François Pellegrino
email: francois.pellegrino@univ-lyon2.fr

Emmanuel Ferragne
email: emmanuel.ferragne@univ-paris-diderot.fr

Fanny Meunier
email: fanny.meunier@ish-lyon.cnrs.fr

2010, a speech oddity: Phonetic transcription of reversed speech

Keywords: reversed speech, speech perception, phonetics

Time reversal is often used in experimental studies on language perception and understanding, but little is known on its precise impact on speech sounds. Strikingly, some studies consider reversed speech chunks as "speech" stimuli lacking lexical information while others use them as "non speech" control conditions. The phonetic perception of reversed speech has not been thoroughly studied so far, and only impressionistic evaluation has been proposed. To fill this gap, we give here the results of a phonetic transcription task of time-reversed French pseudo-words by 4 expert phoneticians. Results show that for most phonemes (except unvoiced stops), several phonetic features are preserved by time reversal, leading to rather accurate transcriptions of reversed words. Other phenomena are also investigated, such as the emergence of epenthetic segments, and discussed with insight from the neurocognitive bases of the perception of time-varying sounds.

Introduction

'Reversed speech' (RS) is the term used to describe signals resulting from time reversal of speech excerpts, without any other alteration. RS was used as early as 1953 by C. Cherry as a competing signal in a dichotic perception task where subjects' attention was drawn to a natural speech target presented to the other ear [START_REF] Cherry | Some experiments on the recognition of speech with one, and with two ears[END_REF]. After the experiment, most subjects reported that these distractors sounded like normal speech, though a few individuals said that there was "something queer about it". Since then, time-reversed stimuli have been intensively used as a control condition in neuroimaging (e.g. [START_REF] Binder | Human Temporal Lobe Activation by Speech and Nonspeech Sounds[END_REF][START_REF] Dehaene | Anatomical variability in the cortical representation of first and second language[END_REF][START_REF] Howard | The cortical localization of the lexicons[END_REF]) and in behavioral studies on speech perception, both for humans and animals [START_REF] Ramus | Language discrimination by human newborns and by cottontop tamarin monkeys[END_REF][START_REF] Toro | Effects of backward speech and speaker variability in language discrimination by rats[END_REF]. Besides, a few studies have investigated human word recognition of RS per se, [START_REF] Saberi | Cognitive restoration of reversed speech[END_REF][START_REF] Greenberg | The relation between speech intelligibility and the complex modulation spectrum[END_REF][START_REF] Meunier | Mesure d'intelligibilité de segments de parole à l'envers en français[END_REF][START_REF] Grataloup | Cognitive restoration of reversed speech in French[END_REF] but the phonetic aspects of RS perception have been neglected so far, with the limited exception of a pilot study in [START_REF] Binder | Human Temporal Lobe Activation by Speech and Nonspeech Sounds[END_REF].

As a consequence, the exact effects of time reversal on phonetic perception are still unknown, leading to an ambiguous situation where RS is either regarded as non speech or speech-like. As an experimental control condition, if RS is taken to be non speech, it should contrast with natural speech and activate only low level auditory neurocognitive processing (e.g. [START_REF] Strand | Phonological working memory with auditory presentation of pseudo-words--An event related fMRI Study[END_REF]). On the contrary, if RS is seen as a kind of delexicalized speech, it should trigger mechanisms of speech perception (identification of phonetic cues, etc.), and potentially higher level processes (e.g. [START_REF] Binder | Human Temporal Lobe Activation by Speech and Nonspeech Sounds[END_REF]).

To date, it has not been possible to settle this issue and to determine where to put RS between non speech and speech since precise knowledge of what happens at the phonetic level is still lacking. This paper therefore aims at assessing to what extent time reversal preserves phonetic cues and to study how time-reversed phonemes are perceived. It is based on a task of detailed phonetic transcription of time-reversed French target words by four expert phoneticians.

Section 2 provides some landmarks on RS perception. Section 3 details the experiment and gives the perceptual results. Section 4 is a discussion.

The perception of reversed speech

Spectro-temporal impact of time reversal

Time reversal is assumed to alter the temporal properties of the original speech signal while preserving its spectral properties. It is true from a static standpoint since both long-term and short-term power spectra of a signal are invariant under time reversal, but this view underestimates the impact of signal dynamics on speech perception. The time derivates of spectral features in speech are of major importance in human speech perception [e.g. [START_REF] Rosen | Temporal information in speech: acoustic, auditory and linguistic aspects[END_REF][START_REF] Divenyi | Frequency change velocity detector: A bird or a red herring?[END_REF][START_REF] Carré | Speech Dynamics: epistemological aspects[END_REF]. Under time reversal, steady parts of speech may be almost invariant but transient parts are dramatically altered: abrupt onsets (that are common in normal speech: stop bursts, vowel onsets, etc.) give rise to abrupt offsets, that are unlikely to occur in real environments because of reverberation; smooth damping (i.e. decaying) result in smoothly ramping (i.e. increasing in amplitude) segments, potentially disturbing duration perception [START_REF] Phillips | Central auditory onset responses, and temporal asymmetries in auditory perception[END_REF]. Consequently, RS is a chimera mixing speech-like chunks with speech oddities that alter major properties of speech, at both spectro-temporal and distributional levels (e.g. proportion of segments with rising vs. falling intensity).

Little is known on the consequences of these alterations on human perception, but several studies have observed that time reversal strongly impairs the ability to discriminate between experimental conditions. For instance, while language discrimination based on rhythmic patterns seems possible for human newborns, it has not been observed for RS [START_REF] Ramus | Language discrimination by human newborns and by cottontop tamarin monkeys[END_REF] and the same holds for cotton-top tamarin monkeys [START_REF] Ramus | Language discrimination by human newborns and by cottontop tamarin monkeys[END_REF] and rats [START_REF] Toro | Effects of backward speech and speaker variability in language discrimination by rats[END_REF]. No clear explanation for such differences is available yet, but in line with differences revealed at the neural level (e.g. [START_REF] Galbraith | Brain stem evoked response to forward and reversed speech in humans[END_REF]), they suggest that the neurocognitive processing of RS may significantly differ from that of normal speech, even at the acoustic-phonetic and rhythmic levels, disregarding lexical information.

Landmarks in reversed speech perception

Two kinds of time reversals have been considered in the literature. In the first one, initiated by Cherry, stimuli result from a global reversal: the reversed stimulus is similar to playing the original sound backward. In the second, more recent approach, time reversal is used as a way to locally degrade speech. The signal is viewed as a sequence of short frames, and reversal is applied within one or several frames, independently from the others.

This local approach aims at evaluating the resistance of speech understanding to degradation by time reversal for increasing time spans. In [START_REF] Saberi | Cognitive restoration of reversed speech[END_REF][START_REF] Greenberg | The relation between speech intelligibility and the complex modulation spectrum[END_REF] for English, and [START_REF] Meunier | Mesure d'intelligibilité de segments de parole à l'envers en français[END_REF] for French, speech signals were divided in sequences of constant duration frames and each frame was locally time-reversed. The rate of correct word recognition as a function of frame duration was evaluated. These studies converge to show that a high intelligibility is preserved for short reversal frames (the 50% intelligibility rate corresponds to 66 ms in English and 100 ms in French). For reversal windows longer than 150~200 ms, word intelligibility is reduced to zero. In [START_REF] Grataloup | Cognitive restoration of reversed speech in French[END_REF], only one frame of a disyllabic target word (or pseudo-word) was time-reversed with the syllable as the reference unit (½ syllable, 1 syllable, 1½ syllable, or 2 syllables were reversed). Behavioral and electrophysiological measures demonstrated the existence of a lexical bias that compensates for the degradation at the phonetic level. However, none of these studies provide any detailed analysis at the phonemic level since they focus on the word level.

To the best of the authors' knowledge, the only study that has to some extent analyzed the perception of RS at the phonetic level is a pilot study in [START_REF] Binder | Human Temporal Lobe Activation by Speech and Nonspeech Sounds[END_REF]. It was used as a control condition before an fMRI experiment, and thus focused on a global reversal procedure. Ten naïve subjects were asked to orthographically transcribe reversed monosyllabic words in order to assess the degree of phoneme recognition in RS. Between-subject agreement was evaluated in terms of identical 'letters' used in the transcription; it reached 72.7%. The between-subject agreement in terms of transcription length (allowing a ±1 letter tolerance) was close to 90%. These preliminary results did not come along with a more thorough phonetic analysis and no comparison between subjects' transcriptions and the phonemic content of the stimuli was performed. Moreover, the use of an orthographic transcription system forced the subjects to cast what they heard into the English orthographic system, i.e. to disregard phonetic cues that they were not able to transcribe.

All the experiments reported in this section tend to confirm that RS is close enough to natural speech to trigger the perceptual identification of phonemes, but they do not provide any information regarding which phonetic cues or phonemes are preserved or altered.

Experiment

Material and subjects

The stimuli were the globally time-reversed versions of 47 pseudo-words agreeing with French phonotactics. They were digitally recorded by a French female speaker in a soundproof booth (PCM, 44,100 Hz; 16 bits). The phonological structure of all pseudo-words is CVC, but in 13 stimuli, the speaker produced a final phonetic schwa that we decided to keep. The 47 stimuli correspond to 154 phonemes (44 oral vowels, 3 nasal vowels, 13 schwas, and 94 consonants).

Table 1 provides the abbreviations used in the paper.

.V.N 4 F.V.S0 5 L.V.F 3 N.V.F 5 N.V.S1 6 R.Ṽ.S0 3 S0.L.V 3 S0.V.F 5 S0.V.S1 4 S1.V.F 5 S1.V.S0 4 Total 47
Four male expert phoneticians from Paris and Lyon (France) enrolled voluntarily in the experiment. They were not aware of the nature and language of the stimuli they would have to transcribe.

