Raj Pandya

Antonios M Alvertis

Qifei Gu

Jooyoung Sung

Laurent Legrand

David Kréher

Thierry Barisien

Alex W Chin

Christoph Schnedermann

Akshay Rao

Exciton Diffusion in Highly-Ordered One Dimensional Conjugated Polymers: Effects of Back-Bone Torsion, Electronic Symmetry, Phonons and Annihilation

Keywords: conjugated polymers, transient absorption microscopy, exciton diffusion, exciton-exciton annihilation, exciton trapping

Introduction

Organic optoelectronic devices ranging from light-emitting diodes to photovoltaic cells and transistors 1- 3 are frequently based on linear polymers with a C2h point group symmetry. Here, the electronic ground state (S0) exhibits 1 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Ag + symmetry. Depending on the polymer conjugation length and backbone geometry the first (S1) and second (S2) excited electronic states are either of 2 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Ag -or 1 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Bu + symmetry [START_REF] Barford | Excitons in Conjugated Polymers: Wavefunctions, Symmetries, and Quantum Numbers[END_REF][START_REF] Barford | Electronic and Optical Properties of Conjugated Polymers[END_REF] .

Irrespective of the exact state ordering, photoexcitations in conjugated polymers rapidly form excitons in the lowest energy excited S1 state. When the S1 state is of 2 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Ag -symmetry the materials are typically non-fluorescent, the S1 state supports some triplet-pair character [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] (TT) [START_REF] Race | Density Matrix Renormalization Calculations of the Relaxed Energies and Solitonic Structures of Polydiacetylene[END_REF][START_REF] Ren | Role of the Dark 2Ag State in Donor-Acceptor Copolymers as a Pathway for Singlet Fission: A DMRG Study[END_REF][START_REF] Aryanpour | Theory of Primary Photoexcitations in Donor-Acceptor Copolymers[END_REF] and frequently shows a short electronic lifetime. On-the-other hand polymers with an S1 state of 1 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Bu + character are often luminescent with long electronic lifetimes [START_REF] Hu | New Insights into the Design of Conjugated Polymers for Intramolecular Singlet Fission[END_REF][START_REF] Busby | A Design Strategy for Intramolecular Singlet Fission Mediated by Charge-Transfer States in Donor-Acceptor Organic Materials[END_REF] .

Despite our considerable understanding of the electronic structure of conjugated polymers [START_REF] Ho | Molecular-Scale Interface Engineering for Polymer Light-Emitting Diodes[END_REF][START_REF] Barford | Excitons in Conjugated Polymers: Wavefunctions, Symmetries, and Quantum Numbers[END_REF][START_REF] Laquai | Excitation Energy Transfer in Organic Materials: From Fundamentals to Optoelectronic Devices[END_REF][START_REF] Markov | Accurate Measurement of the Exciton Diffusion Length in a Conjugated Polymer Using a Heterostructure with a Side-Chain Cross-Linked Fullerene Layer[END_REF][START_REF] Gaab | Anomalous Exciton Diffusion in the Conjugated Polymer MEH-PPV Measured Using a Three-Pulse Pump-Dump-Probe Anisotropy Experiment[END_REF][START_REF] Cirera | Tailoring Topological Order and π-Conjugation to Engineer Quasi-Metallic Polymers[END_REF][START_REF] Toussaint | Linear Polyenes: The Interplay between Electronic Structure, Geometric Structure, and Nonlinear Optical Properties[END_REF] , the impact of the electronic symmetry of the S1 state on exciton diffusion behaviour remains largely unexplored. In other words, is it not known whether a 2 1 Ag -or 1 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Bu + symmetry state would provide better exciton transport properties? At first glance it might appear that the longer lifetimes of 1 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Bu + states would enable longer exciton diffusion lengths. But the significantly altered electronic structure and many-body character of 2 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Ag -states may provide access to unique diffusion properties. The experimental study of exciton diffusion in molecular systems is challenging due to the difficulties associated with measuring ultrafast, nanoscale exciton diffusion properties in systems that exhibit sub-100 ps electronic lifetimes coupled with sub-100 nm exciton diffusion lengths [START_REF] Markov | Accurate Measurement of the Exciton Diffusion Length in a Conjugated Polymer Using a Heterostructure with a Side-Chain Cross-Linked Fullerene Layer[END_REF][START_REF] Gaab | Anomalous Exciton Diffusion in the Conjugated Polymer MEH-PPV Measured Using a Three-Pulse Pump-Dump-Probe Anisotropy Experiment[END_REF] . Critically challenges associated with systematically tuning polymer electronic structure without introducing detrimental effects such as interchain disorder, cross-linking, etc. have until now prevented a systematic investigation of electronic symmetry on exciton diffusion.

