Bob L Sturm

Rolf Bardeli

Thibault Langlois

Valentin Emiya

FORMALIZING THE PROBLEM OF MUSIC DESCRIPTION

The lack of a formalism for "the problem of music description" results in, among other things: ambiguity in what problem a music description system must address, how it should be evaluated, what criteria define its success, and the paradox that a music description system can reproduce the "ground truth" of a music dataset without attending to the music it contains. To address these issues, we formalize the problem of music description such that all elements of an instance of it are made explicit. This can thus inform the building of a system, and how it should be evaluated in a meaningful way. We provide illustrations of this formalism applied to three examples drawn from the literature.

INTRODUCTION

Before one can address a problem with an algorithm (a finite series of well-defined operations that transduce a wellspecified input into a well-specified output) one needs to define and decompose that problem in a way that is compatible with the formal nature of algorithms [START_REF] Sedgewick | Algorithms[END_REF]. A very simple example is the problem of adding any two positive integers. Addressing this problem with an algorithm entails defining the entity "positive integer", the function "adding", and then producing a finite series of well-defined operations that applies the function to an input of two positive integers to output the correct positive integer.

A more complex example is "the problem of music description." While much work in music information retrieval (MIR) has proposed systems to attempt to address the problem of music description [START_REF] Bertin-Mahieux | Automatic tagging of audio: The state-of-the-art[END_REF][START_REF] Fu | A survey of audio-based music classification and annotation[END_REF][START_REF] Yang | Music Emotion Recognition[END_REF], and much work attempts to evaluate the capacity of these systems for addressing that problem [START_REF] Stephen Downie | Ten years of ISMIR: Reflections on challenges and opportunities[END_REF][START_REF] Sturm | Classification accuracy is not enough: On the evaluation of music genre recognition systems[END_REF], we have yet to find any work that actually defines it. (The closest we have found is that of [START_REF] Turnbull | Semantic annotation and retrieval of music and sound effects[END_REF].) Instead, there are many allusions to the problem: predict the "genre" of a piece of recorded music [START_REF] Tzanetakis | Musical genre classification of audio signals[END_REF]; label music with "useful tags" [START_REF] Aucouturier | Introduction -from genres to tags: A little epistemology of music information retrieval research[END_REF]; predict what a listener will "feel" when "listening" to some music [START_REF] Yang | Music Emotion Recognition[END_REF]; find music "similar" to some other music [START_REF] Urbano | Evaluation in Audio Music Similarity[END_REF]. These allusions are deceptively simple, however, since behind them lie many c Bob L. Sturm problems and questions that have major repercussions on the design and evaluation of any proposed system. For example, What is "genre"? What is "useful"? How is "feeling" related to "listening"? "Similar" in what respects?

With respect to the problem of music description, some work in MIR discusses the meaningfulness, worth, and futility of designing artificial systems to describe music [START_REF] Wiggins | Semantic gap?? Schemantic schmap!! Methodological considerations in the scientific study of music[END_REF]; the idea of and the difficulty in "ground truth" [START_REF] Barthet | Multidisciplinary perspectives on music emotion recognition: Implications for content and context-based models[END_REF][START_REF] Craft | How many beans make five? The consensus problem in musicgenre classification and a new evaluation method for single-genre categorisation systems[END_REF][START_REF] Law | Tagatune: A game for music and sound annotation[END_REF]; the size of datasets [START_REF] Bertin-Mahieux | The million song dataset[END_REF], a lack of statistics [START_REF] Flexer | Statistical evaluation of music information retrieval experiments[END_REF], the existence of bias [START_REF] Pampalk | Improvements of audio-based music similarity and genre classification[END_REF], and the ways such systems are evaluated [START_REF] Sturm | Making explicit the formalism underlying evaluation in music information retrieval research: A look at the MIREX automatic mood classification task[END_REF][START_REF] Sturm | The state of the art ten years after a state of the art: Future research in music information retrieval[END_REF][START_REF] Urbano | Evaluation in music information retrieval[END_REF]. Since a foundational goal of MIR is to develop systems that can imitate the human ability to describe music, these discussions are necessary. However, what remains missing is a formal definition of the problem of music description such that it can be addressed by algorithms, and relevant and valid evaluations can be designed.

In this work, we formalize the problem of music description and try to avoid ambiguity arising from semantics. This leads to a rather abstract form, and so we illustrate its aspects using examples from the literature. The most practical benefit of our formalization is a specification of all elements that should be explicitly defined when addressing an instance of the problem of music description.

FORMALISM

We start our formalization by defining the domain of the problem of music description. In particular, we discriminate between the music that is to be described and a recording of it since the former is intangible and the latter is data that a system can analyze. We then define the problem of music description, a recorded music description system (RMDS), and the analysis of such a system. This leads to the central role of the use case.

Domain

Denote a music universe, Ω, a set of music, e.g., Vivaldi's "The Four Seasons", the piano part of Gershwin's "Rhapsody in Blue", and the first few measures of the first movement of Beethoven's Fifth Symphony. A member of Ω is intangible. One cannot hear, see or point to any member of Ω; but one can hear a performance of Vivaldi's "The Four Seasons", read sheet music notating the piano part of Gershwin's "Rhapsody in Blue", and point to a printed score of Beethoven's Fifth Symphony. Likewise, a recorded performance of Vivaldi's "The Four Seasons" is not Vivaldi's "The Four Seasons", and sheet music notating the piano part of Gershwin's "Rhapsody in Blue" is not the piano part of Gershwin's "Rhapsody in Blue".

In the tangible world, there may exist tangible recordings of the members of Ω. Denote the tangible music recording universe by R Ω . A member of R Ω is a recording of an element of ω ∈ Ω. A recording is a tangible object, such as a printed CD or score. Denote one recording of ω ∈ Ω as r ω ∈ R Ω . There might be many recordings of an ω in R Ω . We say the music ω is embedded in r ω ; it enables for a listener an indirect sense of ω. For instance, one can hear a live or recorded performance of ω, and one can read a printed score of ω. The acknowledgment of and distinction between intangible music and tangible recordings of music is essential since systems cannot work with intangible music, but only tangible recordings.

Music Description and the Use Case

Denote a vocabulary, V, a set of symbols or tokens, e.g., "Baroque", "piano", "knock knock", scores employing common practice notation, the set of real numbers R, other music recordings, and so on. Define the semantic universe as

S V,A := {s = (v 1 , . . . , v n )|n ∈ N, ∀1 ≤ i ≤ n[v i ∈ V] ∧ A(s)} (1)
where A(•) encompasses a semantic rule, for instance, restricting S V,A to consist of sequences of cardinality 1. Note that the description s is a sequence, and not a vector or a set. This permits descriptions that are, e.g., time-dependent, such as envelopes, if V and A(•) permit it. In that case, the order of elements in s could be alternating time values with envelope values. Descriptions could also be timefrequency dependent.

We define music description as pairing an element of Ω or R Ω with an element of S V,A . The problem of music description is to make the pairing acceptable with respect to a use case. A use case provides specifications of Ω and R Ω , V and A(•), and success criteria. Success criteria describe how music or a music recording should be paired with an element of the semantic universe, which may involve the sanity of the decision (e.g., tempo estimation must be based on the frequency of onsets), the efficiency of the decision (e.g., pairing must be produced under 100 ms with less than 10 MB of memory), or other considerations.

To make this clearer, consider the following use case. The music universe Ω consists of performances by Buckwheat Zydeco, movements of Vivaldi's "The Four Seasons", and traditional Beijing opera. The tangible music recording universe R Ω consists of all possible 30-second digital audio recordings of the elements in Ω. Let the vocabulary V = {"Blues", "Classical"}; and define A(s) := [|s| ∈ {0, 1}]. The semantic universe is thus, S V,A = {(), ("Blues"), ("Classical")}. There are many possible success criteria. One is to map all recordings of Buckwheat Zydeco to "Blues", map all recordings of Vivaldi's "The Four Seasons" to "Classical", and map all recordings of traditional Beijing opera to neither. Another is to map no recordings of Buckwheat Zydeco and Vivaldi's "The Four Seasons" to the empty sequence, and to map any recording of traditional Beijing opera to either non-empty sequence with a probability less than 0.1.

Recorded Music Description Systems

A recorded music description system (RMDS) is a map from the tangible music recording universe to the semantic universe:

S : R Ω → S V,A . (2)
Building an RMDS means making a map according to wellspecified criteria, e.g., using expert domain knowledge, automatic methods of supervised learning, and a combination of these. An instance of an RMDS is a specific map that is already built, and consists of four kinds of components [START_REF] Sturm | Making explicit the formalism underlying evaluation in music information retrieval research: A look at the MIREX automatic mood classification task[END_REF]: algorithmic (e.g., feature extraction, classification, pre-processing), instruction (e.g., description of R Ω and S V,A ), operator(s) (e.g., the one inputting data and interpreting output), and environmental (e.g., connections between components, training datasets). It is important to note that S is not restricted to map any recording to a single element of V. Depending on V and A(•), S V,A could consist of sequences of scalars and vectors, sets and sequences, functions, combinations of all these, and so on. S could thus map a recording to many elements of V.

One algorithmic component of an RMDS is a feature extraction algorithm, which we define as

E : R Ω → S F,A ′ (3)
i.e., a map from R Ω to a semantic universe built from the vocabulary of a feature space F and semantic rule A ′ (•).

For instance, if

F := C M , M ∈ N, and A ′ (s) := [|s| = 1],
then the feature extraction maps a recording to a single M -dimensional complex vector. Examples of such a map are the discrete Fourier transform, or a stacked series of vectors of statistics of Mel frequency cepstral coefficients.

Another algorithmic component of an RMDS is a classification algorithm, which we define:

C : S F,A ′ → S V,A (4)
i.e., a map from one semantic universe to another. Examples of such a map are k-nearest neighbor, maximum likelihood, support vector machine, and a decision tree.