Experimental design

The experiment was designed and run with Praat [START_REF] Boersma | Praat: doing phonetics by computer[END_REF]. Subjects were seated in a quiet room and heard the stimuli, preceded by a beep and a 500 ms silence, through headphones.

A break was proposed after each ten stimuli. The experiment lasted less than 20 minutes. The 47 stimuli were randomized for each subject, and the latter was prompted to give an accurate phonetic transcription of what he had heard. Each stimulus could be reheard as often as judged necessary by the subject, and answers were given either by handwriting or typing according to subject's preference. Three subjects chose to transcribe their answers in IPA and the fourth one used SAMPA. After the experiment, subjects were asked for informal comments.

Results

Subjects' answers have been recoded in terms of broad phonetic categories by the first author. Moreover, the accuracy of the answer was also reported for several phonetic features (manner of articulation and voicing, vowel quality). Results are given in this section using the rate of exact retrieval of the original phonetic feature or segment as measures.

Overview of the results

3 of the 4 experts transcribed geminate consonants and length marks for vowels. To save space, this factor is not further mentioned in the results, but it will be discussed in Section 5. Besides, 3 of the 4 experts used non native French symbols in their transcription. An important result is that for more than 25% of the stimuli, the experts exactly retrieved the original segments despite the reversal process (see Table 3, first row). For instance, the stimulus resulting from time-reversal of the pseudo-word /mif/ was transcribed as [fim] by the 4 experts, which matches both the correct CVC structure and segments. Most of the correctly retrieved stimuli were continuant waveforms reflecting the higher invariance to time-reversal of continuant segments (or nasals) compared to the very asymmetric temporal nature of stops (see Table 3, second row).

In a very high proportion, oral vowels were perfectly identified (more than 90% on average), while reversed nasal vowels were often transcribed as N+V or N+Ṽ sequences.

The phonetic schwas uttered at the end of 13 pseudowords and consequently present at the beginning of 13 stimuli were very often detected (more than 92% on average). Generally, they were transcribed as either a vowel (e.g. schwa) or a complex sequence of fricatives and vowels, often arising from the misperception of the number of phonemic segments in the sequence (see below).

Number of segments detected

A manual inspection of the subject individual results shows good agreement for the transcription of most broad phonetic classes, except for schwas and unvoiced stops. We have thus pooled together their results in the rest of the paper.

Table 4. Mean difference between the number of phonemes in the original stimuli and in their transcriptions. Standard deviations are also given.

Original Structure

Mean difference in number of segments S.V.F.ə -0.2 (0.5) L.V.F + 0.0 (0.0) N.V.F + 0.1 (0.3) F.V.N + 0.1 (0.4) S.V.F + 0.1 (0.7) N.V.S.ə + 0.5 (0.5) S.L.V + 0.6 (0.5) S.V.S.ə + 0.6 (0.6) S.V.S + 1.1 (0.4) F.V.N.ə + 1.3 (0.9) F.V.S + 1.4 (1.0) R.Ṽ.S + 1.6 (0.7)

Depending on the original structure of the stimuli, transcription length varied up to an overestimation of the number of segments by 1.6 segments for R.Ṽ.S pseudo-words. Stimuli formed from L.V.F pseudo-words were always transcribed as 3-segment chunks. N.V.F and F.V.N reversed waveforms were generally detected as 3-segment chunks too. With the exception of S.V.F.(ə) stimuli, the presence of stops disturbed the perception of the number of segments (up to 1.4 segment for F.V.S stimuli). An additional inspection of the results reveals a tendency to interpret fine phonetic details as cues for additional segments and a trend to decompose reversed stops into a sequence of segments.

Broad phonetic classes accuracy

Accuracy in the detection of broad phonetic classes is given in Table 5. Detection is considered as correct if the original segment was transcribed as one and only one segment (no insertion, no deletion) of the same broad phonetic nature, disregarding consonantal places of articulation and vowel qualities. Even with this narrow definition, most classes reach very high detection accuracies (liquids, nasals, oral vowels, unvoiced and voiced fricatives). Inaccurate detections concentrate on unvoiced stops (9.4%), nasal vowels (16.7%), and schwas (25.0%). Voiced stops are intermediate (61.8%). The global detection accuracy reaches 66.9% for a total of 616 transcribed segments. Not surprisingly, the most inaccurate transcription was for unvoiced stops. The distribution of the transcriptions given for the 96 segments of this nature is given in Figure 1. 30% were transcribed as fricatives and 25% as stops, but their identities varied widely from glottal stops to unreleased voiced stops, depending on the expert and on the stimulus. 28% were decomposed as a cluster (10% including a stop, 18% without stops), 7% were transcribed as a sonorant, and 10% of the segments were simply not detected and transcribed by the experts.

Figure 1. Distribution of the transcriptions for reversed unvoiced stops (rounded to nearest integer).

Epenthetic segments

One consequence of time reversal is that it gives rise to unfamiliar transitions between phonemes. The interpretation of some acoustic cues may thus give rise to epenthetic segments. The stimulus generated by reversing the pseudo-word /sat/ is illustrated in Figure 2

General Discussion

This paper aims at filling a gap between the common use of RS in neurocognitive research and the very limited knowledge of the way humans process such stimuli at the phonetic level. The procedure is based on transcription by experts, under the twofold assumption that they would pay more attention to phonetic details and that they would be able to accurately transcribe such details. Both assumptions are met in the subjects' answers. Moreover, a good degree of agreement is reached for most broad phonetic categories and more than 25% of the stimuli were perfectly retrieved. At the broad phonetic level, the accuracy rate is also high (66.9%).

However, the experiment also reveals that, for rapidly changing sounds such as stops, subjects differ in their transcription strategy and a wide range of transcriptions is proposed, often based on a decomposition of the stop release into several segments. Besides, that the subjects explicitly transcribed vowel length or gemination (for fricatives) is fully compatible with the neurophysiological and psychophysical evidence on the asymmetric perception of duration depending on whether the amplitude envelope of a sound has a ramping or damping shape [START_REF] Phillips | Central auditory onset responses, and temporal asymmetries in auditory perception[END_REF][START_REF] Digiovanni | Mechanisms responsible for differences in perceived duration for risingintensity and falling-intensity sounds[END_REF][START_REF] Giraud | Representation of the temporal envelope of sounds in the human brain[END_REF].

The present study strongly suggests that most phonemes present in RS are intelligible. It means that RS could trigger lexical access if the phoneme sequence corresponds to words in a language intelligible to the subjects, potentially interfering in experiments where RS is the non-speech control condition. Additionally, the presence of abnormal transitions (e.g. abnormal intensity contours) would trigger additional neurocognitive mechanisms, such as Mismatch Negativity [START_REF] Grataloup | Cognitive restoration of reversed speech in French[END_REF][START_REF] Näätänen | The perception of speech sounds by the human brain as reflected by the mismatch negativity (MMN) and its magnetic equivalent (MMNm)[END_REF], when compared to normal speech.

  (waveform and spectrogram). The 4 experts transcribed this signal as [snas] (with an initial glottal stop for one of the expert). The [n] segment arose from the slowly damping oscillation of the [a], corresponding to a smooth ramping in the time-reversed signal (arrows on the figure).

Figure 2 .

 2 Figure 2. Waveform and spectrogram of the reversed version of the pseudo-word [sat]. The section giving rise to an epenthetic [n] segment is shown by arrows.

Table 2

 2 displays the phonological structure of the 47 stimuli. 0 and 1 indicate respectively a unvoiced vs. voiced segment. Dots '.' encode segment boundaries.

Table 1 .

 1 Broad phonetic classes

	Class	Label	Class	Label
	Fricative	F	Schwa	ə
	Liquid	L	Stop	S
	Nasal Consonant	N	Vowel (Nasal)	Ṽ
	Rhotic	R	Vowel (Oral)	V

Table 2 .

 2 Phonological structure of the pseudo-words (before time reversal).

	Structure	Number of stimuli
	F

Table 3 .

 3 Phoneme retrieval. Total numbers of tokens are given in brackets, with the number of successfully retrieved tokens by each expert.

	Index	#1	Expert #2 #3	#4
	All stimuli (47)	9	13	15	15
	Continuant stimuli (12)	8	10	9	8
	Oral vowels (44)	40	39	39	42
	Schwas (13)	12	11	12	13

Table 5 .

 5 Detection accuracy for each broad class.

	Class	Accuracy (%)	No. of segments
	Stop (unvoiced)	9.4	96
	Vowel (nasal)	16.7	12
	Schwa	25.0	52
	Stop (voiced)	61.8	76
	Rhotic	66.7	12
	Vowel (oral)	88.6	176
	Nasal	90.0	60
	Fricative (voiced)	91.7	48
	Fricative (unvoiced)	93.3	60
	Liquid	95.8	24
	All	66.9	616

Acknowledgement

The authors thank the four expert phoneticians for their contribution. This research was supported by a European Research Council grant to the SpiN project N°209234.


Jaromir Mrazek

Eva Simeckova

Radomir Behal

Vadim Glagolev

František Vesely

Milan Stefanik

Mitja Majerle

Jan Novak

Martin Ansorge

Ivan Sivacek

Jitka Vrzalova

Xavier Ledoux

Francois De Oliveira Santos

Ulrich Fischer

Axel Klix

Marilena Avrigeanu

Vlad Avrigeanu

Charged particle activation facility in NPI CAS and in future GANIL/SPIRAL2-NFS

The proton, deuteron and alpha induced reactions are of a great interest for the assessment of induced radioactivity of accelerator components, targets and beam stoppers as well as isotope production for medicine and also to nuclear astrophysics. We present a new irradiation chamber for activation measurements, that forms a prolongation of long-term experimental activities using stacked-foil activation technique in NPI CAS, Řež. The chamber is based on an airlock system and is coupled to a pneumatic transfer system delivered by KIT Karlsruhe. This system is installed in GANIL/SPIRAL2-NFS and will be used for proton, deuteron and alpha particle activation measurements with long-and short-lived isotopes.