Results

Here, we overcome these problems by using femtosecond transient absorption spectroscopy and microscopy [START_REF] Zhu | Exciton Transport in Singlet Fission Materials: A New Hare and Tortoise Story[END_REF][START_REF] Schnedermann | Ultrafast Tracking of Exciton and Charge Carrier Transport in Optoelectronic Materials on the Nanometer Scale[END_REF][START_REF] Pandya | Ultrafast Long-Range Energy Transport via Light-Matter Coupling in Organic Semiconductor Films[END_REF][START_REF] Sung | Long-Range Ballistic Propagation of Carriers in Methylammonium Lead Iodide Perovskite Thin Films[END_REF] to measure exciton transport in different topochemically (i.e. near defect-free as compared to solution processed materials) polymerised polydiacetylene (PDA) chains. PDAs offer an ideal platform to study the effect of electronic properties on exciton transport for several key reasons.

Structurally, PDAs consist of extended (2 to 5 µm) one-dimensional polymer chains that are distinguished by the degree of back-bone torsion. Perfectly planar polymer chains are typically referred to as 'blue' PDA, while polymer chains with a defined back-bone torsional angle (40°) are known as 'red' PDA. We note that although the 'blue' PDA actually has redder absorption peaks than the 'red' PDA, the naming convention is historical and based on the colour of the crystals [START_REF] Schott | The Colors of Polydiacetylenes: A Commentary[END_REF] . In this study the chains are dilutely (polymer content 10 -4 -10 -3 w/w) and homogeneously distributed (50 -100 nm separation) in a ~500 nm thick crystal of their monomer; the size of the domains within the macroscopic crystal are ~20 μm 2 (see SI, S1). We refer to these samples henceforth as (thin) films. PDAs have found widespread use in a range of optoelectronic applications such as temperature sensors [START_REF] Lee | Recent Progress in Stimuli-Induced Polydiacetylenes for Sensing Temperature, Chemical and Biological Targets[END_REF] , lithographic resists [START_REF] Dobisz | High Resolution Electron Beam Lithography with a Polydiacetylene Negative Resist at 50 KV[END_REF] , dye sensitised solar cells [START_REF] Pootrakulchote | A Polydiacetylene-Nested Porphyrin Conjugate for Dye-Sensitized Solar Cells[END_REF] , photoelectric devices [START_REF] Dubas | Ultrathin Polydiacetylene-Based Synergetic Composites with Plasmon-Enhanced Photoelectric Properties[END_REF] and transistors. Although it has found less use than other conjugated polymer systems such as P3HT or polyfluorene the absence of significant interchain disorder, crosslinking, and integration into nanostructures allows the theoretical properties of correlated 1D electronic systems to be tested in the idealized polymer limit in PDA, as such as it is a model system to investigate exciton transport in polymer systems. The electronic structure of PDAs are very similar to other modern conjugated polymers such as polyacenes, iso-indigo-based polymers [START_REF] Hu | New Insights into the Design of Conjugated Polymers for Intramolecular Singlet Fission[END_REF][START_REF] Busby | A Design Strategy for Intramolecular Singlet Fission Mediated by Charge-Transfer States in Donor-Acceptor Organic Materials[END_REF][START_REF] Busby | Fast Singlet Exciton Decay in Push-Pull Molecules Containing Oxidized Thiophenes[END_REF] , carotenoids [START_REF] Buckup | Ultrafast Multiphoton Transient Absorption of β-Carotene[END_REF] and singlet-fission materials making the results here applicable to many other systems.

The torsional backbone angle is controlled by the side-chains of the diacetylene monomer and exact synthetic preparation method as shown in a previous study [START_REF] Al Choueiry | Twisted Polydiacetylene Quantum Wire: Influence of Conformation on Excitons in Polymeric Quasi-One-Dimensional Systems[END_REF] . Other than the backbone torsion both PDAs show near-identical structural and chemical material properties, i.e. degrees of polymer content, polymer alignment, structural disorder, etc. (SI, Figure S1). Electronically, both PDA phases display strong coupling between the monomer units and exhibit large exciton coherence lengths [START_REF] Schott | The Colors of Polydiacetylenes: A Commentary[END_REF][START_REF] Al Choueiry | Twisted Polydiacetylene Quantum Wire: Influence of Conformation on Excitons in Polymeric Quasi-One-Dimensional Systems[END_REF][START_REF] Holcman | Coherent Control of the Optical Emission in a Single Organic Quantum Wire[END_REF][START_REF] Dubin | Macroscopic Coherence of a Single Exciton State in an Organic Quantum Wire[END_REF] . Owing to the planar backbone geometry, 'blue' PDA exhibits an optically accessible 1 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Bu + (S2) which lies above a 2 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Ag -(S1) state. This ordering gives rise to a broad visible absorption spectrum dominated by vibrational progressions at 1400-1500 and 2100 cm -1 (C=C and C≡C stretching mode, blue spectrum; Figure 1b). By contrast, the energetic ordering in 'red' PDA is reversed due to backbone torsions, resulting in a dark 2 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Ag -(S2) state lying above the optically-accessible 1 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Bu + (S1) state. As a consequence, 'red' PDA exhibits a significantly blue-shifted absorption spectrum compared to 'blue' PDA (Figure 1b).