To make this clearer, consider the RMDS named "RT GS" built by Tzanetakis and Cook [START_REF] Tzanetakis | Musical genre classification of audio signals[END_REF]. E maps sampled audio signals of about 30-s duration to S F,A ′ , defined by single 19-dimensional vectors, where one dimension is spectral centroid mean, another is spectral centroid variance, and so on. C maps S F,A ′ to S V,A , which is defined by V = {"Blues", "Classical", "Country", "Disco", "Hip hop", "Jazz", "Metal", "Pop", "Reggae", "Rock"}, and A(s) := [|s| = 1]. This mapping involves maximizing the likelihood of an element of S F,A ′ among ten multivariate Gaussian models created with supervised learning.

Supervised learning involves automatically building components of an S, or defining E and C, given a training recorded music dataset: a sequence of tuples of recordings sampled from R Ω and elements of S V,A , i.e.,

D := {(r i , s i ) ∈ R Ω × S V,A |i ∈ I} (5)
The set I indexes the dataset. We call the sequence (s i ) i∈I the ground truth of D. In the case of RT GS, its training recorded music dataset contains 900 tuples randomly selected from the dataset GTZAN [START_REF] Sturm | The state of the art ten years after a state of the art: Future research in music information retrieval[END_REF][START_REF] Tzanetakis | Musical genre classification of audio signals[END_REF]. These are selected in a way such that the ground truth of D has no more than 100 of each element of S V,A .

Analysis of Recorded Music Description Systems

Given an RMDS, one needs to determine whether it addresses the problem of music description. Simple questions to answer are: does Ω and R Ω of the RMDS encompass those of the use case? Does the S V,A of the RMDS encompass that of the use case? A more complex question could be, does the RMDS meet the success criteria of the use case? This last question involves the design, implementation, analysis, and interpretation of valid experiments that are relevant to answering hypotheses about the RMDS and success criteria [START_REF] Sturm | Making explicit the formalism underlying evaluation in music information retrieval research: A look at the MIREX automatic mood classification task[END_REF][START_REF] Urbano | Evaluation in music information retrieval[END_REF]. Answering these questions constitutes an analysis of an RMDS. Absent explicit success criteria of a use case, a standard approach for evaluating an RMDS is to compute a variety of figures of merit (FoM) from its "treatment" of the recordings of a testing D that exemplify the input/output relationships sought. Examples of such FoM are mean classification accuracy, precisions, recalls, and confusions. An implicit belief is that the correct output will be produced from the input only if an RMDS has learned criteria relevant to describing the music. Furthermore, it is hoped that the resulting FoM reflect the real world performance of an RMDS. The real world performance of an RMDS are the FoM that result from an experiment using a testing recording music dataset consisting of all members in R Ω , rather than a sampling of them. If this dataset is out of reach, statistical tests can be used to determine significant differences in performance between two RMDS (testing the null hypothesis, "neither RMDS has 'learned better' than the other"), or between the RMDS and that of picking an element of S V,A independent of the element from R Ω (testing the null hypothesis, "The RMDS has learned nothing"). These statistical tests are accompanied by implicit and strict assumptions on the measurement model and its appropriateness to describe the measurements made in the experiment [START_REF] Bailey | Design of comparative experiments[END_REF][START_REF] Dougherty | Scientific knowledge is possible with small-sample classification[END_REF].

As an example, consider the evaluation of RT GS discussed above [START_REF] Tzanetakis | Musical genre classification of audio signals[END_REF]. The evaluation constructs a testing D from the 100 elements of the dataset GTZAN not present in the training D used to create the RMDS. They treat each of the 100 recordings in the testing D with RT GS, and compare its output with the ground truth. From these 100 comparisons, they compute the percentage of outputs that match the ground truth (accuracy). Whether or not this is a high-quality estimate of the real world accuracy of RT GS depends entirely upon the definition of Ω, R Ω , S V,A , as well as the testing D and the measurement model of the experiment.

There are many serious dangers to the interpretation of the FoM of an RMDS as reflective of its real world performance: noise in the measurements, an inappropriate measurement model [START_REF] Bailey | Design of comparative experiments[END_REF], a poor experimental design and errors of the third kind [START_REF] Kimball | Errors of the third kind in statistical consulting[END_REF], the lack of error bounds or error bounds that are too large [START_REF] Dougherty | Scientific knowledge is possible with small-sample classification[END_REF], and several kinds of bias. One kind of bias comes from the very construction of testing datasets. For instance, if the testing dataset is the same as the training dataset, and the set of recordings in the dataset is a subset of R Ω , then the FoM of an RMDS computed from the treatment may not indicate its real world performance. This has led to the prescription in machine learning to use a testing dataset that is disjoint with the training dataset, by partitioning for instance [START_REF] Hastie | The Elements of Statistical Learning: Data Mining, Inference, and Prediction[END_REF]. This, however, may not solve many other problems of bias associated with the construction of datasets, or increase the relevance of such an experiment with measuring the extent to which an RMDS has learned to describe the music in Ω.

Summary

Table 1 summarizes all elements defined in our formalization of the problem of music description, along with examples of them. These are the elements that must be explicitly defined in order to address an instance of the problem of music description by algorithms. Central to many of these are the definition of a use case, which specifies the music and music recording universe, the vocabulary, the desired semantic universe, and the success criteria of an acceptable system. (Note that "use case" is not the same as "user-centered.") If the use case is not unambiguously specified, then a successful RMDS cannot be constructed, relevant and valid experiments cannot be designed, and the analysis of an RMDS cannot be meaningful. Table 1 can serve as a checklist for the extent to which an instance of the problem of music description is explicitly defined.

APPLICATION

We now discuss two additional published works in the MIR literature in terms of our formalism.

Dannenberg et al. [7]

The use cases of the RMDS employed by Dannenberg et al. [START_REF] Dannenberg | A machine learning approach to musical style recognition[END_REF] are motivated by the desire for a mode of communication between a human music performer and an accompanying computer that is more natural than physical interaction. The idea is for the computer to employ an RMDS to describe the acoustic performance of a performer in terms of several "styles." Dannenberg et al. circumvent the need to define any of these "styles" by noting, "what really matters is the ability of the performer to consistently produce intentional and different styles of playing at will" [START_REF] Dannenberg | A machine learning approach to musical style recognition[END_REF]. As a consequence, the use cases and thus system analysis are centered on the performer.

One use case considered by Dannenberg et al. defines V = {"lyrical", "frantic", "syncopated", "pointillistic", "blues", "quote", "high", "low"}, and the semantic rule A(s) := [|s| ∈ {1}]. The semantic universe S V,A is then all single elements of V. The music universe Ω is all possible music that can be played or improvised by the specific performer in these "styles." The tangible music recording universe R Ω is all possible 5-second acoustic recordings of the elements of Ω. Finally, the success criteria of this particular problem of music description includes the following requirements: reliable for a specific performer in an interactive performance, classifier latency of under

N, ∀1 ≤ i ≤ n[vi ∈ V] ∧ A(s)}, i.e.,
the set of all sequences of symbols from V permitted by the semantic rule A(•)

{("Robert", 1), ("Robert", "Moog", 4.3), ("french woman", 104.3), ("french woman", "Moog", 459), . . .} semantic rule (A(s))

a Boolean function that defines when sequence s is "permissible"

A(s) := (|s| ∈ {2, 3, 4, 5}) ∧ ({v1, . . . , v |s|-1 } ⊆ {"Robert", "french woman", "bass in other room", "Moog"} ∪ {}) ∧ (v |s| ∈ [0, 2760]) music description the pairing of an element of Ω or RΩ with an element of SV,A label the events (character, time) in recording LCD 1002 of "Automatic Writing" by R. Ashley the problem of music description make this pairing acceptable with respect to the success criteria specified by the use case make this pairing such that F-score of event "Robert" is at least 0. "Training.m" and "Classifying.m" MAT-LAB scripts in "AP1", etc. environment connections between components, external databases, the space within which the system operates, its boundaries folds 2 and 3 of LMD [START_REF] Silla | The Latin music database[END_REF], MIREX computer cluster, local MATLAB license file, etc. recorded music description system (RMDS) (S) S : RΩ → SV,A, i.e., a map from RΩ to SV,A "RT GS" evaluated in [START_REF] Tzanetakis | Musical genre classification of audio signals[END_REF] feature extraction algorithm (E) E : RΩ → S F,A ′ , i.e., a map from RΩ to an element of a semantic universe based on the feature vocabulary F and semantic rule A ′ (s)

compute using [START_REF] Slaney | Auditory toolbox[END_REF] the first 13 MFCCs (including zeroth coefficient) from a recording feature vocabulary (F) a set of symbols R 13 classification algorithm (C) C : S F,A ′ → SV,A, i.e., a map from S F,A ′ to the semantic universe single nearest neighbor recorded music dataset D := ({rω ∈ RΩ, s ∈ SV,A} i )i∈I, i.e., a sequence of tuples of recordings and elements of the semantic universe, indexed by I GTZAN [START_REF] Sturm | The state of the art ten years after a state of the art: Future research in music information retrieval[END_REF][START_REF] Tzanetakis | Musical genre classification of audio signals[END_REF] "ground truth" of D (si)i∈I, i.e., the sequence of "true" elements of the semantic universe for the recordings in D in GTZAN: {"blues", "blues", . . . , "classical", . . . , "country", . . .} analysis of an RMDS answering whether an RMDS can meet the success criteria of a use case with relevant and valid experiments designing, implementing, analyzing and interpreting experiments that validly answer, "Can RT GS [START_REF] Tzanetakis | Musical genre classification of audio signals[END_REF] address the needs of user A?" experiment principally in service to answering a scientific question, the mapping of one or more RMDS to recordings of D, and the making of measurements apply RT GS to GTZAN, compare its output labels to "ground truth", and compute accuracy figure of merit (FoM) performance measurement of an RMDS from an experiment classification accuracy of RT GS in GTZAN real world performance of an RMDS the figure of merit expected if an experiment with an RMDS uses all of RΩ classification accuracy of RT GS Table 1. Summary of all elements defined in the formalization of the problem of music description, with examples. 5 seconds. The specific definition of "reliable" might include high accuracy, high precision in every class, or only in some classes.

Dannenberg et al. create an RMDS by using a training dataset of recordings curated from actual performances, as well as collected in a more controlled fashion in a laboratory. The ground truth of the dataset is created with input from performers. The feature extraction algorithm includes algorithms for pitch detection, MIDI conversion, and the computation of 13 low-level features from the MIDI data. One classification algorithm employed is maximum likelihood using a naive Bayesian model.