Introduction

A good knowledge of charged particle reaction cross sections is important in number of cases, we name two, here:

(a) construction, operation and decommissioning devices like ITER/DONES for IFMIF/DEMO -activation of construction materials, activation of accelerator components (targets, accelerator cavities, beamstops, etc.).

(b) improvements of the reaction models and understanding the underlying physics, that helps to make predictions in combinations of projectiles, targets and energies, where the data are missing or that are not experimentally accessible.

The experimental data are organized into large open access databases (e.g. EXFOR [1]) and are used -together with complex calculations based on theoretical models -to build open access libraries (e.g. TENDL [2]). An important part of the efforts, the underlying model parameters, obtained from interpretations are organized and kept in libraries for building databases and other use (e.g. RIPL [3]). These efforts offer a framework of systematic and consistent approach also for other investigations in nuclear reactions.

A specific case, where more experimental data are needed and a theoretical interpretation is necessary to shed light on the reaction mechanism are reactions with a deuteron projectile. Where a direct processes compete with compound ones and where a breakup contributions play an important role. Number of works were already done to coherently describe the complex activation data in * e-mail: mrazek@ujf.cas.cz the past (e.g. [4][5][6][7][8]). Another theoretical challenge is a prediction of population of isomeric states. Their importance is emphasized in cases, where they decay to a different daughter nucleus than the ground state. Such experimental measurements by activation can become difficult, when the isomer half-life is shorter or comparable to the time necessary for manipulation with the activated sample.

While the number of systematic measurements and theoretical explanations were performed in the last decades, that have improved our knowledge and extended the available databases, there are still reactions, where the actual systematic measurements are difficult and thus very limited or non-existing. One of the important cases are the reactions leading to short-lived products. The aim of this paper is to introduce the complex system for measurements of both short-lived and long-lived isotopes in NFS [9] (Neutrons For Science) laboratory in GANIL-SPIRAL2.

Activation setup in NPI CAS

Nuclear Physics Institute of the Academy of Sciences of Czech Republic operates two cyclotrons: light particle, variable energy U-120M cyclotron (currents from nA to tens of µA) and a new high current (up to 300µA) proton cyclotron TR24 (see the table 1). The operation of the cyclotrons is supported by the project CANAM (Center of Accelerators and Nuclear Analytical Methods) [10] of MEYS and RDI OP of EU.

The traditional charged particle activation setup used in NPI CAS consists of a stack of foils to be measured that are interleaved by monitoring foils (e.g. aluminium Left panel: a stacked foil simulation -an illustrative case of a proton beam on stack of 50 µm nat Fe foils. Foil number is on x-axis, mean energy in the foil (black diamonds) is on the left y-axis), a foil thickness in terms of energy (blue triangles) and an energy straggling (red circles) are on the right-side y-axis. Right panel: A proton beam deposition simulation for Irradiation Chamber beamtarget setup after 1 mm thick carbon degrader. The deposition is drawn at the backside of the sample holder. The distance between the degrader and the activation foil was 28 mm. The beam after the carbon degrader looses 5 MeV and is concentrated at the center (red dots). The beam spread for tantalum degrader (blue crosses) is shown for a comparison. The anti-scattering size is only depicted (a green circle) and was not included in the simulation. ), that (a) make sure that there is no unobserved particle exchange between neighbouring foils and that (b) the complete chain of procedures -from the beam energy and current measurement till detector efficiency calibration -is correct. This method enables a simultaneous measurement of a number of data points. The energy losses and energy straggling have an impact on precision of (especially) low energy data points (see the left panel of figure 1). In measurements of short-lived isotopes with a stacked foils method, one needs to find a compromise between handling possibility of the accumulated activity and accessibility of shorter-lived isotopes. The stacked foils method was successfully applied in a number of experiments ( [4][5][6][7][8]).

NFS at GANIL/SPIRAL2

A laboratory NFS (Neutrons For Science) in GANIL/SPIRAL2 was already presented elsewhere [9]. Despite its name, NFS laboratory will open new opportunities also in activation by charged particles. Initially, light ion beams (p, d, α) are expected to be available with a maximum current 50 µA (a limit allowed for NFS, the maximum reachable current is 5 mA). The available energies are displayed in the table 2.

In the next section, we present the system for irradiation by charged particles for SPIRAL2/NFS, that was built

Irradiation Chamber system

The need of construction of the new system for charged particle activation stems primarily from the limitations of deuteron beams in NPI CAS (see the tables 1, 2) and a simultaneous need of extension of the experimental possibilities. The demands for the construction of the system were based on long term experiences from NPI CAS and specific mode of operation in GANIL/SPIRAL2 NFS. A transport of a stack of foils between a beam in vacuum and storage was initially requested. Further demands were defined, as a new technological solution and thus new abilities appeared:

• an irradiation of a single foil to measure short half-lives • a sub-minute transport time from the irradiation to the detector

• a possibility of the fast change of the beam energy

• a basic information on the beam quality on-line

• a remote operation due to imposed safety restrictions

The selected solution was based on an airlock system coupled to a pneumatic transport system. The airlock system (in figure 2) is composed of two vacuum chambers: one is equipped by a beam-target system and is permanently a part of the NFS beam-line. The other one is equipped by a sample receive/send system and a manipulator, that transfers a sample to/from the beam-target system of the first chamber.

Beam-target system

The beam-target system (see figure 3) consists of • an entrance collimator • a cooled rotating degrader with twelve selectable positions • an anti-scattering collimator -electrode

• a sample lock system with a suspension

• a cooled Faraday cup

The beam is designed to have pencil-like characteristics with 4 mm diameter of the core. The entrance collimator is isolated and has a current readout, so that the information on beam position/focus change is available online. The degrader will allow for a fast change of beam energy (as the time needed for the energy change of the accelerator and subsequent magneto-optic tuning is yet unknown ) at the expense of the energy (and angular -see right panel of figure 1) straggling. The degrader is connected to the same cooling loop as the Faraday cup and the isolation better than 500 MΩ is maintained. The degrader has its own electrical readout electrode. The third element downstream is an anti-scattering collimator-electrode, that has several functions: (a) to remove eventual particles scattered by collimator or degrader to large angles (angular straggling, see figure 1), (b) to pose as an electrode for the sample+Faraday cup block, to repel eventual kicked-out electrons in Faraday cup measurement, (c) keep the electrons coming from the degrader material on the degrader side or deflect them by a pair of magnets that are installed around the collimator hole.

Collimator diameters are 5 mm both.

Cooling system was tested successfully with alcohol, but for safety constraints it will operate with much more viscous ethylene-glycol. The expected typical power on the system is in the order of Watts, the system was tested at 40 W.

The beam-target system is remotely retractable from the beam, so that after an irradiation, another experiment can take a place without a need of operator intervention.

Manipulator system and a sample holder

The second -send/receive -chamber is based on a 6-way cross with five valves. It is equipped by a manipulator arm, that can be positioned in the range of 0-350 mm. Two valves operate when a sample is needed to be received and sent, other two valves serve for the venting and evacuat- ing. The interlock valve (between the chambers) can be operated, when the same pressure is on its both sides.

For practical and operational speed reasons, the samples are received from top (see the middle part of figure 2) and sent out (also exploiting the gravity) downwards. To use the stacked-foil method, the key property of a manipulator system is to align the sample holder not only to the beam axis, but also direction-wise. To ensure this, we decided for the design of the sample holder as shown in figure 4. The sample holder is from aluminium, 19 mm in diameter and 40 mm long. It is pierced through so that it can house a foil (a stack of foils) of 15 mm diameter. The inner diameter of the steel bolt fixing the foil(s) is 11 mm. The correct orientation in the beam (or detector) is ensured by two (symmetrically placed) magnets fixed in the body of the sample holder. The orientation of the sample -once it reaches the manipulator (in the airlock system or at the detector) -is immediate.

Further operation details

The operation of the airlock system has a slow phase of evacuating down to 10 -5 mbar and a fast pressurized N 2 venting using pressure guards to avoid eventual critical conditions for the valves.

The Faraday cup is inclined by approx. 30 • from the beam axis towards the manipulator. When the sample is being translated to the beam position, it comes into an electrical and thermal contact with the Faraday cup and then the Faraday rotates to the beam axis. This way the cooling and charge readout of the sample holder and the Faraday cup is connected.

There were performed several tests on U-120M cyclotron of NPI CAS. These tests allowed to solve number of issues, define degrader material (see figure 1) and proved the possibility to transfer the sample from the beam position to the HPGe detector (including venting) well bellow one minute. The operational tests were performed with proton beam and nat Fe foils and the measured data follow the previous experimental data and TENDL curves of different reactions ( nat Fe(p,x) 52 Mn, nat Fe(p,x) 53m+g Fe).

Pneumatic Transfer System

The Pneumatic Transport System (PTS) for neutron irradiations is based on a system of TU Dresden and it was delivered by KIT Karlsruhe. Sample holders were made of plastic (lighter than aluminium) and the activation foil was located on top of the sample holder.

The PTS system was adapted for the charge particle sample holders: it was equipped with air-brakes at every ending point to keep the deceleration of the aluminium sample holders at reasonable limits.

The control software was built in LabView, that communicates with PLC system of the PTS system, software of the PTS, controllers of the IC system and control system of the HPGe detector positioning.

Summary

The complex system of Irradiation Chamber (IC) and PTS for charged particle activation was installed in NFS laboratory of GANIL/SPIRAL2 facility, where the access is open and excellence based.

The Irradiation Chamber was built, tested and installed also with the efforts of J.Diviš, J.Záluský, J. Červeňanský, L.Király, S.Slovák, M.Gschray, J.Gabrhel, M.Götz. Authors are indebted to GANIL/SPIRAL2 technical staff, that provided help and cooperation during various phases of the project. The authors acknowledge the MEYS project SPIRAL2-CZ, CZ.02.1.01/0.0/0.0/16_013/0001679.