The ensemble excited state properties of 'blue' and 'red' PDA films following photo-excitation with a 10 fs pump pulse centred at 560 nm [START_REF] Liebel | Broad-Band Impulsive Vibrational Spectroscopy of Excited Electronic States in the Time Domain[END_REF] are shown Figure 1c,d. In 'blue' PDA, the pump-probe spectrum is characterised by a sharp stimulated emission (SE) band at 650 nm, a broad photoinduced absorption (PIA) at 750 -900 nm and a second narrower PIA at 680 -700 nm (Figure 1c). By using spectral decomposition methods outlined in a previous study [START_REF] Pandya | Optical Projection and Spatial Separation of Spin Entangled Triplet-Pairs from the S1 (2 1 Ag -) State of Pi-Conjugated Systems[END_REF] and fitting the electronic decays (exponential decay convoluted with instrument response) we find that the SE (1 1 Bu + ) decays mono-exponentially to the ground state with a lifetime of 630 ± 10 fs, while the PIA at 750 -900 nm decays with a lifetime 1300 ± 40 fs and the PIA at 680 -700 nm with 2300 ± 130 fs. The observed excited-state features and dynamics agree excellently with previous reports [START_REF] Schott | The Colors of Polydiacetylenes: A Commentary[END_REF][START_REF] Pandya | Optical Projection and Spatial Separation of Spin Entangled Triplet-Pairs from the S1 (2 1 Ag -) State of Pi-Conjugated Systems[END_REF][START_REF] Kraabel | Singlet Exciton Relaxation in Isolated Polydiacetylene Chains Studied by Sub-Picosecond Pump-Probe Experiments[END_REF] . Following this agreement, we can assign the SE band to 1 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Bu + (S2), the broad PIA is attributed to the 2 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Ag -(S1) state and the narrow PIA at 680 -700 nm to a hot, vibrationally excited, ground state [START_REF] Kraabel | Singlet Exciton Relaxation in Isolated Polydiacetylene Chains Studied by Sub-Picosecond Pump-Probe Experiments[END_REF][START_REF] Kraabel | Triplet Exciton Generation, Transport and Relaxation in Isolated Polydiacetylene Chains: Subpicosecond Pump-Probe Experiments[END_REF][START_REF] Buckup | Pump-Probe and Pump-Deplete-Probe Spectroscopies on Carotenoids with N = 9-15 Conjugated Bonds[END_REF] . Photoexcitation of 'blue' PDA to 1 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Bu + (S2) is thus followed by rapid internal conversion within 90 fs to 2 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Ag -(S1) (i.e. rise time for 2 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Ag -kinetic as derived following spectral decompostion [START_REF] Pandya | Optical Projection and Spatial Separation of Spin Entangled Triplet-Pairs from the S1 (2 1 Ag -) State of Pi-Conjugated Systems[END_REF] ) as determined by spectral decomposition methods [START_REF] Pandya | Optical Projection and Spatial Separation of Spin Entangled Triplet-Pairs from the S1 (2 1 Ag -) State of Pi-Conjugated Systems[END_REF] , which decays back to the ground state. In 'red' PDA the transient absorption spectrum shows a single broad PIA in the 650 -900 nm range. The decay of this PIA is 13.8 ± 0.2 ps and has been suggested to correspond to the transition between 1 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Bu + (which is the S1 state in this system) and a state lying at ~3-3.5 eV, of n Ag or charge transfer (CT) character [START_REF] Al Choueiry | Twisted Polydiacetylene Quantum Wire: Influence of Conformation on Excitons in Polymeric Quasi-One-Dimensional Systems[END_REF] . We emphasise that we carried out both fs-TA (and fs-TAM measurements, see below) as a function of carrier density (excitations per unit length of polymer chain (n0) in a range from n0 = 0.1 -2 nm -1 ; SI, S4). In both cases these results showed that for 'blue' PDA the kinetics do not vary with carrier concentration whereas annihilation was observed for 'red' PDA. In order to extract the exciton diffusion characteristics of 'blue' and 'red' PDA we performed widefield femtosecond transient absorption microscopy (fs-TAM) with a similar 10 fs pump pulse centred at 560 nm [START_REF] Schnedermann | Ultrafast Tracking of Exciton and Charge Carrier Transport in Optoelectronic Materials on the Nanometer Scale[END_REF][START_REF] Sung | Long-Range Ballistic Propagation of Carriers in Methylammonium Lead Iodide Perovskite Thin Films[END_REF][START_REF] Pandya | Femtosecond Transient Absorption Microscopy of Singlet Exciton Motion in Side-Chain Engineered Perylene-Diimide Thin Films[END_REF] . In fs-TAM a pump pulse is focussed via a high-numerical aperture (NA=1.1) objective to near the diffraction limit Gaussian spot (σ = 143 nm), which locally creates excitons within the sample.