The system analysis performed by Dannenberg et al. involve experiments measuring the mean accuracy of all sys-tems created and tested with 5-fold cross validation. Furthermore, they evaluate a specific RMDS they create in the context of a live music performance. From this they observe three things: 1) the execution time of the RMDS is under 1 ms; 2) the FoM of the RMDS found in the laboratory evaluation is too optimistic for its real world performance in the context of live performance; 3) using the confidence of the classifier and tuning a threshold parameter provides a means to improve the RMDS by reducing its number of false positives.

Turnbull et al. [24]

Turnbull et al. [START_REF] Turnbull | Semantic annotation and retrieval of music and sound effects[END_REF] propose several RMDS that work with a vocabulary consisting of 174 unique "musically relevant" words, such as "Genre-Brit Pop", "Usage-Reading", and "NOT-Emotion-Bizarre / Weird". A(s) := [|s| = 10 ∧ ∀i = j(v i = v j )], and so the elements of S V,A are tuples of ten unique elements of V. The music universe Ω consists of at least 502 songs (the size of the CAL500 dataset), such as "S.O.S." performed by ABBA, "Sweet Home Alabama" performed by Lynyrd Skynyrd, and "Fly Me to the Moon" sung by Frank Sinatra. The tangible music recording universe R Ω is composed of MP3-compressed recordings of entire music pieces. The RMDS sought by Turnbull et al. aims "[to be] good at predicting all the words [in V]", or "produce sensible semantic annotations for an acoustically diverse set of songs." Since "good", "sensible" and "acoustically diverse" are not defined, the success criteria is ambiguous. Ω is also likely much larger than 502 songs.

The feature extraction algorithm in the RMDS of Turnbull et al. maps a music recording to a semantic universe built from a feature vocabulary F := R 39 , and the semantic rule ′ (s) := [|s| = 10000]. That is, the algorithm computes from an audio recording 13 MFCC coefficients on 23ms frames, concatenates the first and second derivatives in each frame, and randomly selects 10000 feature vectors from all those extracted. The classification algorithm in the RMDS uses a a maximum a posteriori decision criterion, with conditional probabilities of features modelled by a Gaussian mixture model (GMM) of a specified order. One RMDS uses expectation maximization to estimate the parameters of an 8-order GMM from a training dataset.

Turnbull et al. build an RMDS using a training dataset of 450 elements selected from CAL500. They apply this RMDS to the remaining elements of CAL500, and measure how its output compares to the ground truth. When the ground truth of a recording in CAL500 does not have 10 elements per the semantic rule of the semantic universe, Turnbull et al. randomly add unique elements of V, or randomly remove elements from the ground truth of the recording until it has cardinality 10.

Turnbull et al. compute from an experiment FoM such as mean per-word precision. Per-word precision is, for a v ∈ V and when defined, the percentage of correct mappings of the system from the recordings in the test dataset to an element of the semantic universe that includes v. Mean per-word precision is thus the mean of the |V| perword precisions. Turnbull et al. compare the FoM of the RMDS to other systems, such as a random classifier and a human. They conclude that their best RMDS is slightly worse than human performance on "more 'objective' semantic categories [like instrumentation and genre]" [START_REF] Turnbull | Semantic annotation and retrieval of music and sound effects[END_REF]. The evaluation, measuring the amount of ground truth reproduced by a system (human or not) and not the sensibility of the annotations, has questionable relevance and validity to the ambiguous use case.

CONCLUSION

Formalism can reveal when a problem is not adequately defined, and how to explicitly define it in no uncertain terms. An explicit definition of a problem shows how to evaluate solutions in relevant and valid ways. It is in this direction that we move with this paper for the problem of music description, the spirit of which is encapsulated by Table 1. The unambiguous definition of the use case is central for addressing an instance of the problem of music description.

We have discussed several published RMDS within this formalism. The work of Dannenberg et al. [START_REF] Dannenberg | A machine learning approach to musical style recognition[END_REF] provides a good model since its use case and analysis are clearly specified -both center on a specific music performerand through evaluating the system in the real world they actually complete the research and development cycle to improve the system [START_REF] Urbano | Evaluation in music information retrieval[END_REF]. The use cases of the RMDS built by Tzanetakis and Cook [START_REF] Tzanetakis | Musical genre classification of audio signals[END_REF] and Turnbull et al. [START_REF] Turnbull | Semantic annotation and retrieval of music and sound effects[END_REF] are not specified. In both cases, a labeled dataset is assumed to provide sufficient definition of the problem. Turnbull et al. suggest a success criterion of annotations being "sensible," but the evaluation only measures the amount of ground truth reproduced. Due to the lack of definition, we are thus unsure what problem either of these RMDS is actually addressing, or whether either of them is actually considering the music [START_REF] Sturm | A simple method to determine if a music information retrieval system is a "horse[END_REF]. An analysis of an RMDS depends on an explicit use case. The definition of the use case in Dannenberg et al. [START_REF] Dannenberg | A machine learning approach to musical style recognition[END_REF] renders this question irrelevant: all that is needed is that the RMDS meets the success criteria of a given performer, which is tested by performing with it.

While we provide in this paper a formalization of the problem of music description, and a checklist of the components necessary to define an instance of such a problem, it does not describe how to solve any specific problem of music description. We do not derive restrictions on any of the components of the problem definition, or show how datasets should be constructed to guarantee an evaluation can result in good estimates of real world performance. Our future work aims in these directions. We will incoprorate the formalism of the design and analysis of comparative experiments [START_REF] Bailey | Design of comparative experiments[END_REF][START_REF] Sturm | Making explicit the formalism underlying evaluation in music information retrieval research: A look at the MIREX automatic mood classification task[END_REF], which will help define the notions of relevance and validity when it comes to analyzing RMDS. We seek to incorporate notions of learning and inference [START_REF] Hastie | The Elements of Statistical Learning: Data Mining, Inference, and Prediction[END_REF], e.g., to specify what constitutes the building of a "good" RMDS using a training dataset (where "good" depends on the use case). We also seek to explain more formally two paradoxes that have been observed. First, though an RMDS is evaluated in a test dataset to reproduce a large amount of ground truth, it appears to not be a result of the consideration of characteristics in the music universe [START_REF] Sturm | Classification accuracy is not enough: On the evaluation of music genre recognition systems[END_REF]. Second, though artificial algorithms have none of the extensive experience humans have in music listening, description, and culture, they can reproduce ground truth consisting of extremely subjective and culturally centered concepts like genre [START_REF] Frow | Genre[END_REF].

  , Rolf Bardeli, Thibault Langlois, Valentin Emiya. Licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). Attribution: Bob L. Sturm, Rolf Bardeli, Thibault Langlois, Valentin Emiya. "Formalizing the Problem of Music Description", 15th International Society for Music Information Retrieval Conference, 2014.

ACKNOWLEDGMENTS

The work of BLS and VE is supported in part by l'Institut franc ¸ais du Danemark, and ARCHIMEDE Labex (ANR-11-LABX-0033).


Daisuke Taura

Heejun Min

Claudine Katan

Eiji Yashima

Synthesis of a double-stranded spiroborate helicate bearing stilbene units and its photoresponsive behaviour †

Introduction

The design and synthesis of functional molecules and polymers that exhibit a unidirectional molecular motion, such as rotary and elastic (extension and contraction) motions induced by external stimuli, 1 in particular by irradiation of light have attracted a great deal of attention in the past two or three decades, [START_REF] Feringa | For reviews on photoresponsive molecular machines[END_REF] because such photoresponsive molecular and polymeric systems can be applied for developing molecular motors, 2a-2d,2g,3 artificial muscles, 1i,1j,4 and molecular devices and materials. 2d-2f,5 We previously reported a series of ortho-linked tetraphenol strands bearing a biphenol, 6 biphenylene (1), 7 bipyridine, 8 or bisporphyrin 9 unit in the middle that formed unique doublestranded helicates bridged by two spiroborates in which one 6,7 or two Na + cations 8 were accommodated in the center except for a large bisporphyrin-linked spiroborate helicate. 9 Conventional optical resolution of these helicates, such as diastereomeric salt formation or chiral high-performance liquid chromatography (HPLC) separation, provided pairs of optically pure enantiomers being stable toward racemisation. Among the helicates prepared so far, the optically active biphenylene-linked spiroborate helicate a Department of Molecular Design and Engineering, Graduate School of Engineering, Nagoya University, Chikusa-ku, Nagoya 464-8603, Japan. Fax: +81-52-789-3185; Tel: +81-52-789-4495; E-mail: yashima@apchem.nagoya-u.ac.jp b Institut des Sciences Chimiques de Rennes, UMR 6226 CNRS-Université de Rennes 1, France. † Electronic Supplementary Information (ESI) available: Additional spectroscopic data. See DOI: 10.1039/b000000x/ (1) showed an intriguing ion-triggered reversible extension and contraction motion coupled with a unidirectional twisting motion upon release and binding of Na + ions (Scheme 1A); 7a the contracted helicate (1 C ) significantly extended through addition 50 of [2.2.1]cryptand ([2.2.1]), resulting in the formation of extended helicate (1 E ) with a B-B distance of 13.0 Å that is more than two-fold when compared with that of the contracted 1 C (6.0 Å). One might anticipate that if this unique spring-like motion could be controlled by irradiation of light, this would contribute 55 to developing feasible nanoscale mechanical devices. 1g,2,5e To this end, we have designed and synthesized three novel ortho-linked tetraphenol strands linked by photoresponsive cisand trans-stilbene 4b,10 and trans-azobenzene 10b,11 units in the middle (Scheme 2) and investigated their double-stranded 60 helicate formations stabilised by spiroborates with NaBH 4 . We found that only the cis-stilbene-containing tetraphenol strand yielded a desired double-stranded spiroborate helicate (cis-3), whereas the tetraphenol strands with a trans-stilbene or transazobenzene unit did not give such a helicate. We expected that 65 the contracted helicate cis-3 C would undergo an extension (cis-3 E ) and contraction (cis-3 C ) motion by the Na + ion release and binding (chemical stimulus) as well as by light irradiation (physical stimulus) through cis-to-trans photoisomerisation of the stilbene units, thus producing cis-3 E and more extended trans-3 E , 70 respectively (Scheme 1B). Although this cis-trans photoisomerisation was irreversible because of the photocleavage of the trans-stilbene moieties under irradiation of light, to the best of our knowledge, wholly-artificial double helices showing a light-triggered spring-like motion are hitherto 75 unknown. 4d,12,13 Scheme 1 Extension (E) and contraction (C) motions of spiroborate-based double-stranded helicates bearing biphenyl units (1) (A) and stilbene units (3) (B) in the middle. The contracted helicates (1C and cis-3C) including a sodium ion in their helical cavities extend upon the addition of [2.2.1] (1E and cis-3E). Further addition of NaPF6 causes the formation of contracted forms. The cis-3E is further extended upon cis-to-trans photoisomerisation.