Figure 1 .

 1 Figure1. Left panel: a stacked foil simulation -an illustrative case of a proton beam on stack of 50 µm nat Fe foils. Foil number is on x-axis, mean energy in the foil (black diamonds) is on the left y-axis), a foil thickness in terms of energy (blue triangles) and an energy straggling (red circles) are on the right-side y-axis. Right panel: A proton beam deposition simulation for Irradiation Chamber beamtarget setup after 1 mm thick carbon degrader. The deposition is drawn at the backside of the sample holder. The distance between the degrader and the activation foil was 28 mm. The beam after the carbon degrader looses 5 MeV and is concentrated at the center (red dots). The beam spread for tantalum degrader (blue crosses) is shown for a comparison. The anti-scattering size is only depicted (a green circle) and was not included in the simulation.

Figure 2 .

 2 Figure2. The Irradiation Chamber. It is composed of two vacuum chambers separated by an interlock valve (light gray). The rightmost part is permanently in the beamline and it contains a beam-target system. The left part contains a system for receiving/sending samples and is equipped with a manipulator. Two sample holders are depicted in green -one is waiting for receiving (in the middle) and the other is in the beam position (on the right).

Figure 3 .

 3 Figure 3. Beam-target system. Beam direction is from the left, collimator, twelve position degrader, anti-scattering-electrode.

Figure 4 .

 4 Figure 4. Sample holder -a cut. The red circles are the magnets (see the text), the foil (in brown) is fastened by a steel bolt (in gray).

Table 1 .

 1 Beam energies in [MeV] of U-120M and TR24 cyclotrons (the second row is for TR24) in NPI CAS.

	protons deuterons 3 He	4 He
	6-38	11-20	16-53 22-40
	18-24	-	-	-
	[4]

Table 2 .

 2 Beam energies in [MeV] of LINAC accelerator of GANIL/SPIRAL2.

	protons	deuterons 3 He	4 He
	0.75 -33 1.5 -40	2.25 -72 3 -80
	within a program SPIRAL2-CZ, that is a framework of a
	Czech participation in GANIL/SPIRAL2.

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).


Tomasz Jakubczyk

Miroslav Bartos Cd

Lorenzo Scarpelli

Karol Nogajewski Cf

Wolfgang Langbein

Marek Potemski

Jacek Kasprzak
email: jacek.kasprzak@neel.cnrs.fr

Coherent dynamics of resonantly excited excitons in monolayers of transition metal dichalcogenides

Keywords: coherent nonlinear spectroscopy, transition metal dichalcogenides, exciton dynamics four-wave mixing micro-spectroscopy

We investigate dynamics of resonantly excited excitons in single-layers of MoSe 2 and WS 2 down to 4.5 K. To this end, we measure the delay dependence of the heterodyne four-wave mixing (FWM) amplitude induced by three, short laser pulses. This signal depends not only on the population of optically active excitons, which affects the absorption of the probe, but also on the population of optically inactive states, by interaction-induced energy shift, influencing the refractive index experienced by the probe. As such, it offers insight into density dynamics of excitons which do not directly couple to photons. Reproducing the coherent signal detected in amplitude and phase, the FWM delay dependence is modeled by a coherent superposition of several exponential decay components, with characteristic time constants from 0.1 picosecond up to 1 nanosecond. With increasing excitation intensity and/or temperature, we observe strong interference effects in the FWM field amplitude, resulting in progressively more complex and nonintuitive signal dynamics. We attribute this behaviour to increasingly populated exciton dark states, which change the FWM field phase by the relative effect on absorption and refractive index. We observe that exciton recombination occurs on a significantly longer timescale in WS 2 with respect to MoSe 2 , which is attributed to the dark character of exciton ground state in the former and the bright in the latter.

INTRODUCTION

Carrier relaxation dynamics in semiconductor devices, for example in optical amplifiers, is a relevant issue in modern optoelectronics and has been intensively studied using heterodyne pump-probe spectroscopy . [START_REF] Borri | Heterodyne pump-probe and four-wave mixing in semiconductor optical amplifiers using balanced lock-in detection[END_REF][START_REF] Dommers | Complete ground state gain recovery after ultrashort double pulses in quantum dot based semiconductor optical amplifiers[END_REF][START_REF] Cesari | Refractive index dynamics and linewidth enhancement factor in p-doped InAs/GaAs quantum-dot amplifiers[END_REF][START_REF] Langbein | Ultrafast gain dynamics in InP quantum-dot optical amplifiers[END_REF] Such methods of ultrafast coherent spectroscopy are also employed in fundamental studies of excitons (EX), bound complexes of electrons and holes, hosted by emerging material systems and their nanostructures, such as colloidal quantum dots , 5 nanoplatelets , [START_REF] Naeem | Giant exciton oscillator strength and radiatively limited dephasing in two-dimensional platelets[END_REF] perovskites , [START_REF] Becker | Long exciton dephasing time and coherent phonon coupling in CsPbBr 2 Cl perovskite nanocrystals[END_REF] single-layers (SLs) of semiconducting transition metal dichalcogenides [START_REF] Scarpelli | Resonantly excited exciton dynamics in two-dimensional MoSe 2 monolayers[END_REF] (TMDs) and their van der Waals heterostructures . [START_REF] Jakubczyk | Coherence and density dynamics of excitons in a single-layer MoS 2 reaching the homogeneous limit[END_REF] In the SL limit, TMDs become direct band gap materials, with the gap located at K +/-points of the hexagonal Brillouin zone. The optical response, even at room temperature, is dominated by EXs with a binding energy of several hundreds meV. With such a high binding energy -stemming from the two-dimensional confinement, reduced dielectric screening and high carrier effective masses -light-matter interaction is expected to be enhanced with respect to conventional semiconductors, like GaAs. This is experimentally confirmed by a strong excitonic absorption in TMD SLs of around 10%. Also, they accommodate circularly polarized EXs, where the helicity of the transition (σ + / σ-) is linked with the index (+/-) of the K valley. Thus, apart from spin-allowed (bright) transitions, there are also spin-forbidden (dark) ones, between different valleys. EXs with the in-plane momentum beyond the light cone cannot recombine radiatively either and are also called dark. The multitude of available dark states and their various feeding channels , [START_REF] Moody | Exciton dynamics in monolayer transition metal dichalcogenides[END_REF] as well as the energetic structure varying from material to material, is expected to strongly affect the EX dynamics in the family of TMDs. Time-resolved photoluminescence experiments [START_REF] Robert | Exciton radiative lifetime in transition metal dichalcogenide monolayers[END_REF][START_REF] Godde | Exciton and trion dynamics in atomically thin MoSe 2 and WSe 2 : Effect of localization[END_REF][START_REF] Plechinger | Valley dynamics of excitons in monolayer dichalcogenides[END_REF] performed on TMD SLs have shown bi-exponential or multi-exponential decays on a timescale of a few hundred picosecond. Owing to non-resonant excitation, the modeling and interpretation of photoluminescence is blurred due to the numerous intermediate states in the scattering pathway from the initially excited electron-hole pairs to the emission of the bright EX states. Conversely, resonant driving with a short laser pulse, directly imprints a well controlled density of bright EXs within the light cone into a well defined valley (K + , K -or both).

This can be conveniently accomplished by performing four-wave mixing (FWM) spectroscopy . [START_REF] Langbein | Heterodyne spectral interferometry for multidimensional nonlinear spectroscopy of individual quantum systems[END_REF] The latter was recently employed to investigate dynamics of resonantly driven EX population in MoSe 2 SLs . [START_REF] Scarpelli | Resonantly excited exciton dynamics in two-dimensional MoSe 2 monolayers[END_REF] Considering the excitonic landscape in TMD SLs, these initial FWM studies have established several dominating processes in the EX density dynamics, like direct radiative recombination competing with scattering and evolution of density in dark state reservoirs (out of the light cone direct spin-allowed, direct spin-forbidden, indirect spin-allowed, and indirect spin-forbidden). These experiments were however performed only down to 77 K and measurements at lower temperatures down to 4.2 K have not been carried out. Recently, an example of EX population dynamics measured via FWM was reported and discussed in Ref. [9] using MoS 2 heterostructure.

In this work, we study dynamics of resonantly excited EXs in bare SLs of MoSe 2 and WS 2 down to 4.5 K. We present a comprehensive set of FWM measurements on a timescale from 0.1-1300 ps, combining excitation power (P ) and temperature (T ) dependencies. Increasing P and/or T , the dark states become increasingly more populated, resulting in more and more complex FWM delay traces. In these experiments, we observe prominent interference effects in the FWM amplitude, revealing different scattering channels between several EX reservoirs. We also conclude that exciton relaxation occurs on a significantly longer timescale in WS 2 with respect to MoSe 2 , which is tentatively attributed to dark character of the exciton ground state in WS 2 and the bright in MoSe 2 . Following Ref. [8], the FWM amplitude versus delay is modeled by a coherent superposition of up to five components of complex exponential decays, generating highly non-intuitive delay traces, yet reproducing the ensemble of our measurements very well.

EXPERIMENT

Samples

We here use two kinds of TMD SLs: MoSe 2 and WS 2 , the same as the ones as explored in Ref. [15] and Ref. [16], respectively. In both cases, the SLs were produced by means of polydimethylsiloxane-based exfoliation of bulk crystals, purchased from HQ Graphene, and transferred onto a Si substrate with a 90 nm SiO 2 surface layer and then inserted into an optical He-flow cryostat. In MoSe 2 , the lowest conduction band and the highest valence band have the same spin, therefore the EX with the lowest energy is bright (optically active). It is worth to note that, the next conduction band hosting an electron with the opposite spin, and thus contributing to the dark EX, lies only 1.5 meV away , [START_REF] Lu | Magnetic field mixing and splitting of bright and dark excitons in monolayer MoSe[END_REF] around twenty times closer than previous theoretical predictions . [START_REF] Dery | Polarization analysis of excitons in monolayer and bilayer transition-metal dichalcogenides[END_REF] In WS 2 SLs though the configuration is inverse, i.e., the lowest conduction band and the highest valence band have opposite spin, therefore the lowest lying EX is dark. The next conduction band, sharing the same spin projection as the valence band, is around 30 meV higher, producing the spin-allowed bright exciton. The corresponding energy diagrams for both materials are depicted in Fig. 1 a.