After a variable time delay, a broadband, 7 fs probe pulse centered at 780 nm is loosely focussed on the sample to image the pump-induced signal changes in the material. The transmitted probe is subsequently imaged onto a two-dimensional detector. By recording the transmission image with and without the pump pulse incident on the sample, we calculated the spatially-resolved transient absorption image of the locally excited exciton distribution as a function of pump-probe time delay. Tracking its temporal evolution then allows us to extract information about exciton diffusion properties. For 'blue' PDA we probed in the centre of the 2 1 Ag -PIA band at 800 nm (blue stripes in Figure 1c), while the probe wavelength in 'red' PDA was set to 790 nm to be centred with the 1 1 Bu + PIA (orange stripes in Figure 1d). Spectral decomposition based on previous studies [START_REF] Pandya | Optical Projection and Spatial Separation of Spin Entangled Triplet-Pairs from the S1 (2 1 Ag -) State of Pi-Conjugated Systems[END_REF] demonstrates that this wavelength ensures that we exclusively monitor the S1 exciton population, free of other spectral features. This point is discussed further in SI, S2 where we report the probe wavelength dependence of the fs-TAM demonstrating the 'hot' exciton and S1 diffusion properties to be distinct. Although 'blue' PDA undergoes faster electronic relaxation as compared to its 'red' counterpart it is important to note that the electronic relaxation and diffusion coefficient are not necessarily linked i.e. dark channels, internal conversion, etc, will change the lifetime but not necessarily the diffusion coefficient.

To quantify the spatial extent of the signal, we extracted the spatial standard deviation for all transient absorption images using a 2D Gaussian fit (see SI, S3). We subsequently computed the mean-squaredisplacement (MSD) for the underlying spatial exciton profile, which is a measure of the change in the spatial extent of the exciton population as a function of time. Fitting is performed with a 2D Gaussian as opposed to Bessel function (which results from the far-field radiation pattern of a transmitted planewave probe). This is due to the computational complexity associated with the latter and the fact that fitting with a 2D Gaussian shows that the diffraction rings are reproduced faithfully. This implies that the deviation from a true two dimensional spatial profile in our measurements, due to the diffraction ring, has no effect on the retrieved spatial standard deviation (see SI,S3). As shown in Figure 2a,b, both samples exhibit a pronounced increase in their MSD over the first 5 ps with 'blue' PDA displaying MSD values exceeding 'red' PDA. In both PDA phases, the exciton propagation is highly anisotropic and proceeds exclusively along a single axis (see SI, S3). This behaviour is expected based on the onedimensional structure of PDA and the low polymer content (10 -4 -10 -3 w/w) of the samples remains, preventing inter-chain transport. Several chains will be excited by the pump pulse with the pump probe signal taken as the average of all chains within the exciting spot. Our results therefore suggest that exciton propagation proceeds along the long-axis of the PDA chains (monoclinic b axis of crystals) [START_REF] Schott | The Colors of Polydiacetylenes: A Commentary[END_REF][START_REF] Al Choueiry | Twisted Polydiacetylene Quantum Wire: Influence of Conformation on Excitons in Polymeric Quasi-One-Dimensional Systems[END_REF] .

We remark that varying the probe wavelength within the broad PIA did not alter the observed transport behaviour (SI, Figure S2).

Although the PDA chains are highly ordered, their thin films are known to contain some underlying nanoscale inhomogeneity. We therefore repeated fs-TAM measurements on 30 sample locations in 'blue' PDA and 21 in 'red' PDA, to build up a statistical picture of singlet exciton transport. Furthermore, to accurately compare the spatial dynamics of both samples, we restricted our analysis to time-delays >250 fs in order to avoid effects due to S2-S1 internal conversion processes in 'blue' PDA. A salient feature of the MSD curves displayed in Figure 2a and 2b is that for 'blue' PDA the maximum MSD value reached is typically greater than 'red' PDA. In order to extract the relevant diffusion coefficients, we constructed the associated differential diffusion equation [START_REF] Crank | The Mathematics of Diffusion[END_REF] !"($,&)

!& = 𝐷 + ! , "($,&) !$ , -- "($,&) / - 0 "($,&) , √/ . ( 1
)
Here 𝑛(𝑥, 𝑡) is the electronic population at a time t and position x, D is the diffusion coefficient, τ is the electronic lifetime and 𝛾 is a coefficient which accounts for exciton-exciton annihilation. In the absence of exciton annihilation this equation can be solved to show that in one dimension,