Scheme 2 Synthesis of spiroborate-based helicates bearing trans-azobenzene (trans-2) (a), trans-stilbene (trans-3) (b), and cis-stilbene (cis-3) (c) units in the middle.

Results and discussion

Synthesis of tetraphenol strands linked by photoresponsive units and the corresponding spiroborate helicates

The trans-azobenzene-containing tetraphenol (trans-8) was synthesised by Suzuki-Miyaura coupling of a boronic acid (5) with trans-4,4'-diiodoazobenzene (trans-6), subsequently followed by deprotection of the methoxy groups using BBr 3 according to a previously reported method [6][7][8] (Scheme S1 in ESI †). The trans-and cis-stilbene-containing tetraphenols (trans-11 and cis-11, respectively) were also synthesised in a similar way by Suzuki-Miyaura coupling of a boronic acid ester (9) 9 with the corresponding trans-and cis -4,4'-diiodostilbenes (trans-10 and cis-10, respectively)

(Scheme S2 in ESI †).
The obtained tetraphenol strands were then allowed to react with an equimolar amount of NaBH 4 in 1,2-dichloroethane and ethanol (5/1, v/v) at 80 °C for 20 h under similar conditions used 25 for the synthesis of the spiroborate helicates reported previously 6- 8 (Scheme 2). However, the products from trans-8 and trans-11 helicates prepared so far. [6][7][8][9] In order to assign these signals, 2 equiv. of [2. resulting in the formation of the extended helicate (1 E ) (Scheme 1A). 7a Based on these observations, the two sets of nonequivalent signals (B and B, b in Fig. 1B) could be reasonably assigned to 25 those of the contracted (cis-3 C ) and extended (cis-3 E ) forms, respectively, which indicates that the Na + ion binding and release rates, in other words, the corresponding contracted and extended motions of cis-3 are slow in CD 3 CN on the 1 H NMR time scale, giving separated signals under equilibrium (b in Fig. 1B), from 30 which the ratio of cis-3 C to cis-3 E and its equilibrium constant (K) were estimated to be 30 : 70 and 2 x 10 3 M -1 , respectively. In the presence of [2.2.1] (2 equiv.), the ratio of cis-3 C to cis-3 E changed to be approximately 10 : 90 (c in Fig. 1B). The further addition of [2.2.1] (3 equiv.) brought about the complete shift of the equilibrium to the extended helicate formation (cis-3 E ).

were
According to this assignment, the upfield shifts of some aromatic protons for cis-3, in particular those for cis-3 C (b in Fig. 1B), were explained by the shielding effect of the benzene rings of the other strand as observed in the other spiroborate helicates. 6- 9 The upfield shift of the stilbene protons (g') was more significant. The calculated structures of cis-3 C and cis-3 E in CH 3 CN by using the Hartree-Fock method (HF, 6-31G*) support the upfield shifts of the stilbene protons (g') of cis-3 C , which are favourably positioned above the benzene rings of the twisted stilbene residues into one direction (Fig. 1A). The energy difference (ΔG Cal ) between the calculated structures in CH 3 CN revealed that the cis-3 E is 2 kcal mol -1 more stable than the cis-3 C (Fig. 1A), which is consistent with the major species (cis-3 E ) observed by the 1 H NMR spectrum of cis-3 in CD 3 CN at 298 K at equilibrium (b in Fig. 1B).

The equilibrium between cis-3 C and cis-3 E was also unambiguously confirmed by the concentration-dependent 1 H NMR and 2D NOESY (2D exchange spectroscopy (EXSY)) spectra of cis-3 in the absence and presence of [2.2.1] (2 equiv.), respectively. 14 The intensities of the signals (B) derived from cis-3 C gradually decreased with the decrease in the concentration of cis-3 (0.31-0.080 mM), whereas those of the signals (B) due to cis-3 E relatively increased and the signals became sharper without any changes in their chemical shifts (Fig. S4 in ESI †), from which the K value was also estimated to be 2 x 10 3 M -1 , indicating that the equilibrium involves the sodium ion release and binding. In addition, the 2D NOESY (EXSY) spectrum of cis-3 with 2 equiv. of [2.2.1] showed clear positive exchange cross-peaks between the two species, contracted cis-3 C (B) and extended cis-3 E (B) forms as well as negative intrastrand NOE cross-peaks for cis-3 E (Figs. S6-S8 in ESI †). Although interstrand NOE cross-peaks were not observed for cis-3 E , a similar tendency was reported for the other extended spiroborate helicates. 7a The subsequent addition of NaPF 6 to the solution of cis-3 E in the presence of 2 equiv. of [2.2.1] (d in Fig. 1B) regenerated the original helicates as a mixture of cis-3 C and cis-3 E , while new sharp signals different from those of the ligand cis-11 and the helicates (cis-3 C and cis-3 E ) appeared in the aromatic and t-Bu proton resonance regions; the content of the new species was ca. 6% (d in Fig. 1B). We found that the identical new species also formed when a CD 3 CN solution of cis-3 was allowed to stand for a long time (Fig. S9A in ESI †); its ESI mass spectrum (Fig. S10 in ESI †) was identical to that measured just after dissolving cis-3 in CH 3 CN (Fig. S3 in ESI †). These results together with the COSY spectrum of a CD 3 CN solution of cis-3 after standing for ca. 36 h (Fig. S9B in ESI †) suggest that the new species is a structural isomer of the racemo-helicates (cis-3 C and cis-3 E ) and 55 is most likely a double-stranded meso-helicate with an extended structure (meso-cis-3 E ) that will be generated from the racemohelicates in solution and reached an equilibrium within a few days. Indeed, calculations predict that meso-cis-3 E is 2.7 kcal mol -1 more stable than cis-3 E (Table S1 in ESI †). Although the 60 mechanism for the present racemo-to-meso-cis-3 isomerisation, which requires the breaking and reformation process of the spiroborate groups, remains unknown, these results revealed that the racemo-cis-3 exists as a mixture of the interconvertible contracted (cis-3 C ) and extended (cis-3 E ) forms in solution and 65 that the ion-triggered extension and contraction motion mostly proceeds reversibly; however, the present system seems to be more complicated than before, and involves the other chemical equilibrium,

i.e., racemo-to-meso-cis-3 isomerisation. Apparently, a further study is necessary to unambiguously 70 elucidate the structure of the meso-cis-3 E by single-crystal X-ray analysis and the mechanism of this unexpected isomerisation.

Photoisomerisation of cis-11

Cis/trans photoisomerisation of the single strand cis-11 was first investigated by 1 H NMR and absorption spectroscopies in 75 CD 3 CN (Figs. S11-S12 and S13-S14, respectively in ESI †). Upon irradiation of UV light at 295 nm, the intensities of the cis-11 protons decreased with time, while the new peaks due to trans-11 appeared with a gradual increase in their intensities and the cis/trans ratio reached a constant value (8 : 92) after irradiation 80 for 120 min as estimated from the integral ratio of the cis-and trans-stilbene protons (f c and g t ) (Table 1 and Fig. S11a in ESI †). Subsequent irradiation with UV light at 360 nm, which induces the trans-to-cis isomerisation, produced a mixture of cis-and trans-11 (Fig. S11b in ESI †); its ratio changed with time and 85 reached an almost constant value (42 : 56) at the photostationary state after 120 min irradiation (Table 1 and Fig. S12 in ESI †), indicating that the trans-to-cis photoisomerisation did not take place efficiently as reported for typical stilbene derivatives. 15 The changes in the absorption spectra of cis-11 in a dilute 90 CD 3 CN solution during the course of the cis-to-trans followed by the trans-to-cis photoisomerisations were also measured (Fig.

S13 in ESI †

). An absorption peak at 335 nm due to the formation of trans-11 gradually increased with time accompanied with a

Photoisomerisation of cis-3

With all the above photoisomerisation results of cis-11 in mind, we then investigated the photoisomerisation behaviour of the double-stranded helicate cis-3 composed of two cis-11 strands with 1 H NMR and absorption spectroscopies in a manner similar to that for cis-11. Fig. 2 shows the 1 H NMR spectral changes of cis-3 in CD 3 CN in the presence of 2 equiv. of [2.2.1] before and after irradiation of UV light at 295 nm followed by 360 nm for 60 min each (for more detailed time-dependent 1 H NMR spectral changes and the product ratio changes, see Fig. S15 in ESI † and Fig. 3, respectively).

In sharp contrast to cis-11, the 1 H NMR spectrum of cis-3 became quite complicated after irradiation of UV light at 295 nm for 60 min (Fig. 2b) because of newly generated cis,trans-3 E and trans-3 E as well as the unreacted cis-3 E (Scheme 3), suggesting that the cis-to-trans isomerisation in this system was less efficient presumably because of the highly strained, covalently-bonded spiroborated duplex structure (Fig. 1A); the product ratio estimated by its 1 H NMR spectrum (Fig. 2b) was cis,trans-3 E (39%) and trans-3 E (38 %) along with a small amount of trans-4 (2%) (Table 1, see below). However, it should be emphasised that the trans-3 E , which was not directly synthesised by the reaction of trans-11 with NaBH 4 under the conditions used for the synthesis of spiroborate helicates [6][7][8][START_REF] Nishimura | Calculation results suggest that relative Gibbs free energy between cis-3E and trans-3E is not significantly different (Table S1 in ESI †). 17 For photooxidation of stilbene derivatives resulting in the formation of aldehydes[END_REF] (Scheme 2), was successfully produced via cis-to-trans photoisomerisation of the corresponding cis-3. Subsequent irradiation of UV light at 360 nm further produced a new set of proton resonances with a characteristic singlet at 9.84 ppm (Fig. 2c), which could be derived from an aldehyde species (trans-4) most likely generated by photooxidation of trans-3 E in the presence of trace amounts of 30 oxygen. 17 The dialdehyde (trans-4) formation that was evidenced by the negative ESI-mass spectrum (Fig. S16B in ESI †) unfortunately caused the irreversible trans-cis photoisomerisation of this system, and its content increased with time (Fig. S15 in ESI † and Table 1).