Please verify that (1) all pages are present, (2) all figures are correct, (3) all fonts and special characters are correct, and (4) all text and figures fit within the red margin lines shown on this review document. Complete formatting information is available at http://SPIE.org/manuscripts Return to the Manage Active Submissions page at http://spie.org/submissions/tasks.aspx and approve or disapprove this submission. Your manuscript will not be published without this approval. Please contact author_help@spie.org with any questions or concerns.

11278 -47 V.

Why four-wave mixing microscopy ?

SL-TMDs explored in these experiments have been fabricated as bare flakes deposited directly on SiO 2 /Si substrates. They are in direct contact with the environment, thus are prone to various microscopic and macroscopic disorder factors, such as: strain, wrinkling, flake deformations and cracks, lattice defects and vacancies, variation of charge state and of dielectric constant in the substrate. Their optical response is therefore affected by spectral inhomogeneous broadening , [START_REF] Jakubczyk | Radiatively limited dephasing and exciton dynamics in MoSe 2 monolayers revealed with four-wave mixing microscopy[END_REF][START_REF] Moody | Intrinsic homogeneous linewidth and broadening mechanisms of excitons in monolayer transition metal dichalcogenides[END_REF] σ, which varies across the flake [START_REF] Jakubczyk | Impact of environment on dynamics of exciton complexes in a WS 2 monolayer[END_REF] from nearly zero to a few tens of meV. When examining optical properties of TMDs it is desirable to use a spectroscopic tool capable to separate homogeneous and inhomogeneous contributions within the EX's spectral lineshape. In order to avoid adding up responses from areas exhibiting different optical properties, one would also preferentially employ microscopy, where diffraction limited excitation beams restrict spatial averaging to sub-micron areas.

The heterodyne FWM technique, exploited in this work, complies with these two requirements. FWM is a nonlinear polarization in the material, proportional to µ 4 E 1 E 2 E 3 , where µ is the transition oscillator strength and E 1,2,3 are three femto-second laser pulses. To study EXs in MoSe 2 SL at a light wavelength around 750 nm, we use a Ti:Sapphire laser as pulse source (Tsunami-Femto, from Spectra-Physics). Conversely, for EXs in WS 2 SL at around 590 nm, we use optical parametric oscillator (Inspire, provided by Radiantis). We use a different set of optics adapted for these two wavelengths. The temporal chirp at the sample was pre-compensated using a grating-based pulseshaper to achieve Fourier-limited pulses.

FWM is here detected through optical heterodyning, enabling co-linear propagation of E 1,2,3 . This permits microscopy configuration, which has been required to measure coherent nonlinear response of individual excitons . [START_REF] Kasprzak | Coherent response of individual weakly confined exciton-biexciton systems[END_REF][START_REF] Kasprzak | Vectorial nonlinear coherent response of a strongly confined exciton-biexciton system[END_REF][START_REF] Mermillod | Harvesting, coupling, and control of single-exciton coherences in photonic waveguide antennas[END_REF] First, E 1,2,3 are modulated by distinct radio frequencies Ω 1, 2, 3 around 80 MHz using acousto-optic deflectors. E 1,2,3 propagate co-linearly and are focussed onto the sample down to the diffraction limit using a microscope objective of NA=0.65. In the detection path, the reflected signal is interfered with the reference beam E R modulated at the phase-locked linear combination of radio-frequencies Ω 3 + Ω 2 -Ω 1 , picking up the FWM response from the reflection. Then, background-free interference between E R and FWM is recorded as non-oscillating spectral interference on a CCD camera, which is installed at the output of the imaging spectrometer . [START_REF] Langbein | Heterodyne spectral interferometry for multidimensional nonlinear spectroscopy of individual quantum systems[END_REF] FWM amplitude and phase are obtained via spectral interferometry. Owing to the limited phase stability and lack of phase referencing [START_REF] Delmonte | Coherent coupling of individual quantum dots measured with phasereferenced two-dimensional spectroscopy: Photon echo versus double quantum coherence[END_REF] in the current experiments, only the FWM amplitude is exploited.

As depicted in Fig. 1 b, in a presence of σ, FWM forms a photon echo centered at t = τ 12 , where the latter is the delay between E 1 and E 2 . The time-integrated (TI) FWM amplitude as a function of τ 12 measures the EX coherence dynamics, which in the simple case decays exponentially. From the decay constant, one retrieves the microscopic dephasing time T 2 , which can be translated to the full-with at half-maximum of the homogeneous broadening γ = 2h/T 2 . An example of such a measurement performed on MoSe 2 at T=4.5 K, yielding T 2 0.7 ps, is presented in Fig. 1 b.

There are two principal channels causing EX dephasing: radiative decay (occuring in sub-ps range) and pure dephasing. The latter is due to the EX-EX scattering, such that the phase imprinted onto the EX cloud by E 1,2,3 is rapidly lost. As a result the EX coherence withstands only a few ps. Instead, a part of EX density transferred to the dark states is present and evolves on a much longer timescale; up to at least 1 ns, as presented in Fig. 1 c. To measure EX density dynamics we set τ 12 = 0. E 1 and E 2 induce EX density E 1 E 2 , slowly oscillating at the frequency Ω 2 -Ω 1 . The EX density in different states affects via EX-EX interaction the EX polarization created by E 3 , inducing the detected FWM. The amplitude and phase of the FWM signal is therefore a probe of the EX-EX interaction between the optically probed transition and the EX states containing the density. To rephrase, after the delay τ 23 , the last pulse E 3 arrives and triggers the FWM, which is proportional to the remaining EX density weighted by their mutual interactions. The dynamics of this combined effect is probed by reading out the TI-FWM amplitude versus τ 23 , as depicted in the inset of Fig. 1 c. The figure also shows that, at low T and weak P , more than 90% of the EX population is gone already after only a few ps, owing to the radiative decay and simultaneous scattering to the EX dark states.

Another possible channel leading to the loss of density is the EX diffusion: during τ 23 , EXs can be ejected out of the sub-µm excitation spot . [START_REF] Perea-Causín | Exciton propagation and halo formation in two-dimensional materials[END_REF] We have looked for such propagation by performing spatially-resolved FWM experiments. Namely, by spatially displacing (typically, by several µm) E 1,2 with respect to the probe beam E 3 while detecting FWM, one becomes sensitive to the EX in-plane propagation effects. We have performed numerous experiments in such configurations, providing no evidence for EX diffusion in our samples. In fact, owing to the inhomogeneous broadening in these bare SLs, EXs are expected to be spatially localized and thus exhibiting nearly no diffusion at low temperatures.

The following of this work is devoted to presentation and modeling of such τ 23 -dependences measured at MoSe 2 and WS 2 for different P and T .

Modeling exciton density dynamics measured via heterodyne four-wave mixing.

To describe the loss of EX density when increasing delay τ 23 , the proposed fitting function R is a multi-exponential decay for τ 23 > 0 and zero otherwise: R(τ 23 ) ∝

n i=1 A i θ(τ 23 ) exp (ıφ i -τ23
τi ). Because we employ a coherent detection scheme, it is sensitive to the phase of each of the decay processes. Therefore, R is a complex function, providing characteristic times, amplitudes and phases of n processes that are involved, (τ i , A i , φ i ).

To mimic the experiments, the temporal form of the pulse train, employed to drive FWM, must be taken into account. Firstly, to take into account the finite pulse duration, R is convoluted with a Gaussian characterized by a FWHM of 2 ln(2)τ 0 given by the laser autocorrelation. Secondly, owing to the lase repetition period T R 13 ns, the signal might not fade away completely before the arrival of the following pulse sequence, but the leftover proportional to exp(-T R τi ) is present. When adding up such contribution from the second pulse sequence onward over an infinite number of pulses in the heterodyne detection scheme, the pile-up term is obtained as the geometric sum exp(-

T R τi ) + exp(-2T R τi ) + exp(-3T R τi ) + exp(-4T R τi ) + ... = exp(- T R τ i ) 1-exp (- T R τ i
)

. R therefore takes the form:

Please verify that (1)

R(τ

23 ) = n i=1 A i {[exp( T R τ i ) -1] -1 + 1 2 [(1 + erf( τ 23 τ 0 - τ 0 2τ i )]} × exp (ıφ i - τ 23 τ i + τ 2 0 4τ 2 i ) (1)
Finally, to fit the FWM amplitude we take absolute value of R(τ 23 ), namely |R(τ 23 )|, yielding a rather lengthy, yet still analytical form.

RESULTS

MoSe 2 single layer

In Fig. 2 a we present the EX density dynamics measured on MoSe 2 SL at T = 4.5 K for three different driving powers P , as indicated. We estimate that exciting with 1 µW generates around 10 11 EX/cm 2 . Qualitatively, for the ensemble of our data we distinguish: i) signal at τ 23 < 0 owing to the pile-up effect, ii) when approaching τ 23 = 0 from negative values, we see a rise on a 100 fs scale, due to the finite duration of the laser pulses, iii) strong, initial decay, by around an order of magnitude in the FWM amplitude over a few hundreds fs scale, attributed to the radiative decay and EX scattering to dark EX states. iv) followed after τ 23 > 1 ps by a much longer decay, due to the population of the dark EX states and their subsequent relaxation and interaction with the bright ones, i.e., EXs within the light cone, obeying spin selection rules for radiative recombination.