MSD(𝑡) = 𝐴 (𝑡 -𝑡 7 ) 8 , (2)
where A is proportionality factor and α accounts for non-linearity in time. In the case of α = 1 we retrieve normal diffusion, while α > 1 is known as super-diffusive behavior and α < 1 is referred to as subdiffusive behavior. In this latter case a time-dependent diffusivity D(t) can be defined as,

D(t) = 9 : 𝐴 (𝑡 -𝑡 7 ) 8;9 . (3)
In the presence of annihilation no analytical solution exists and equation ( 1) must be fit to extract D and 𝛾 [START_REF] Crank | The Mathematics of Diffusion[END_REF] (see SI, S3 for further details). We emphasize that in the data shown in Figure 2,3, α is an empirical fix applied to the equation to best match the experimental data as opposed to being derived from the analytical solution to equation 1. In the case of 'blue' PDA (a) there is no dependence of the MSD on the carrier density, whereas in 'red' PDA (b) the diffusion coefficient increases at higher pump fluences. In this latter case exciton-exciton annihilation effects can be considered to be playing a role. n0 represents the number of (initial) excitations per nm of polymer chain.

To identify, which model should be applied, we carried out a fluence dependence on the same sample spot for both PDAs, as shown in Figure 3. Here, we find that the transient spatio-temporal dynamics of 'blue' PDA do not vary with power (Figure 3a), while 'red' PDA displays a pronounced power dependence (Figure 3b), indicative of a higher annihilation rate. This increased annihilation rate of 1 1 Bu + compared to 2 1 Ag -excitons is an important observation that is not well understood. While beyond the scope of this article, we note that the 2 1 Ag -state exhibits a lower transition dipole moment than 1 1 Bu + 7,37 . Given that the annihilation cross-section of a state is proportional to its transition dipole moment 38- 40 this may explain why the 2 1 Ag -symmetry excitons do not exhibit significant annihilation effects [START_REF] Tempelaar | Exciton-Exciton Annihilation Is Coherently Suppressed in H-Aggregates, but Not in J-Aggregates[END_REF] . A similar effect has been observed in H-and J-aggregate systems. In the former, where S1 is dark, the phase relationship between two dipoles leads to destructive interference and suppressed annihilation, whereas in J-aggregates where S1 is bright the opposite occurs [START_REF] Tempelaar | Exciton-Exciton Annihilation Is Coherently Suppressed in H-Aggregates, but Not in J-Aggregates[END_REF][START_REF] Ryzhov | Low-Temperature Kinetics of Exciton-Exciton Annihilation of Weakly Localized One-Dimensional Frenkel Excitons[END_REF][START_REF] Scheblykin | Strong Nonmonotonous Temperature Dependence of Exciton Migration Rate in J Aggregates at Temperatures from 5 to 300 K[END_REF] . The 2 1 Ag -excitons can still scatter in such a way that would lead to annihilation (impact ionisation, Auger effects, etc), but this would require the excitons to be very close, or at high densities. We note that it is possible that local geometry/structure around the excitons prevents them from approaching each other, and that the excitation density does not reach the magnitudes required for short-range annihilation mechanisms to be operative (on the timescales of internal conversion). Computing the inter-exciton separation (at the highest excitation densities used in the study) shows that excitons remain separated by at least 1 monomer unit in both 'blue' and 'red' PDA suggesting Auger and impact ionisation events can indeed be ignored (see SI, S5). Other effects such as disorder or local microstructure may in addition to the symmetry of S1 play a role in supressing annihilation in 'blue' PDA. Study of other polyene systems such as carotenoids which have been shown to have similar dynamics to PDA [START_REF] Pandya | Optical Projection and Spatial Separation of Spin Entangled Triplet-Pairs from the S1 (2 1 Ag -) State of Pi-Conjugated Systems[END_REF] , may aid this, however independently investigating the effects of symmetry remains challenging.

Based on the observed power dependence, we applied equation 1 to directly extract diffusion constant, D, and annihilation parameter, 𝛾 for 'red' PDA. In contrast, for 'blue' PDA we fit equation 2 to the MSD trace to extract A and α and then extract the diffusion coefficient from the time-dependent diffusivity D(t) according to equation 3. Following this analysis, we derive a mean diffusion coefficient of 34 ± 10 cm 2 s -1 for 'blue' PDA (value at 2500 fs, see SI, S3 for full D(t)), which is significantly larger and more varying across different sample locations as compared to 'red' PDA with 7 ± 6 cm 2 s -1 . We remark that the spread in diffusion coefficients for 'blue' PDA is larger than for 'red', however further studies are required to ascertain the origin of such behaviour. The exponent value, α, for 'blue' PDA lies between 0.7 -0.9 and indicates sub-diffusive exciton transport. For 'red' PDA the annihilation coefficient, 𝛾, ranged between 0.1 -1.1 cm s -0.5 (SI, S5). The units of cm s -0.5 reflect the one-dimensional nature of polymer. The annihilation coefficient, like the diffusion coefficient, will be time dependent, however is found not to vary in the time range studied here nor is found to vary significantly with sample location (SI, S5). We note that to obtain D for 'red' PDA the fluence dependent traces were fit independently. The data in Figure 2 captures the heterogeneity (between sample locations) at a single fluence. The variation in the annihilation coefficient is further captured by repeating fluence dependent measurements at four different sample locations; 𝛾 is found to have a maximum spread of 0.21 cm s -0.5 .