35

The photo-induced cis/trans isomerisation of cis-3 E was also investigated by absorption spectroscopy. Similar time-dependent absorption spectral changes of cis-3 E in a dilute CD 3 CN solution containing 3 equiv. of [2.2.1] were observed during the course of the cis-to-trans and trans-to-cis photoisomerisations, but the 40 changes were not significant as compared to those of cis-11 because of inefficient cis-to-trans isomerisation and oxidation that took place during the trans-to-cis photoisomerisation, giving the oxidized trans-4 (Fig. S20a,b in ESI †). The fluorescence intensity of cis-3 E was also enhanced upon irradiation of UV light 45 (295 nm), resulting in the formation of a mixture of cis,trans-3 E , trans-3 E , and trans-4 (Fig. S20c,d in ESI †).

Optical resolution of helicates

As mentioned above, the cis-3 just after preparation undergoes the ion-triggered extension and contraction motion in acetonitrile, which further changes its structure to a new species in acetonitrile under equilibrium. We tentatively assigned the cis-3 and the new species as racemo-and meso-cis-3 E helicates, respectively, on the basis of their NMR spectra. Obviously, the racemo-cis-3 helicate is chiral, whereas the meso-cis-3 E helicate is achiral, and in order to confirm these structural assignments, we investigated if the chiral cis-3 could be resolved into enantiomers by HPLC using a chiral column (Chiralpak IB, Daicel).

Fig. 4a shows the chromatogram of the resolution of cis-3, showing a single peak detected by UV, although the circular dichroism (CD)-detected chromatogram indicated a partial separation with negative and positive CDs; their CD intensities were too weak to evidence the enantioseparation (Fig. 4b). Next, a CH 3 CN solution of a mixture of cis,trans-3 E , trans-3 E , and trans-4 prepared from cis-3 E through the photoisomerisation under air was injected into the chiral HPLC system (Fig. S21 in ESI †). As shown in Fig. 4d, the trans-3 E and trans-4 were not completely but rather were partially separated into a pair of enantiomers with the first positive and second negative CD signs. Importantly, the trans-3 E and trans-4 were converted from cis-3 E via photoisomerisation without any breaking and reformation process of the spiroborate groups, leading to the conclusion that the cis-3 is a racemo-helicate with a double-stranded spiroborate structure and the new species formed from the cis-3 in CD 3 CN may possess a meso structure.

Conclusions

In summary, we have successfully synthesised the photoresponsive spiroborate-based double-stranded helicate (cis-3) bearing cis-stilbene units in the middle, which underwent an extension (cis-3 E ) and contraction (cis-3 C ) motion by the Na + ion release and binding. Although photo-induced cis/trans isomerisation of cis-3 did not efficiently nor reversibly take place because of its highly strained spiroborated duplex structure in addition to the oxidative photo-cleavage of the trans-stilbene moieties under irradiation of UV light, the present results provide further design strategy for developing novel photoresponsive spiroborate-based helicates that exhibit light-triggered spring-like motions by introducing an overcrowded double bond into the helicates as a photo-and thermo-responsive linker. 2a,2c,2g,3 Experimental section

Materials and instruments

Materials. All starting materials and dehydrated solvents were 45 purchased from commercial suppliers and were used without further purification unless otherwise noted. Silica gel (SiO 2 ) for the flash chromatography was purchased from Merck. The boronic acid (5), 18 (E)-4,4'-diiodoazobenzene ( 6), 19 the boronic acid pinacol ester (9), 9 and (Z)-4,4'-diiodostilbene (10) 20 were 50 prepared according to the literature.

Instruments. The melting points were measured on an MPA100 OptiMelt automated melting point apparatus or a Yanaco MP-500D melting point apparatus (Kyoto, Japan), and were uncorrected. The NMR spectra were obtained using a 55 Varian UNITY INOVA 500AS spectrometer operating at 500 MHz for 1 H and 125 MHz for [START_REF] Khan | For examples of photoresponsive helical foldamers[END_REF] C or a Varian INOVA 700 spectrometer operating at 700 MHz for 1 H and 175 MHz for [START_REF] Khan | For examples of photoresponsive helical foldamers[END_REF] C using tetramethysilane (TMS) or a solvent residual peak as the internal standard. The electron spray ionization (ESI) mass 60 spectra were recorded using a JEOL JMS-T100CS spectrometer (Akishima, Japan). The absorption spectra were measured in a 1or 10-mm quartz cell on a JASCO V-570 spectrophotometer. The photoirradiation for cis/trans isomerisation was performed on a JASCO FP-6500 spectrofluorometer with a 150 W xenon lamp 65 under argon unless otherwise noted. The slit width used in the experiments was 20 nm. The chiral HPLC analyses were performed on a JASCO PU-2080 liquid chromatograph equipped with UV-visible (JASCO MD-2010) and CD detectors (JASCO CD-2095) using a Chiralpak IB column (0.46 (i.d.) x 25 cm, 70 Daicel, Osaka, Japan).

Synthetic procedures

trans-7. To a mixture of 5 (427 mg, 1.15 mmol), trans-6 (200 mg, 0.461 mmol), and Pd(PPh 3 ) 4 (133 mg, 0.115 mmol) in toluene (7.2 mL) was added 2 M aqueous K 2 CO 3 (7.2 mL). The 75 reaction mixture was stirred at 110 °C for 36 h under nitrogen. After being cooled to room temperature, the mixture was extracted with EtOAc. The organic layer was washed with H 2 O and dried over anhydrous MgSO 4 . After filtration, the solvent was evaporated under reduced pressure. The crude product was 18H, C(CH 3 ) 3 ), 1.34 (s, 18H, C(CH 3 ) 3 ). [START_REF] Khan | For examples of photoresponsive helical foldamers[END_REF] C NMR (125 MHz, CDCl 3 , r.t.): δ 154. 79, 153.46, 151,78, 146.06, 143.12, 142.62, 133.36, 132.33, 130.25, 129.30, 129.05, 127.61, 127.34, 125.32, 122.86, 110.69, 60.73, 55.90, 34.64, 34.30, 31.74, 31. trans-8. To a solution of trans-7 (200 mg, 0.241 mmol) in CH 2 Cl 2 (5.0 mL) was added a CH 2 Cl 2 solution of BBr 3 (1.0 M, 2.41 mL, 2.41 mmol) at -78 °C under nitrogen. After being warmed to room temperature, the reaction mixture was stirred for 19 h. The mixture was then quenched with H 2 O (10 mL) at 0 °C and stirred at room temperature for 1 h. After the solvent was evaporated, the residue was extracted with EtOAc. The organic layer was washed with H 2 O and dried over anhydrous MgSO 4 . After filtration, the solvent was evaporated under reduced chromatography with n-hexane/EtOAc (10/0 to 10/1 (v/v)) to afford trans-8 (170 mg, 90.9% yield) as a dark orange solid. Mp: 156.5-158.1 ºC (dec). 1 H NMR (500 MHz, CDCl 3 , r.t.): δ 8.06 (d, J = 8.5 Hz, 4H, ArH), 7.77 (d, J = 8.5 Hz, 4H, ArH), 7.44 (d, J = 2.5 Hz, 2H, ArH), 7.38 (dd, J = 2.5, 8.5 Hz, 2H, ArH), 7.33 (m, 4H, ArH), 7.01 (d, J = 8.5 Hz, 2H, ArH), 5.55 (s, 2H, ArOH), 5.47 (s, 2H, ArOH), 1.39 (s, 18H, C(CH 3 ) 3 ), 1.35 (s, 18H, C(CH 3 ) 3 ). [START_REF] Khan | For examples of photoresponsive helical foldamers[END_REF] C NMR (125 MHz, CDCl 3 , r.t.): δ 152. 03, 151.13, 147.67, 144.58, 144.49, 141.02, 130.35, 128.38, 128.15, 128.08, 128.00, 127.23, 124.26, 123.42, 123.41, 116.56, 34.55, 34.43, 31.71 (19 signals trans-11. To a solution of 9 (93 mg, 0.22 mmol), trans-10 (43 mg, 0.10 mmol), and Pd(PPh 3 ) 4 (23 mg, 0.020 mmol) in toluene (5.0 mL) was added 2 M aqueous K 2 CO 3 (5.0 mL). The reaction mixture was stirred at 110 °C for 22 h under nitrogen. After being cooled to room temperature, the mixture was extracted with EtOAc. The organic layer was washed with H 2 O and aqueous HCl, and dried over anhydrous MgSO 4 . After filtration, the solvent was evaporated under reduced pressure. The crude product was purified by flash chromatography (SiO 2 , 12 g) with n-hexane/EtOAc (10/0 to 6/4 (v/v)) to afford trans-11 (27 mg, 35% yield) as a white solid. ε 298 : 2.73 x 10 5 M -1 cm -1 . Mp: 152.8-155.6 ºC. 1 H NMR (500 MHz, CDCl 3 , r.t.) δ 7.67 (d, J = 8.5 23, 147.34, 144.54, 144.32, 137.24, 136.89, 129.86, 128.72, 128.36, 128.15, 127.81, 127.20, 127.05, 124.20, 123.91, 116.67, 34.53, 34.41, 31.72, 31.71 55 cis-11. To a solution of 9 (216 mg, 0.509 mmol), cis-10 (100 mg, 0.231 mmol), and Pd(PPh 3 ) 4 (58.8 mg, 0.0509 mmol) in toluene (5.0 mL) was added 2 M aqueous K 2 CO 3 (5.0 mL). The reaction mixture was stirred at 105 °C for 29 h under nitrogen. After being cooled to room temperature, the mixture was 60 extracted with EtOAc. The organic layer was washed with H 2 O and dried over anhydride MgSO 4 . After filtration, the solvent was evaporated under reduced pressure. The crude product was purified by flash chromatography (SiO 2 , 12 g) with nhexane/EtOAc (10/0 to 8/2 (v/v)) to afford cis-11 (80.0 mg, 65 44.7% yield) as a white solid. ε 298 : 2.73 x 10 5 M -1 cm -1 . Mp: 136.5-138.9 ºC. 1 H NMR (500 MHz, CDCl 3 , r.t.) δ 7.48-7.45 (m, 8H, ArH), 7.36 (d, J = 2.5 Hz, 2H, ArH), 7.35 (dd, J = 2.5, 8.5 Hz, 2H, ArH), 7.31 (d, J = 2.4 Hz, 2H, ArH), 7.29 (d, J = 2.5 Hz, 2H, ArH), 6.99 (d, J = 8.5 Hz, 2H, ArH), 6.68 (s, 2H, HC=CH), 70 5.57 (s, 2H, OH), 5.54 (s, 2H, OH), 1.35 (s, 18H, C(CH 3 ) 3 ), 1.33 (s, 18H, C(CH 3 ) 3 ). [START_REF] Khan | For examples of photoresponsive helical foldamers[END_REF] C NMR (125 MHz, CDCl 3 , r.t.): δ 151. 22, 147.28, 144.49, 144.26, 136.74, 136.73, 130.27, 129.59, 129.38, 128.30, 128.13, 128.11, 127.80, 127.00, 124.14, 123.93, 116.62, 34.51, 34.39, 31.71, 31.70 trans-2. To a solution of trans-8 (3.0 mg, 3.9 µmol) in 1,2dichloroethane (625 µL) was added a solution of NaBH 4 in ethanol (31.2 mM, 125 µL, 3.90 µmol). The reaction mixture was stirred at 80 °C for 20 h under argon. After being cooled to room 80 temperature, the solvents were evaporated under reduced pressure and the residue was dried in vacco. The negative ESI mass spectrum of the residue dissolved in CD 3 CN was then measured using CH 3 CN as the eluent (Fig. S1 After being cooled to room temperature, the solvents were evaporated under reduced pressure and the residue was dried in vacco. The negative ESI mass spectrum was measured after the residue was dissolved in DMSO-d 6 and then diluted with CH 3 CN (Fig. S2 in ESI †). cis-3. To a solution of cis-11 (20.5 mg, 26.5 µmol) in 1,2dichloroethane (3.0 mL) was added a solution of NaBH 4 in ethanol (44.5 mM, 0.60 mL, 27 µmol). The reaction mixture was stirred at 80 °C for 20 h under argon. The precipitate obtained from the mixture during the reaction was collected by filtration, washed with 1,2-dichloroethane, and dried in vacco to afford cis-3 as a white solid in 51.6% yield (11.0 mg). Mp: > 300 ºC. 1 H NMR (500 MHz, DMSO-d 6 , 25 °C) δ 7.33 (d, J = 8.5 Hz, 8H, ArH), 7.19 (d, J = 2.0 Hz, 4H, ArH), 7.18 (d, J = 2.5 Hz, 4H, ArH), 7.09 (dd, J = 2.5, 8.5 Hz, 4H, ArH), 7.06 (d, J = 2.5 Hz, 4H, ArH), 6.79 (d, J = 8.5 Hz, 8H, ArH), 6.56 (d, J = 8.5 Hz, 4H, ArH), 6.27 (s, 4H, CH=CH), 1.30 (s, 36H, C(CH 3 ) 3 ), 1.25 (s, 36H, C(CH 3 ) 3 ). [START_REF] Khan | For examples of photoresponsive helical foldamers[END_REF] C NMR (175 MHz, DMSO-d 6 , 40 °C): δ 154. 90, 152.16, 140.82, 140.38, 139.23, 133.79, 131.23, 130.86, 20 130.78, 129.00, 126.92, 125.54, 124.93, 124.34, 124.80, 120.67, 33.57, 33.52, 31.47, 31.34 (20 signals