The dynamics is fitted with |R(τ 23 )| yielding several decay constants from 0.2 ps to 200 ps, as reported in Fig. 2 b. It is worth to point out that at the highest excitation power, after a dominating initial decay, for τ 23 > 10 ps a strong modulation is measured and reproduced by the model. This is due to the interference between signals of different decay processes, whose phases are given by the relative effect on absorption and Please verify that (1) all pages are present, (2) all figures are correct, (3) fonts and special characters are correct, and ( 4) all text and figures fit within the red margin lines shown on this review document. Complete formatting information is available at http://SPIE.org/manuscripts Return to the Manage Active Submissions page at http://spie.org/submissions/tasks.aspx and approve or disapprove this submission. Your manuscript will not be published without this approval. Please contact author_help@spie.org with any questions or concerns.

11278 -47 V. 2 (p.5 of 10) / Color: No / Format: Letter / Date: 2/10/2020 3:59:32 AM SPIE USE: ____ DB Check, ____ Prod Check, Notes: refractive index dynamics. We expect that the phases are largely modified with increasing excitation powers, due to a larger population of dark states reservoir, consistently with our data.

To study the influence of phonon scattering and of thermal distribution across the states, we measure the EX density dynamics for increasing T at fixed P = 0.5 µW, as shown in Fig. 3 a. Note that for each temperature the pulse wavelength was adjusted to match the EX transition, which was monitored with the reflectance of E 1 . Also, the temperature induced spatial drifts were corrected, such that the same sample spot was probed. The dynamics are fitted with |R(τ 23 )| involving five decay components, explicitly given in Fig. 3 b. For the fastest decay, we observe a decrease from around 300 fs down to 100 fs, when increasing T from 5 K to 100 K. While no clear trends are observed for other decay components, the interplay of their relative amplitudes induces important changes in the FWM amplitude for τ 23 > 2 ps. For the lowest temperatures a monotonous decay of the FWM is observed when increasing τ 23 , as exemplified for T = 4.5 K. From T = 50 K onward, a bump starts to build up at τ 23 100 ps. With increasing T , the bump progressively shifts to shorter τ 23 . At T = 100 K, after the initial decay, the FWM amplitude increases, reaches maximum at τ 23 = 20 ps, to finally decay generating another bump at τ 23 = 200 ps. Again, such a surprising form of the τ 23 -delay dependence is due to the interference between different decay processes, building up when increasing T owing to increasingly larger amplitude of processes exhibiting longer decay. This is a strong indication of increased population in a multiplicity of dark EX reservoirs with increasing temperature, mainly enabled by increased phonon scattering at higher temperatures.

WS 2 single layer

We now discuss complementary results obtained on a WS 2 SL. In Fig. 4 a we present the EX density dynamics measured at fixed P = 1.5 µW for different temperatures. At low temperatures a strong pile up effect is observed. Namely, at T = 5 K after a dominating initial decay within 0 < τ 23 < 1 ps, a quasi-stationary signal with a minor modulation at around 100 ps is observed, around an order of magnitude above the noise level. Interestingly, with Please verify that (1) all pages are present, (2) all figures are correct, (3) all fonts and special characters are correct, and (4) all text and figures fit within the red margin lines shown on this review document. Complete formatting information is available at http://SPIE.org/manuscripts Return to the Manage Active Submissions page at http://spie.org/submissions/tasks.aspx and approve or disapprove this submission. Your manuscript will not be published without this approval. Please contact author_help@spie.org with any questions or concerns. increasing temperature, the initial loss of the signal becomes more pronounced, followed by the recovery, which here is most pronounced for T = 80 K. At higher temperatures the recovery at longer τ 23 is suppressed, such that a broad bump is visible at T = 160 K. Qualitatively, we observe shifting of modulations to shorter delays with increasing T , as indicated with red and blue arrows.

The data are again modeled with the |R(τ 23 )| function containing four components. The fitted decay constants are plotted in Fig. 4 b, from which we conclude that with increasing T the dynamics strongly accelerates: when increasing T , values of τ 1 , τ 2 and τ 3 decrease. To interpret the data we remind that the bright EX in WS 2 is located around 32 meV above the optically inactive (spin-forbidden) one. The initial decay corresponds to the radiative decay and the scattering to dark states, also toward the optically inactive EX. The lower the temperature, the more efficiently EX population gets trapped in this state. In particular, at T = 5 K the Boltzmann factor exp(-32meV/0.43meV)=exp(-74.4) entirely isolates the population from the thermal activation toward the bright EX. Alternative scattering channels are required to bring the population back to the bright EX state. Such a long relaxation time explains a very strong pile up effect observed at low temperatures. With increasing T the thermal scattering from the spin-forbidden to spin-allowed EX state becomes relevant, for example at T = 160 K the Boltzmann factor rises to exp(-32meV/13.7meV) 0.1, enabling faster relaxation processes. This results in a faster decay and negligible pile-up at high temperatures, consistent with our findings.

Finally, we note that even at elevated temperatures, accumulation of the EX density for long τ 23 and the pile up effect can be restored by increasing P and thus occupancy of the dark EX states. This can be concluded from Fig. 5, showing intensity dependence of the FWM delay traces for different temperatures.

CONCLUSIONS

To conclude, using heterodyne FWM micro-spectroscopy, we measured temporal dynamics of EXs from 0.1 ps to 1.3 ns in MoSe 2 and WS 2 SLs after resonant excitation with a short laser pulse. The rich data set, including Please verify that (1) all pages are present, (2) all figures are correct, (3) all fonts and special characters are correct, and (4) all text and figures fit within the red margin lines shown on this review document. Complete formatting information is available at http://SPIE.org/manuscripts Return to the Manage Active Submissions page at http://spie.org/submissions/tasks.aspx and approve or disapprove this submission. Your manuscript will not be published without this approval. Please contact author_help@spie.org with any questions or concerns. excitation intensity and temperature dependence for both materials, is consistently modeled with a coherent superposition of several exponential decays. They are attributed to direct radiative decay, EX scattering out of the light cone, and spin-flip processes between spin-allowed and spin-forbidden EXs. Owing to the complex character of the data, unconventional interference effects arise in the FWM delay dependence, when different EX reservoirs display comparable occupancy. This permits to gain direct insight into interaction of bright excitons with the dark ones. As an outlook, we propose to use this approach to measure EX radiative lifetimes and evolution of EX dark states in van der Walls heterostructures with engineered thicknesses of encapsulating layers . [START_REF] Fang | Control of the exciton radiative lifetime in van der waals heterostructures[END_REF]

Figure 1 .

 1 Figure1. Direct exciton transitions in single-layers of transition metal dichalcogenides and their observables probed with four-wave mixing spectroscopy. a) Schema of band structure of TMD SLs, showing the highest lying valence band and the lowest lying pair of conduction bands, at the K + and K -points. Owing to the reverse energy splitting, in MoSe2 (WS2) the transition of the lowest energy is optically (in)active, bright (dark). b, c) Representative measurements on MoSe2 SL at T=4.5K. b) Time-integrated FWM amplitude as a function of delay τ12, for τ23=0, as depicted in the inset, reflecting EX coherence dynamics. c) Time-integrated FWM amplitude as a function of delay τ23, for τ12 = 0, as depicted in the inset, reflecting EX population dynamics.

  e n s i t y d e p e n d e n c e a t T = 4 . 5 K (σ,σ,σ,σ) a y ( p s ) e x c i t a t i o n ( n W )

Figure 2 .

 2 Figure2. Exciton density dynamics in MoSe2 SL measured with four-wave mixing. Excitation power P as indicated at T = 4.5 K. For the highest P = 1200 nW an oscillation observed between 10 ps and 1000 ps is due to the interference between different decay components, enabled by increased amplitude of the ones featuring longer decay constants. In all figures, fits are given with semi-transparent traces.

Figure 3 .

 3 Figure 3. Exciton density dynamics in MoSe2 SL measured with four-wave mixing. Temperature dependence, as indicated, at fixed excitation power P = 0.5 µW. With increasing temperature unusual shapes of the delay traces are measured and reproduced by the modeling. In the fit we observe that the interference sets in when the components showing a longer decay increase their relative amplitudes.

  11278 -47 V. 2 (p.6 of 10) / Color: No / Format: Letter / Date: 2/10/2020 3:59:32 AM SPIE USE: ____ DB Check, ____ Prod Check, Notes: a y c o n s t a n t ( p s ) T e m p e r a t u r e ( K )

Figure 4 .

 4 Figure 4. Exciton density dynamics in WS2 SL measured with four-wave mixing. Temperature dependence, as indicated, at fixed excitation power P = 1.5 µW. The modeling indicates shortening of the second and the third decay constants with increasing temperature. It is visible in the data as a shift of the bumps towards shorter delays τ23 when increasing the temperature.

  11278 -47 V. 2 (p.7 of 10) / Color: No / Format: Letter / Date: 2/10/2020 3:59:32 AM SPIE USE: ____ DB Check, ____ Prod Check, Notes:

Figure 5 .

 5 Figure 5. Exciton density dynamics in WS2 SL measured with four-wave mixing. Power dependence measured at different temperatures as indicated. The used excitation powers are the following: a) (0.2, 0.4, 0.8, 2.7, 5.2) µW, b) (0.08, 0.4, 1.6) µW, c) (0.08, 0.4, 1.5) µW, d) (0.08, 0.4, 1.5) µW, e) (0.4, 1.5, 3.6, 7.5) µW, f) 6 µW.

  all pages are present,(2) all figures are correct, (3) all fonts and special characters are correct, and (4) all text and figures fit within the red margin lines shown on this review document. Complete formatting information is available at http://SPIE.org/manuscripts Return to the Manage Active Submissions page at http://spie.org/submissions/tasks.aspx and approve or disapprove this submission. Your manuscript will not be published without this approval. Please contact author_help@spie.org with any questions or concerns.