Critically, while this annihilation factor is substantial, the ratio of the first and third term in equation 1 is ~3 -5 at early times, rising to ~10 -20 after 3 ps (SI, S5). This suggests that the annihilation is still only a minor contributor to the sub-diffusive transport [START_REF] Deng | Long-Range Exciton Transport and Slow Annihilation in Two-Dimensional Hybrid Perovskites[END_REF] . In comparison to other conjugated polymer systems the annihilation coefficient in 'red' PDA is around 10 times lower e.g. as compared to polyfluorene [START_REF] Sneyd | Highly Efficient Exciton Transport via Transient Delocalization in Films of Poly(3-Hexylthiophene) Nanofibers Prepared by Seeded Growth[END_REF] where 𝛾 was measured to be ~3 cm s -0.5 . However, 𝛾 is also 2 -3 orders of magnitude larger than in these systems [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] and more comparable to that observed in 2D TMDC materials such as MoS2 [START_REF] Sun | Observation of Rapid Exciton-Exciton Annihilation in Monolayer Molybdenum Disulfide[END_REF] , black phosphorous [START_REF] Surrente | Onset of Exciton-Exciton Annihilation in Single-Layer Black Phosphorus[END_REF] and 2D perovskites [START_REF] Delport | Exciton-Exciton Annihilation in Two-Dimensional Halide Perovskites at Room Temperature[END_REF] . Comparison of absolute values should however be performed with care due to the time dependent nature of the diffusion coefficient and the difference in dimensionality between these systems. At early times (sub-1 ps) 𝛾 will be higher due to the radiative decay of excitons, the 𝑡 -1/2 dependence of the annihilation parameter and static annihilation; we do not consider this latter contribution at the fluences used here in-line with other studies [START_REF] Sneyd | Highly Efficient Exciton Transport via Transient Delocalization in Films of Poly(3-Hexylthiophene) Nanofibers Prepared by Seeded Growth[END_REF] . We note for 'red' PDA, the diffusion coefficients extracted in Figure 2c are from traces at the lowest fluence used in this study (n0 (carriers per unit length of polymer) ~ 0.1 nm -1 ).

Discussion

Having discussed the origin of the observed exciton dynamics we can determine the overall diffusion length from

𝐿 > = ?2𝐷𝜏 BCBD , ( 5
)
where τelec is the intrinsic lifetime of the underlying process. Previously we have determined for 2 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Ag -, τelec is ~600 fs, whereas for 1 1 Bu + , τ ~9000 fs [START_REF] Pandya | Optical Projection and Spatial Separation of Spin Entangled Triplet-Pairs from the S1 (2 1 Ag -) State of Pi-Conjugated Systems[END_REF] . Based on these values we can extract diffusion lengths of 26 ± 6 nm and 33 ± 5 nm, for 'blue' and 'red' PDA, respectively. It is noteworthy to mention that the diffusion length is similar for both systems, despite their stark difference in diffusion coefficient and annihilation characteristics. Long-range exciton transport thus requires not only highly mobile excitons with minimized annihilation, but also long state lifetimes. We re-emphasise at the polymer concentrations used here, the PDA chains do not interact, with the separation between chains (~100 nm)

being larger than the exciton diffusion length. Indeed, several studies have shown that inter-chain hopping and the formation of inter-chain charger transfer states does not occur in PDA [START_REF] Brabec | The Spin Signature of Charged Photoexcitations in Carbazolyl Substituted Polydiacetylene[END_REF][START_REF] Sariciftci | Absence of Photoinduced Electron Transfer from the Excitonic Electron-Hole Bound State in Polydiacetylene Conjugated Polymers[END_REF] . In a previous study it was also shown that exciting at the band-edge of PDA, whose band-edge absorption is dominated by partially polymerised states, does not show any new spectroscopic signatures or electronic lifetime changes [START_REF] Pandya | Optical Projection and Spatial Separation of Spin Entangled Triplet-Pairs from the S1 (2 1 Ag -) State of Pi-Conjugated Systems[END_REF] . This again furthers the conclusion that hopping between chains does not occur.