Computational calculations

for dianions, monoanions and neutral helicates. For all optimised geometries, a subsequent frequency calculation was performed to verify that geometries correspond to energy minimums. The effect of basis set on geometries and properties was checked by increasing the basis set from the standard 3-21G basis up to the 6-31G*. Calculations were conducted both in vacuum and in the presence of solvent (CH 3 CN) using the Polarizable Continuum Model (PCM) as implemented in Gaussian 09.

For several helicates, we have first verified at the HF/3-21G level of theory in vacuum that substitution of the t Bu groups with 40 hydrogen atoms does not significantly change the optimized geometries and respective stability of the structures. For these simplified structures, we were able to increase the basis set up to 6-31G* and at the same time afford frequency calculations for solvated structures. In fact, relying solely on differences of electronic energies or on geometries optimized in vacuum appears to be insufficient to properly describe the relative stability of such double-stranded spiroborate helicates. In addition, geometry optimization was found to be more involved with the ωB97X-D functional than at the HF level but leads to 50 consistent results whenever available. We thus retained the HF/6-31G* level of theory in CH 3 CN and summarize the main results in Table S1. Comparison of the monoanion cis-3 C with the dianion cis-3 E and a free sodium cation needs to be corrected for basis set superposition errors (BSSE). This could be qualitatively 55 estimated from electronic energies in vacuum where BSSE amounts to ca. 3 kcal mol -1 . stilbene residues, see: (a) C. A. Stanier, S. J. Alderman, T. D. W. Claridge and H. L. Anderson, Angew. Chem., Int. Ed., 2002, 41, 1769-1772; (b) D.-H. Qu, Q.-C. Wang and H. Tian, Angew. Chem., Int. Ed., 2005, 44, 5296-5299; (c) P. Kuad, A. Miyawaki, Y. Takashima, H. Yamaguchi and A. Harada, J. Am. Chem. Soc., 2007, 10 129, 12630-12631; (d) V. Serreli, C.-F. Lee, E. R. Kay and D. A. Leigh, Nature, 2007, 445, 523-527. 11 For examples of photoresponsive supramolecular systems using azobenzene residues, see: (a) S. Shinkai, T. Nakaji, T. Ogawa, K. Shigematsu and O. Manabe, J. Am. Chem. Soc., 1981, 103, 111-115;

  complicated based on their 1 H NMR spectra, and the electrospray ionisation (ESI) mass spectra in the negative mode indicated the formation of dimers and trimers bridged by one and 30 two spiroborates, respectively, along with unreacted strands (Scheme 2a, b and Figs. S1 and S2 in ESI †), probably because the planar trans-azobenzene and trans-stilbene units hamper such a twisted spiroborate formation due to steric effects. On the other hand, the cis-stilbene strand (cis-11) favourably 35 reacted with NaBH 4 , giving the corresponding helicate (cis-3) in 51.6% yield under the same condition (Scheme 2c) as supported by the ESI mass spectrum, which showed signals due to the

Fig. 1 (

 1 Fig. 1 (A) Optimised structures of left-handed helical cis-3C (monoanion) and cis-3E (dianion) calculated by Hartree-Fock method (6-31G*, CH3CN), in which all the t Bu groups were substituted with hydrogen atoms for simplifying the calculations. (B) 1 H NMR spectra (500 MHz, CD3CN, 25 °C) of cis-11 (0.65 mM) (a), cis-3 (0.31 mM) (b), (b) + 2 equiv. [2.2.1] (c), and (c) + 2 equiv. NaPF6 (d). B and B denote the signals for the contracted (cis-3C) and extended (cis-3E) helicates, respectively. The peak assignments were done on the basis of gCOSY and NOESY spectra (Figs. S5-S8 in ESI †). * denotes the signals tentatively assigned to the meso-cis-3E (see the text and Figs. S9 and S10 in ESI †). monovalent (cis-3 -) and divalent (cis-3 2-) anions at m/z = 1581.75 and 779.36, respectively (Fig. S3 in ESI †). Extension and contraction motion of cis-3 Figs. 1B (a and b) show the 1 H NMR spectra of cis-11 and cis-3, respectively, measured in CD 3 CN at 25 °C. Interestingly, most of the proton resonances for cis-3 appeared as two sets of nonequivalent broadened signals after spiroborate helicate formation (B and B, b in Fig. 1B); such nonequivalent proton resonances have not been observed for the other spiroborate

  2.1] were added to a solution of cis-3. As shown in c in Fig. 1B, the broad signals became sharper ones, while maintaining their chemical shifts (B) along with significant decrease in the signals marked by B in c in Fig. 1B. As reported 20 previously, the Na + ion weakly bound in the contracted helicate (1 C ) could be released quantitatively in the presence of [2.2.1],

a

  Product ratios were estimated by 1 H NMR analyses. b 0.65 mM at 25 °C under air (see Figs. S11 and S12 in ESI †). c 0.25 mM cis-3 in the presence of 2 equiv. [2.2.1] at 25 °C under argon (see Fig. 3 and Fig. S15 in ESI †). d Overall cis and trans ratio of the stilbene units of 3 and 4. e A trace amount of oxidised aldehydes (ca. 2%) was produced during the photoisomerisation (see Fig. S11b in ESI †).

Fig. 2 1

 2 Fig. 2 1 H NMR spectral changes of cis-3 (500 MHz, CD3CN, 25 °C, 0.25 mM) in the presence of 2 equiv. [2.2.1] before (a) and after irradiation of UV light at 295 nm for 60 min (b), followed by irradiation of UV light at 360 nm for 60 min under argon (c) (see Fig. S15 in ESI †). B, H, J, and F denote the signals for cis-3E, cis,trans-3E, trans-3E, and trans-4, respectively. The peak assignments were done on the basis of the ESI-MS spectrum of the mixture (Fig. S16 in ESI †) and gCOSY and NOESY spectra (Figs. S17-S19 in ESI †).