11278 -47 V. 2 (p.4 of 10) / Color: No / Format: Letter / Date: 2/10/2020 3:59:32 AM SPIE USE: ____ DB Check, ____ Prod Check, Notes:

11278 -47 V. 2 (p.9 of 10) / Color: No / Format: Letter / Date: 2/10/2020 3:59:32 AM SPIE USE: ____ DB Check, ____ Prod Check, Notes:


T Carpi

Y Lefèvre

C Hénaux

J F Llibre

D Harribey

3D Hybrid Model of the Axial Flux Motor Accounting Magnet Shape

Keywords: Axial flux, finite difference method, Fourier series, magnetic scalar potential, magnet shape, permanent magnet, separation of variables

This paper presents a generalization of an analytical model of an axial flux permanent magnet machine to any magnet shape. It uses an existing model which computes the 3D magnetic flux density by the separation of variables and finite difference method. The original magnet shape is modified by adding a radial dependency to the Fourier series description of permanent magnets magnetization. The model is then developed for a general complex magnet shape. As an example, the model will be computed for a circular magnet shape and will be compared to a finite element analysis.

I. INTRODUCTION

HE structures of Axial Flux Permanent Magnet (AFPM) machine structures are still under development [START_REF] Shokri | Comparison of performance characteristics of axial-flux permanent magnet synchronous machine with different magnet shapes[END_REF]. Thus, modeling some of their particularities is becoming an issue. In axial flux surface mounted permanent magnet machines, permanent magnets are often considered as sector shaped magnets. However, others magnet shapes can be found in some AFPM structures [START_REF] Shokri | Comparison of performance characteristics of axial-flux permanent magnet synchronous machine with different magnet shapes[END_REF], [START_REF] Pahlavani | Minimisation of torque ripple in slotless axial flux BLDC motors in terms of design considerations[END_REF], [START_REF] Gulec | Magnet asymmetry in reduction of cogging torque for integer slot axial flux permanent magnet motors[END_REF]. Nevertheless, considering 3D analytical modeling, despite the variety of the methods used, only sector shaped magnets have been considered [START_REF] Huang | 3-D analytical modeling of no-load magnetic field of ironless axial flux permanent magnet machine[END_REF], [START_REF] Jin | 3-D analytical magnetic field analysis of axial flux permanent magnet machine[END_REF], [START_REF] Carpi | Hybrid Modeling Method of Magnetic Field of Axial Flux Permanent Magnet Machine[END_REF].

The method developed in [START_REF] Carpi | Hybrid Modeling Method of Magnetic Field of Axial Flux Permanent Magnet Machine[END_REF] consists in a combined resolution with analytical and finite difference method (FDM). The resolution is based on the image method where the geometry is repeated infinitely in the axial direction [START_REF] Jin | 3-D analytical magnetic field analysis of axial flux permanent magnet machine[END_REF]. This way, the problem can be described by a double Fourier series in the axial and angular directions. The Laplacian is then solved by separation of variables. However, Bessel functions given for the radial solutions are not chosen because of their complexity. Instead, the r dependent function is kept unknown and solved by a 1D FDM.

This model is adaptable to any magnets distributions thanks to the Fourier series description. It is also adaptable to multistage machines thanks to the image method. Nevertheless, only sector shaped magnets can be modeled by this method. Benefiting from the FDM in the radial direction, this paper proposes to extend the model to any magnet shapes by modifying the Fourier series description for each discretized radius. Subsequently, the solution will be computed for circular magnet shape and compared to finite element analysis (FEA).

II. GENERALIZATION OF THE SHAPE

A single sided AFPM machine with axially magnetized surface mounted permanent magnets is considered in this paper.

As in [START_REF] Carpi | Hybrid Modeling Method of Magnetic Field of Axial Flux Permanent Magnet Machine[END_REF], the following assumptions are made:

-Because of the air-space between the magnets, we assume that the permeability of magnets and the air is the same and equal to µ 0 .

-Back-irons have infinite permeability so the boundary conditions (BC) at the planes z = 0 and z = h m + g is taken as normal flux boundary conditions. Where g is the airgap width and h m the permanent magnet width.

-The problem is limited in the radial direction with parallel flux boundary conditions on cylinders at r = R 0 and r = R 1 .

Using magnetic scalar potential formulation (MSP), Ω, the partial differential equation to be solved is deduced from Maxwell equations:

∆Ω = 𝑑𝑖𝑣 𝑴 (1)
with M the magnetization vector.

To reduce the number of regions to consider, the image method is used to replace the normal flux BC by a periodical extension in the axial direction [START_REF] Carpi | Hybrid Modeling Method of Magnetic Field of Axial Flux Permanent Magnet Machine[END_REF]. This leads to a double Fourier series description of the magnetization of the permanent magnets in the azimuthal θ and axial z directions.

A complex magnet shape is considered in Fig. 1. To be able to compute this magnet shape, the Fourier series description has to be modified depending on the radial position. The parameters necessary to characterize any shape are the arc pole α p and the phase β between the origin and the center position of the angular opening as shown in Fig. 1. Thus, a general Fourier series describing the magnetization M z of any magnet shape can be expressed as follows.

𝑀 𝑧 (𝑟, 𝜃, 𝑧) = ∑ 𝑛 ∑ 𝑀 𝑛𝑘 (𝑟) cos(𝑛𝑝𝜃 + 𝛽(𝑟)) cos ( 𝑘𝜋 𝜏 𝑧) 𝑘 ( 2
)
where p is the number of pole pairs, τ = h m +g is the half period of the magnetization in the z-direction. The coefficients M nk have to be determined in function of α p (r) and the phase β(r) which represents purely the asymmetrical feature of the magnet shape. Fig. 2 shows the angular and axial dependency of the magnetization for a given radius. The image method repeats the geometry in the axial direction, so M z (z) is not impacted by the magnet shape.

III. HYBRID MODELING METHOD

Thanks to the double Fourier series description, there are three regions to be considered separated by cylindrical surfaces at r = R int and r = R ext . Air regions I (R int ≥ r ≥ R 0 ) and III (R 1 ≥ r ≥ R ext ), and the PM region II (R ext ≥ r ≥ R int ). The new magnet shape must be included in the magnet region between R int and R ext . In order to make the solution easier to handle, the Fourier series of the magnetization (2) is written in terms of cosine and sine functions.

All the magnets are axially magnetized. Therefore, the magnetization 𝑴 has only a component in the axial direction. The second member of the equation ( 1) is reduced to: The separation of variables consists in assuming that the solution Ω is the product of three functions f, g and h: Ω(𝑟, 𝜃, 𝑧) = 𝑓(𝑟). 𝑔(𝜃). ℎ(𝑧)

(5)

Combining ( 1) and ( 5) leads to three ordinary differential equations for each function f, g and h. The solutions for f are Bessel functions and cosine and sine functions both for h and g. To avoid using Bessel functions, f is kept unknown and will be determined by FD method. The principle of superposition allows writing the solution as follows:

Ω(𝑟, 𝜃, 𝑧) = ∑ ∞ 𝑛 ∑ 𝜔 𝑐 𝑛𝑘 (𝑟) cos(𝑛𝑝𝜃) sin ( 𝑘𝜋 𝜏 𝑧) ∞ 𝑘 + ∑ ∞ 𝑛 ∑ 𝜔 𝑠 𝑛𝑘 (𝑟) sin(𝑛𝑝𝜃) sin ( 𝑘𝜋 𝜏 𝑧) ∞ 𝑘 ( 6
)
The function f is now replaced by ω cnk and ω snk which depend on the harmonic rank, np and kπ/τ are respectively the periodicities in the angular and axial direction. Thus, inserting [START_REF] Carpi | Hybrid Modeling Method of Magnetic Field of Axial Flux Permanent Magnet Machine[END_REF] in the partial differential equation (1) yields to three ordinary differential equations in each region I, II and III over the new unknowns ω cnk and ω snk . There are n x k equations to be solved in each region and for each cosine and sine component.

{ 𝑑

The boundary conditions ( 9) and (10) over the cylinders r = R 0 and r = R 1 are taken as parallel flux so that no flux goes out of the region:

𝐵 𝑟 𝐼 | 𝑟=𝑅 0 = 0 (9) 𝐵 𝑟 𝐼𝐼𝐼 | 𝑟=𝑅 1 = 0 ( 10
)
The interface conditions (11) to (16) between the permanent magnet region II and the air regions (I and III) over the cylinders r = R int and r = R ext are the continuity of the normal component of the magnet flux density B n and the continuity of the tangential component of the magnetic field density H t : Continuity of the tangential component of the magnetic field density H t leads to the continuity of the functions ω cnk and ω snk , while the continuity of the normal component of the magnetic flux density B r yields to the continuity of ∂ω cnk /∂r and ∂ω snk /∂r.

𝐵 𝑟 𝐼 | 𝑟=𝑅 𝑖𝑛𝑡 =
Adding the interface and boundary conditions bring about a matrix system for each harmonic rank gathering all equations mentioned before:

𝐴 𝑐 𝑛𝑘 . 𝑣 𝑐 𝑛𝑘 = 𝐵 𝑐 𝑛𝑘 (17) 𝐴 𝑠 𝑛𝑘 . 𝑣 𝑠 𝑛𝑘 = 𝐵 𝑠 𝑛𝑘 ( 18
)
The final expression of the axial magnetic flux density is deduced from the magnetic scalar potential solved by FD method: where v nk are functions of 𝑟. Differential equations are solved by FD method for each azimuthal n and axial k harmonics. The 1D FD method discretizes the problem in the radial direction and is continuous in the angular and axial directions thanks to the Fourier series description. The general method for any magnet shape was presented. In the next section, this model is applied to cylindrical magnet shape as an example.