Finally, the low temperature absorption linewidth of crystals studied here is similar to that reported for isolated chains in previous studies (SI, S1) , this again suggests that inter-chain interactions are not significant [START_REF] Spagnoli | Temperature Dependence of Spectroscopic Properties of Isolated Polydiacetylene Chains Strained by Their Monomer Single Crystal Matrix[END_REF] . These observations further corroborate the claim that defect states generally play little role in our observations here. Performing fs-TAM measurements on thicker PDA crystals which have a lower Urbach energy 31 (30 meV versus 34 meV) shows that the diffusion coefficient and dynamics remain qualitatively unchanged (SI, S4). There is however a small difference (0.6 -2 cm 2 s -1 ) in the diffusion coefficient between the thick and thin crystals which may be related to the ~10% difference in electronic disorder; further studies are needed to fully confirm this. We finally also note that PDA crystals were masked under a polarised optical microscope prior to measurement (SI, S1) to ensure excitation was performed away from grain boundaries which can be well resolved in the materials. As such although the energetic disorder is larger than kBT in the materials our results suggests that trap states are not significant in our observations or responsible for the difference in transport between 'blue' and 'red' PDA.

Despite the high diffusion coefficients obtained at room temperature in these systems (Figure 2), which would be suggestive of a strong tendency of excitons to delocalize, we observe sub-diffusive exciton transport in both 2 1 Ag -and 1 1 Bu + states, which is typical of strong exciton-phonon interactions [START_REF] Vlaming | Subdiffusive Exciton Motion in Systems with Heavy-Tailed Disorder[END_REF] that lead to localized excitons. In order to understand these seemingly inconsistent observations, we employ ab initio GW-BSE calculations to quantify the two competing effects. These calculations on the properties of excitons and exciton-phonon interactions have previously been systematically tested on PDAs as a part of a separate study [START_REF] Alvertis | First Principles Modeling of Exciton-Polaritons in Polydiacetylene Chains[END_REF] . The electronic coupling between excitons residing on neighbouring monomers of the polymer chain (J-coupling) drives the system to a more delocalized state. From a calculation of the exciton bandwidth (the range of the energy-momentum dispersion along the chain, SI, S6) we could determine 𝐽 = 0.54 eV for the 1 1 Bu + state of the planar ('blue') PDA.

Unfortunately, the multi-excitonic character of the 2 1 Ag -state prevents us from obtaining the corresponding J value within GW-BSE, which is a methodology based on Green's functions that only captures single excitations [START_REF] Rohlfing | Electron-Hole Excitations and Optical Spectra from First Principles[END_REF] and cannot describe biexcitons [START_REF] Refaely-Abramson | Origins of Singlet Fission in Solid Pentacene from an Ab Initio Green's Function Approach[END_REF] . Extending this formalism to describe excited states involving more than two particles is an active field of research and beyond the scope of this work. Nevertheless, the larger diffusion coefficient of the 2 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Ag -state makes it likely that it will have a value of 𝐽 that is greater than 0.54 eV. The 1 1 Bu + state of the 'red' PDA will have a smaller excitonic coupling compared to the same state in the 'blue' phase due to the decreased p-orbital overlap that the torsion 𝜃 between subsequent monomers induces, and in particular a simple estimate of its value is 𝐽 • 𝑐𝑜𝑠𝜃 = 0.41 eV. We note that while the 2 1 Ag -state has been shown to possess 1 (TT) character, no free triplets [START_REF] Pandya | Optical Projection and Spatial Separation of Spin Entangled Triplet-Pairs from the S1 (2 1 Ag -) State of Pi-Conjugated Systems[END_REF] have been observed to form in the specific type of red blue PDA crystals studied here, hence we do not consider their contribution here. This in accordance with the low triplet yield via singlet fission in PDA [START_REF] Kraabel | Triplet Exciton Generation, Transport and Relaxation in Isolated Polydiacetylene Chains: Subpicosecond Pump-Probe Experiments[END_REF][START_REF] Lanzani | Triplet-Exciton Generation Mechanism in a New Soluble (Red-Phase) Polydiacetylene[END_REF] and the generally low intersystem crossing yields in conjugated polymers [START_REF] Beljonne | Spin-Orbit Coupling and Intersystem Crossing in Conjugated Polymers: A Configuration Interaction Description[END_REF] . This in agreement with the significantly lower diffusion coefficient reported for triplet excitons [START_REF] Lunt | Exciton Diffusion Lengths of Organic Semiconductor Thin Films Measured by Spectrally Resolved Photoluminescence Quenching[END_REF][START_REF] Mikhnenko | Exciton Diffusion in Organic Semiconductors[END_REF] (2 -3 orders of magnitude lower than observed here for singlets) and predictions of a lower J coupling [START_REF] Refaely-Abramson | Origins of Singlet Fission in Solid Pentacene from an Ab Initio Green's Function Approach[END_REF] .