Fig. 3

 3 Fig. 3 Time-dependent changes in the product ratio during the photoisomerisation of cis-3 (0.25 mM) in the presence of 2 equiv. [2.2.1] upon irradiation of UV light at 295 nm, followed by irradiation of UV light at 360 nm under argon (see Fig. S15 in ESI †).

Fig. 4

 4 Fig. 4 HPLC chromatograms for the resolution of cis-3 in CH3CN (a) and cis-3E in the presence of 7 equiv. [2.2.1] in CD3CN after irradiation of UV light at 295 nm for 90 min under air (c) (see Fig. S21b in ESI †). Magnified chromatograms corresponding to the areas indicated by the squares in (a) and (c) are shown in (b) and(d), respectively. HPLC conditions: column, Chiralpak IB (Daicel, 0.46 (i.d.)×25 cm); eluent: hexane/CHCl3 (6:4, v/v) containing 0.5 mg/mL tetrabutylammonium bromide (TBAB); flow rate, 1.0 mL min -1 ; column temperature, 25 °C. The peak assignments were performed by fractionation of each peak, followed by measurements of the 1 H NMR spectra. We note that photoirradiation of cis-3E in air produced a larger amount of the photooxidised trans-4.

  Scheme 4 Synthesis of trans-azobenzene ligand (trans-8).

  68. HRMS (positive mode ESI): m/z calcd for [M(C 56 H 66 N 2 O 4 )+Na + ] + , 853.4920; found 853.4922.

  Hz, 4H, ArH), 7.59 (d, J = 8.5 Hz, 4H, ArH), 7.39 (d, J = 2.5 Hz, 2H, ArH), 7.37 (dd, J = 2.5, 8.5 Hz, 2H, ArH), 7.33 (d, J = 2.5 Hz, 2H, ArH), 7.31 (d, J = 2.5 Hz, 2H, ArH), 7.23 (s, 2H, HC=CH), 7.01 (d, J = 8.5 Hz, 4H, ArH), 5.56 (s, 2H, OH), 5.52 (s, 2H, OH), 1.37 (s, 18H, C(CH 3 ) 3 ), 1.34 (s, 18H, C(CH 3 ) 3 ). 13 C NMR 50 (125 MHz, CDCl 3 , r.t.): δ 151.

  (20 signals out of 21 expected ones). HRMS(ESI): m/z calcd for [M(C 54 H 60 O 4 )-H + ] -, 771.4413; found 771.4379.

Table 1

 1 Results of the photoisomerisation of cis-11 and cis-3 in CD3CN upon irradiation of UV light at 295 and subsequently at 360 nm

		11 a,b			3 a,c
		cis-11 : trans-11	cis-3 / %	cis,trans-3E / %	trans-3E / %	trans-4 / %	cis : trans d
	Before irradiation	99 : 1	97	3	0	0	98.5 : 1.5
	After irradiation with 295
	nm light	8 : 92	21	39	38	2	40.5 : 58.5
	(cis-to-trans)
	After irradiation with 360
	nm light	42 : 56 e
	(trans-to-cis)

  out of 20 expected ones). HRMS (negative mode ESI): m/z calcd for [M(C 52 H 58 N 2 O 4 )-H + ] -, 773.4318; found 773.4311.

  out of 21 expected ones). HRMS(ESI): m/z calcd for [M(C 108 H 112 B 2 Na 2 O 8 )-Na + ] -, 1581.8470; found 1581.8462.

We used quantum chemical approaches, as implemented in the Gaussian 09 package, 21 to construct model double-stranded spiroborate helicates. Geometry optimisations were performed at different levels of theory, namely Hartree-Fock (HF) and density functional theory (DFT) using the hybrid ωB97X-D functional, 22

Acknowledgements

This work was supported in part by Grant-in-Aids for Scientific Research (S) from the JSPS (E.Y.) and by Program for Leading


Wiktor Walasik

Alejandro Rodriguez

Gilles Renversez

Symmetric plasmonic slot waveguides with a nonlinear dielectric core: bifurcations, size effects, and higher order modes

We study the nonlinear stationary waves propagating in metal slot waveguides with a Kerr-type dielectric core. We develop two independent semi-analytical models to describe these waves in such waveguides. Using those models we compute the dispersion curves for the first ten modes of a nonlinear slot waveguide. For symmetric waveguides we find symmetric, antisymmetric, and asymmetric higher order modes which are grouped in two families. In addition, we study the influence of the slot width on the first symmetric and asymmetric modes. We also show that the dispersion curve of the first asymmetric mode is invariant with respect to the slot width for high propagation constant values and we provide analytical approximations of this curve. Using realistic values for the optogeometric parameters of the structure, we are able to obtain the bifurcation at a realistic power.

I. INTRODUCTION

Nonlinear plasmonics is now a flourishing branch of photonics 1 . Among it, the studies of nonlinear waves combining both plasmon and soliton features started in 1980s 2,[START_REF] Mihalache | Progress in Optics[END_REF] . The interest in the field started to grow again thanks to the results provided in Ref. [START_REF] Feigenbaum | [END_REF] . Even if no experimental demonstration of plasmon-solitons has been published yet 5 , in the mid term, such states can have several applications, e.g. in nonlinear couplers 6 or in a four wave mixing process 7 . Moreover, linear metal slot waveguides with a few hundred nanometers wide silicon core have already been fabricated and characterized 8,9 .

The structure studied in Ref. [START_REF] Feigenbaum | [END_REF] is a nonlinear dielectric core surrounded by two semi-infinite metal regions. It will be called here the nonlinear slot waveguide (NSW). Recent studies of plasmon-solitons in NSWs with Kerrtype focusing nonlinear core have led to a number of interesting predictions, including the existence of symmetric, antisymmetric, and asymmetric modes 10,11 . The asymmetric nonlinear mode appears from the symmetric one through a symmetry-breaking bifurcation at a critical power. This bifurcation phenomenon has already been described in nonlinear slot waveguides, fully dielectric nonlinear waveguides, and waveguides made of nonlinear metal sandwiched between linear dielectrics 10,[12][13][START_REF] Kivshar | Stability of spatial optical solitons[END_REF][START_REF] Davoyan | [END_REF] .

In this article limited to the study of stationary solutions, we demonstrate the existence of a number of previously unknown higher order modes in the NSW. We show that bifurcations also occur for some of the higher order modes. We study the influence of the slot width on the dispersion curves of the first asymmetric mode. We show that these curves have an invariant part that can be described analytically. These results are obtained using two approaches we developed: (i) a semi-analytical model, where field profiles in the core are described by the Jacobi elliptic special functions and (ii) a model where the field profile in the core is obtained by direct integration of Maxwell's equations.

II. MODELS

Let us consider monochromatic transverse magnetic (TM) waves propagating along the z direction (all field components evolve proportionally to (exp[i(k 0 βz -ωt)]) in a symmetric one-dimensional NSW depicted in Fig. 1. Here k 0 = ω/c, where c denotes the speed of light in vacuum, β denotes the effective mode index and ω is the light angular frequency. The nonlinear Kerr-type dielectric is isotropic with ǫ = ǫ l +α|E| 2 and α > 0 (focusing nonlinearity). The relation between the nonlinear parameter α and the coefficient n 2 , that appears in the definition of an intensity dependent refractive index n = √ ǫ l + n 2 I, is α = ǫ 0 cǫ l n 2 where the intensity is defined as I = ǫ 0 c|ǫ||E| 2 /(2β), and the vacuum permittivity is denoted by ǫ 0 .

Our first model uses a semi-analytical approach that provides closed analytical expressions for the nonlinear dispersion relation and for the field profiles of the modes. It employs the fact that Maxwell's equations for TM electromagnetic wave in the Kerr-type nonlinear medium can be reduced to a single nonlinear wave equation for the magnetic field component 2,16 , assuming that the nonlinear permittivity modification is small and depends only on the transverse E x component. This equation is solved using the first integral approach 17,18 yielding

dH y dx 2 -k 2 0 q 2 2 H 2 y + k 2 0 a 2 H 4 y = c 0 , (1)
where q 2 i = β 2 -ǫ l,i (i ∈ {1, 2, 3} enumerates the region), a = β 2 n 2 /(ǫ 0 ǫ l,2 c), and c 0 is the integration constant that in the case of a finite-size nonlinear medium is given by

c 0 = k 2 0 (ǫ 2 /ǫ 1 ) 2 q 2 1 -q 2 2 + aH y (0) 2 /2 H y (0) 2 . ( 2
)
The resulting field profiles in the NSW core are described by Jacobi elliptic special functions [START_REF] Abramowitz | Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables[END_REF] and hence we denote this model as the Jacobi elliptic model (JEM).

The problem is split in two cases depending on the sign of c 0 . For c 0 ≤ 0 only node-less solutions exist. In contrast, for c 0 > 0 only solutions with nodes appear. The field profile for the case c 0 > 0 is given by:

H y (x) = δ + cn s/A (x -x 0 )|m , (3)
in which

δ 2 ± = A 2 Q 2 + 4Ac 0 ± AQ /2, (4)
x 0 = -A/s cn -1 H y (0)/δ + |m . (5)
Here [START_REF] Abramowitz | Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables[END_REF] for the definitions and properties of this family of special functions).

Q = k 2 0 q 2 2 , A = 2/(k 2 0 a), s = δ 2 + + δ 2 -, m = δ 2 + /s, and cn[x|m] is a Jacobi elliptic function of parameter m (see reference
Using the continuity of the tangential field components at the interfaces between the nonlinear medium and the bounding metals, were fields decay exponentially, the nonlinear dispersion relation is obtained in an analytical form (for c 0 > 0):

k 0 q 3 ǫ 2 A/s cn s/A (d -x 0 )|m = ǫ 3 sn s/A (d -x 0 )|m dn s/A (d -x 0 )|m , (6)
in which Jacobi elliptic functions sn and dn appear. For a given structure the solutions of Eq. ( 6) are found fixing the values of the free parameter H y (0) [see Eq. ( 5)]. Similar equations can be derived for the case c 0 < 0 but the field profile and dispersion relations are described by different Jacobi elliptic functions.