𝐵 𝑧 = -𝜇 0 [∑ ∞ 𝑛 ∑ 𝑣 𝑐

IV. CIRCULAR MAGNET SHAPE

The model is applied to an AFPM machine with cylindrical magnets as shown in Fig. 4 and its parameters are given in Table 1. This magnet shape is often used in AFPM machines [START_REF] Shokri | Comparison of performance characteristics of axial-flux permanent magnet synchronous machine with different magnet shapes[END_REF]. Thanks to the symmetrical shape of this type of magnet, the phase β(r)=0 and the Fourier series of the magnetization M z is reduced to (20).

For each discretized radius r, the arc pole is calculated in order to create a circular shape.The arc pole expression is deduced from Fig. 5 (a):

𝛼 𝑝 (𝑟) = (𝑎𝑐𝑜𝑠 𝑅 𝑐 2 + 𝑟 2 -𝑅 2 2. 𝑅 𝑐 . 𝑟 ) 𝜋 2𝑝 ⁄ (23)
As seen in Fig. 4, R c is the radial position of the center of the magnet and R the radius of the circular magnet. This allows creating an exact circular shape from the initial sector shape. The arc pole function is plotted in Fig. 5 (b).

The FEA is carried out on ANSYS/Emag 3D [START_REF]ANSYS Mechanical APDL Low Frequency Electromagnetic Analysis 215 Guide[END_REF] and based on a magnetic scalar potential formulation. The FEA is done under the same condition as the model, that means on one pair of poles of the machine. Also, the same assumptions are made: the permeability of the magnets and BC.

To validate the fact that the magnet shape has changed, both computation methods are compared on a radial line at the middle of the airgap z = h m + g/2 and for several angles θ = 0° (in front of the symmetrical axis), θ = 5.5° and θ = 7.5°.

The results are computed for 16 harmonics. The root mean square (RMS) error between the hybrid model and the FEA on Fig. 6 are about 1.2% for the three plots. The RMS errors are about 1.2% and 0.6% if we consider the influence of θ and z independently.

Back electromotive force and torque can be easily computed from the B z component of the magnetic flux density [START_REF] Carpi | Hybrid Modeling Method of Magnetic Field of Axial Flux Permanent Magnet Machine[END_REF]. If the field created by eddy currents may be neglected, eddy current losses in the magnets can be estimated. The armature reaction field must be computed. The laplacian can be solved in 2D (𝜃, 𝑧) using a MSP formulation [START_REF] Hemeida | Analytical modeling of eddy current losses in Axial Flux PMSM using resistance network[END_REF]. Removing the magnets from this study and adding the boundary condition 𝐻 𝑡 = 𝐾 at the interface with the stator (𝑧 = ℎ 𝑚 + 𝑔), where 𝐾 is the current sheet modeling the winding distribution, allows to solve the problem. Then, the same method used to calculate eddy current losses in [START_REF] Hemeida | Analytical modeling of eddy current losses in Axial Flux PMSM using resistance network[END_REF] can be applied.

V. ACCURACY AND COMPUTATION TIME

The results of the hybrid model presented Fig. 6 are computed for 1) FDM part of the model: a radial discretization of 141 points and n and k equals 16 harmonics 2) analytical part of the model: 91 points and 221 points respectively for the angular and axial discretization. The total number of nodes is then 2,087,787 in the volume considered.

The computer used for the simulation is an Intel Core i7 at 1.9 GHz and 32 Go RAM. The computation time for 16 harmonics is 40 seconds to generate the entire solution. Nevertheless, the purpose here was to validate the model, so the computation time might be optimized. The addition of the radial dependency of the arc pole includes a loop into several initial nested loops, generating an increase of the computation time. It is approximately 4 times bigger than the one in [START_REF] Carpi | Hybrid Modeling Method of Magnetic Field of Axial Flux Permanent Magnet Machine[END_REF], but remains much less than a FEA that took 74 seconds for 83,302 nodes within the same volume. Fig. 7 presents the accuracy of the model and the computation time with respect to the number of harmonics. The RMS error mentioned above are the errors in the rdirection for z = h m + g/2 and θ = 0, θ-direction for r = R int + 0.8(R ext -R int ) and z = h m + g/2 and z-direction for r = R int + 0.8(R ext -R int ) and θ = 0. Therefore, a good accuracy can be reached with a small number of harmonics, and the computation time is convenient to include the model into optimization studies for example. Thus, this model presents the advantage of being faster, easy to set up, and offering larger perspectives over the usual FEA.

VI. CONCLUSION

This paper presents a generalization of a 3D analytical model of AFPM with sector shaped magnets to AFPM with more complex magnet shapes. The method consists in including a radial dependency to the Fourier series description of the magnetization to be able to compute any magnets shape. The model was developed for a general complex magnet shape and the specific case of circular magnet shape was taken as an example, the method was validated by comparison with FEA.

The initial model is suitable to model a lot of particularities that can be found in axial flux machines such as multistage machines, different winding distributions and different magnets distributions. This paper extends the model to possibly any magnets shapes. Thus, this paper provides a method to model a lot of particularities of AFPM machines that usually require FEA. This allows further optimization studies over the magnet shape.

Fig. 1 .

 1 Fig. 1. Complex magnet shapes with the arc pole and the phase depending on radial position.

Fig. 2 .

 2 Fig. 2. Angular (a) and axial (b) dependency of the magnetization.

Fig. 3 .

 3 Fig. 3. Representation of the different regions considered in the problem.

  𝑀 𝑧 (𝑟, 𝜃, 𝑧) = ∑ 𝑛 ∑ 𝑀 𝑐 𝑛𝑘 (𝑟) cos(𝑛𝑝𝜃) cos (

Fig. 4 . 3 -

 43 Fig. 4. 3-D view of the AFPM machine with circular magnets.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Modification of magnets shape (a) and arc pole in function of the radius (b).

Fig. 7 .

 7 Fig. 7. Accuracy and computation time in function of the number of harmonics.

  This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMAG.2019.2950892, IEEE Transactions on Magnetics

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  2 𝜔 𝑐 𝑛𝑘

		𝐼 𝑑𝑟 2 +	1 𝑟	𝑑𝜔 𝑐 𝑛𝑘 𝑑𝑟	𝐼	-(	𝑘 2 𝜋 2 𝜏 2 +	𝑛 2 𝑝 2 𝑟 2 ) 𝜔 𝑐 𝑛𝑘	𝐼 = 0
	𝑑 2 𝜔 𝑐 𝑛𝑘 𝑑𝑟 2 + 𝐼𝐼	1 𝑟	𝑑𝜔 𝑐 𝑛𝑘 𝑑𝑟	𝐼𝐼	-(	𝑘 2 𝜋 2 𝜏 2 +	𝑛 2 𝑝 2 𝑟 2 ) 𝜔 𝑐 𝑛𝑘	𝐼𝐼 = -𝑀 𝑐 𝑛𝑘	𝑘𝜋 𝜏	(7)
		𝑑 2 𝜔 𝑐 𝑛𝑘 𝑑𝑟 2	𝐼𝐼𝐼	+	1 𝑟	𝑑𝜔 𝑐 𝑛𝑘 𝑑𝑟	𝐼𝐼𝐼	-(	𝑘 2 𝜋 2 𝜏 2 +	𝑛 2 𝑝 2 𝑟 2 ) 𝜔 𝑐 𝑛𝑘	𝐼𝐼𝐼 = 0
		𝑑 2 𝜔 𝑠 𝑛𝑘 𝑑𝑟 2 + 𝐼	1 𝑟	𝑑𝜔 𝑠 𝑛𝑘 𝑑𝑟	𝐼	-(	𝑘 2 𝜋 2 𝜏 2 +	𝑛 2 𝑝 2 𝑟 2 ) 𝜔 𝑠 𝑛𝑘	𝐼 = 0
	𝑑 2 𝜔 𝑠 𝑛𝑘 𝑑𝑟 2 + 𝐼𝐼	1 𝑟	𝑑𝜔 𝑠 𝑛𝑘 𝑑𝑟	𝐼𝐼	-(	𝑘 2 𝜋 2 𝜏 2 +	𝑛 2 𝑝 2 𝑟 2 ) 𝜔 𝑠 𝑛𝑘	𝐼𝐼 = -𝑀 𝑠 𝑛𝑘	𝑘𝜋 𝜏	(8)
	{	𝑑 2 𝜔 𝑠 𝑛𝑘 𝑑𝑟 2	𝐼𝐼𝐼	+	1 𝑟	𝑑𝜔 𝑠 𝑛𝑘 𝑑𝑟	𝐼𝐼𝐼	-(	𝑘 2 𝜋 2 𝜏 2 +	𝑛 2 𝑝 2 𝑟 2 ) 𝜔 𝑠 𝑛𝑘	𝐼𝐼𝐼 = 0

  This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMAG.2019.2950892, IEEE Transactions on Magnetics

	𝐵 𝑟 𝐼𝐼 | 𝑟=𝑅 𝑒𝑥𝑡 =𝐵 𝑟 𝐼𝐼𝐼 | 𝑟=𝑅 𝑒𝑥𝑡	(14)
	𝐻 𝜃 𝐼 | 𝑟=𝑅 𝑒𝑥𝑡 =𝐻 𝜃 𝐼𝐼 | 𝑟=𝑅 𝑒𝑥𝑡	(15)
	𝐻 𝑧 𝐼 | 𝑟=𝑅 𝑒𝑥𝑡 =𝐻 𝑧 𝐼𝐼 | 𝑟=𝑅 𝑒𝑥𝑡	(16)
		𝐵 𝑟 𝐼𝐼 | 𝑟=𝑅 𝑖𝑛𝑡	(11)
		𝐻 𝜃 𝐼 | 𝑟=𝑅 𝑖𝑛𝑡 =𝐻 𝜃 𝐼𝐼 | 𝑟=𝑅 𝑖𝑛𝑡	(12)
		𝐻 𝑧 𝐼 | 𝑟=𝑅 𝑖𝑛𝑡 =𝐻 𝑧 𝐼𝐼 | 𝑟=𝑅 𝑖𝑛𝑡	(13)

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