission
	hω 0 (eV)	3.376 ± 0.0002	3.382 ± 0.001
	f	0.0072 ± 0.0002	0.012 ± 0.001
	ω LT (meV)	3	5
	h (μeV)	3 ± 0.2	3 .5 ± 0.5
	h (meV)	<0.5	<1
		PL and reflection
	hω 0 (eV)	3.3758 ± 0.0002	3.382 ± 0.001
	ω LT (meV)	1.8 ± 0.2	6.5
	h * (meV)	0.75	1.5

ACKNOWLEDGMENTS

This work has been supported in part by the RFBR, the Program of the Presidium of RAS, the Dynasty Foundation, and the FP7 ITN "Spin-Optronics" (237252). T.V.S. acknowledges the Université Montpellier 2 for its hospitality.