We also develop a second model, based on the results of Refs [START_REF] Mihalache | [END_REF]21 for a single nonlinear dielectric/metal interface only, where the nonlinear term accounts for both transverse and longitudinal electric field components. This approach allows one to obtain a closed-form analytical expression for the dispersion relations for each of the two interfaces of the NSW. In particular, for the left interface, one obtains:

(ǫ 2,0 /β) 2 -2ǫ 2,0 (ǫ 1 β) 2 / (ǫ 1 β) 2 + (ǫ 2,0 q 1 ) 2 +ǫ l,2 + ǫ 0 cǫ l,2 n 2 E 2 0 /2 E 2 0 = C, (7)
where E 0 = E 2 x (0) + E 2 z (0) denotes the total electric field at the left interface and ǫ 2,0 = ǫ l,2 + αE 2 0 . To get the equation on the right interface at x = d (see Fig. 1) we substitute E 0 by E d and ǫ 2,0 by ǫ 2,d .

For fixed λ, d, and material parameters (ǫ l,1 , ǫ l,2 , n 2 ) the dispersion relation is computed using a shooting method. Generally, the three dimensional phase space of possible solutions spanned by E 0 , E d and β is scanned. For each triplet a set of Maxwell's equations coupling E x and E z components 16,[START_REF] Mihalache | [END_REF]21

dE z dx = k 0 β - ǫ 2 β E x , (8a)
dE x dx = k 0 βǫ 2 E z -2α 1 E z E 2 x β -ǫ2 β ǫ 2 + 2αE 2 x (8b)
is solved by numerical integration inside the waveguide core. If the solution in the core is consistent with the previously assumed field values at the slot interfaces then the corresponding β is accepted as a genuine solution of the problem. This model will be called the interface model (IM).

In our case of a symmetric slot waveguide the problem can be split in two cases: (i) symmetric and antisymmetric modes and (ii) asymmetric modes. In the first case the phase space can be reduced to two dimensions (E 0 and β) using the symmetry argument (E d = ±E 0 ). In the second case the phase space can also be reduced to two dimensions (this time E 0 and E d ) using the following method. Equating right-hand sides of the dispersion equation [Eq. (7)] for the left interface with the one for the right interface, allows one to eliminate the integration constant C. It follows that for the fixed field intensities at both interfaces (E 0 and E d ), this equation can be solved analytically providing the allowed values of the propagation constant β. The resulting equation for β has the form a 4 β 4 + a 2 β 2 + a 0 = 0 which means that we can have up to two different possible solutions for a given pair (E 0 , E d ) (β being real and positive) that have to be verified using the numerical integration of Eqs. (8).

III. RESULTS

Figure 2 shows the NSW dispersion curves for solutions corresponding to different symmetries, obtained by (a) the JEM and (b) the IM, as a function of the averaged nonlinear index modification ∆n = 1/d d 0 n 2 I dx. In the region where ∆n < 0.1 we find good quantitative agreement between results of these two models, which are radically different in this respective approach to deal with NSWs. In this region the electric field component peak ratio max{|E 2 z |}/ max{|E 2 x |} is smaller than 0.25. For values of ∆n above 0.1 both models predict qualitatively the same type of behavior but lead to quantitative differences, that can be understood looking at the assumptions used in the JEM. Such results confirm the validity of both models and of their numerical implementation. It is worth mentioning that even if the IM offers more generality than the JEM, the latter requires ten times less computational time than the former one.

The IM also provides more physical insights since the field profiles are described analytically even if it employs unusual special functions. In what follows, we describe results obtained with the IM. For low light intensity, i.e. small index modification, a symmetric and an antisymmetric mode are obtained 11 . At ∆n ≈ 0.07 an asymmetric mode bifurcates from the symmetric one 10 . The effective index β of the three modes described here grows rapidly with the increase of ∆n . The field profiles of these first order modes are presented in gray and black in Fig. 3. For larger ∆n two groups of higher order modes appear. The first group of node-less modes, appearing for ∆n 3 in the frame of IM, possesses only very high effective indices β. The insets in Fig. 2 show enlarged regions of parameter space where these modes appear, and where bifurcation of asymmetric mode from symmetric higher-order mode occurs. The typical field profiles of these higher order modes are shown in Fig. 3(a). Both the first and the second asymmetric branch are doubly degenerated (i.e. at fixed β there exist two modes with equal powers but shapes inverted with respect to the center axis x = d/2) but for simplicity we show only one of the field plots. The second group of higher order modes with nodes, appearing for ∆n 0.08 in the frame of IM, consists of an alternation of symmetric and anti-symmetric modes. These modes resemble the higher order modes of a linear slot waveguide with ǫ l,2 > 3.46 2 . The field profiles of the two first modes of this second group are provided in Fig. 3(b). In each case, the analytical approximation corresponding to a single interface problem is shown with a black dotted line. All the parameters are the same as in Fig. 2.

I d =

+∞

-∞ I dx as abscissa (data not shown). The overlap of the nonlinear dispersion curves is obtained for large differences of the effective indices between symmetric and asymmetric modes ∆β. This is the consequence of the fact that with the increase of effective index the asymmetric modes concentrate around only one of the interfaces, so that the problem reduces to a single metal/dielectric interface where the width of the core is no longer important.

As it can be expected, the curves corresponding to zero integration constants c 0 = 0 in the JEM [see Eq. ( 2)] provides solutions of the single metal/nonlinear dielectric interface problem. Using Eq. (2) of the JEM, an analytical formula for the approximated effective index of the highly asymmetric modes is derived for large ∆β values

β = ǫ 1 ǫ l,2 (ǫ l,2 -ǫ 1 )/ ǫ 2 l,2 -ǫ 2 1 + n 2 ǫ 2 1 H 2 0 /(2ǫ 0 cǫ l,2 ) , (9)
where ǫ 2 was approximated by ǫ l,2 .

For the IM, to describe the single metal/nonlinear dielectric interface problem, one has to set C = 0 in Eq. (7). The analytical solution of this equation is provided by Eq. ( 14) in Ref. 21 in the frame of the single metal-nonlinear dielectric interface problem. As it can be seen in Fig. 5, both approximations nicely fit their respective computed dispersion curves of asymmetric modes.

Figure 6 shows the dependence of the nonlinear index change ∆n at which the first bifurcation occurs (red curve) on the width of the NSW core. One can see that the thicker the core the lower the value of ∆n where the bifurcation occurs. For d ≈ 1 µm (being ≈ 2λ/3) the bifurcation occurs at ∆n between 10 -5 and 10 -3 which are realistic values for hydrogenated amorphous silicon 22 . The value of ∆n for which ∆β = 0.1 is shown in green. The values of ∆n for which the asymmetric modes become strongly asymmetric, i.e. light intensity of the asymmetric mode at one metal/dielectric interface is five times larger than on the other one, is approximately twice higher than the bifurcation threshold (data not shown). Cladding with higher permittivity also allows to reduce the ∆n where the bifurcation occurs, e.g. for 0.4-µm-wide core sandwiched between metals with permittivity equal to -40 (all other parameters unchanged) ∆n is reduced 4 times and for metals with permittivity equal to -20 ∆n is reduced 100 times.

IV. CONCLUSION

We have developed two independent models describing plasmonic slot waveguides with a nonlinear dielectric core. Qualitatively the results of the simplified, but analytical approach and the exact more numerical approach are the same. Both models predict bifurcation of asymmetric modes from the symmetric node-less modes. We show that the bifurcation occurs not only for the first but also for higher order nonlinear modes. Both the nonlinear dispersion curves and the field profiles for the higher order symmetric and antisymmetric modes are shown for the first time. We show that the first nonlinear asymmetric mode dispersion curves have invariant parts with respect to the core width and we provide analytical formula for this invariant curve. We also obtain that the intensity at which the bifurcation occurs decrease with the increase of the core width or the metal permittivity.

FIG. 1 .

 1 FIG. 1. Symmetric nonlinear slot waveguide (NSW) geometry.

FIG. 2 .

 2 FIG.2. Dispersion curves of the first and higher order symmetric (S -blue), antisymmetric (AN -red), and asymmetric (AS -green) modes for the symmetric NSW obtained with the JEM (a) and the IM (b). The insets show zooms on the regions with bifurcation of the first symmetric higher order mode. Results obtained for a 0.4-µm-wide hydrogenated amorphous silicon core ǫ l,2 = 3.46 2 (n2 = 2 • 10 -17 m 2 /W)22 sandwiched between gold ǫ1 = ǫ3 = -90 at a wavelength λ = 1.55 µm. Losses in metals are neglected.

FIG. 3 .

 3 FIG. 3. Magnetic field component for the first and higher order node-less modes (a) and the modes with nodes (b) corresponding to the points indicated in Fig. 2(b).

Figure 4 FIG. 4 .

 44 Figure4shows the dispersion curves for the parameter space region where the asymmetric mode bifurcates from the symmetric mode for different widths of the NSW core. Notice that the effective mode index is plotted as a function of the power localized in the core P c = d 0 S z dx, in which S z denotes the z-component of the Poynting vector. One can observe that, in this coordinate, the β(P c ) curves for asymmetric modes have invariant parts with respect to the core width. The same conclusion is also obtained if one considers the intensity density

FIG. 5 .

 5 FIG.5. Zoom of the dispersion curves of the first symmetric (S -blue), antisymmetric (AN -red), and asymmetric (AS -green) modes for the symmetric NSW obtained with the JEM (a) and the IM (b). In each case, the analytical approximation corresponding to a single interface problem is shown with a black dotted line. All the parameters are the same as in Fig.2.

FIG. 6 .

 6 FIG. 6. Average nonlinear index change ∆n at the appearance of the asymmetric modes (bifurcation) (red +) and for ∆β = 0.1 (green •) as a function of the core width d.

M. Kauranen and A. Y. V. Zayats, Nature Photon. 6, 737 (2012).

V. ACKNOWLEDGMENTS

This work was supported by the European Commission through the Erasmus Mundus Joint Doctorate Programme Europhotonics (Grant No. 159224-1-2009-1-FR-ERA MUNDUS-EMJD). The authors thank Y. V. Kartashov for helpful comments on this work. G. R. thanks J.-M. Fédéli for fruitful discussions.

VI. REFERENCES