Moreover, we quantify the strength of exciton-phonon coupling by calculating the reorganization energy of the 1 1 Bu + state of PDA to be λ = 0.43 eV (SI, S7), which denotes the driving force of the system to localize after photoexcitation. The carbon-carbon double-and triple-bond stretch have the largest contribution to the reorganization energy (see Huang-Rhys factor, Table S2), resulting in the prominent vibronic progression of the absorption spectrum of Figure 1b. This can be intuitively understood from the fact that these vibrational motions allow the polymer chains to transiently explore configurations closer to the limit of a structure without Peierls distortion and hence to the metallic limit [START_REF] Kivelson | First-Order Transition to a Metallic State in Polyacetylene: A Strong-Coupling Polaronic Metal[END_REF] . Indeed, the exciton wavefunction is significantly affected when these vibrations are displaced compared to other motions, as shown in Figure S6 of the SI. Moreover, phonons are known to lead to a renormalization of exciton properties compared to a 'static' picture [START_REF] Alvertis | Impact of Exciton Delocalization on Exciton-Vibration Interactions in Organic Semiconductors[END_REF] . In order to quantify the vibrationally-induced renormalization of properties relevant to exciton diffusion in PDA we employ a quadratic approximation (SI, S7), which allows us to identify the effect of the individual phonon modes.

We find that phonons at room temperature only lead to a small increase of the exciton energy by 15 meV, and a small decrease of the excitonic coupling by 5 meV. However, we find that the magnitude of the exciton transition dipole moment undergoes a more significant renormalization due to the effect of phonons, from a value of 1.82 a.u. to 1.30 a.u., and it is mainly the carbon-carbon double-and triplebond stretching motions that drive this effect (SI Table S3). This reduction in the magnitude of the transition dipole moment |𝝁| results in reduced exciton-exciton annihilation, since the cross-section of this process is proportional to |𝝁| as already discussed previously. Hence phonons appear to quench the competing to exciton diffusion effect of exciton-exciton annihilation.

Based on the reorganization energy and J-coupling strength, we can now classify the transport regime in which PDAs operate. If J-coupling is small compared to the reorganization energy, we anticipate the system to localize on a single monomer and transport will only occur via incoherent hopping.

Conversely, in our PDA films, we calculated, J ~ λ. This suggests that exciton transport may operate in the coherent regime > where exciton delocalization is dominant [START_REF] Aragó | Regimes of Exciton Transport in Molecular Crystals in the Presence of Dynamic Disorder[END_REF][START_REF] Aragó | Dynamics of the Excitonic Coupling in Organic Crystals[END_REF] . However, the similarity between J and λ also suggests that exciton localization via exciton-phonon couplings can still occur. The subtle interplay between exciton-phonon coupling (localization) and J-coupling (delocalization) demonstrates that exciton transport occurs through the motion of partially delocalized excitons that reside on several monomers. This leads to sub-diffusive exciton transport with high diffusion coefficients. We expect this conclusion to carry to the 2 1 Ag -state, since the 𝐽 coupling of this state is likely larger than that of 1 1 Bu + , as its experimentally measured more rapid delocalization suggests. Moreover, increasing exciton delocalization has been shown to lead to reduced coupling to high-frequency phonons in organic materials [START_REF] Alvertis | Impact of Exciton Delocalization on Exciton-Vibration Interactions in Organic Semiconductors[END_REF] . Since the reorganization energy is dominated by the contribution of such high-frequency modes (Table S2), we expect its value to be smaller for the 2 1 Ag -compared to the 1 1 Bu + state, and for the criterion 𝐽 > 𝜆 that defines coherent transport to still hold.

Conclusion

In summary we have investigated the influence of electronic state symmetry on exciton diffusion in conjugated polymers. We have shown 2 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Ag -excitons have on average ~3 times higher diffusion coefficients than 1 1 Bu + excitons in PDA polymer chains. Where the exciton transport of 1 1 Bu + excitons exhibits annihilation effects, for 2 1 Ag -excitons, the smaller transition dipole moment likely results in no measurable annihilation. The motion of both 2 1 Ag -and 1 1 Bu + excitons however appears to remain sub-diffusive likely due to exciton-phonon mediated trapping effects. Despite its smaller diffusion coefficient, the longer electronic lifetime of 1 1 Bu + means that 2 1 Ag -and 1 1 Bu + excitons have a similar diffusion length (~30 nm). Our results suggest that 2 1 Ag -excitons are able to move across space as effectively as excitations that live orders of magnitude longer. This is in part due to the absence of annihilation losses.

The polymer backbone geometry will also play a role i.e. 'blue' PDA is planar while the 'red' PDA is twisted. This will also influence coupling across monomers along the chain via orbital overlap and the extent of exciton delocalization. However, these effects must be combined with electronic correlation phenomena, such as the formation of the 2 [START_REF] Tamai | Exciton Diffusion in Conjugated Polymers: From Fundamental Understanding to Improvement in Photovoltaic Conversion Efficiency[END_REF] Ag -, such that a simple band picture with different hopping matrix elements cannot explain all of our observations. Indeed, it is not obvious that the dispersion (which will determine the group velocity and later the diffusion rates) is necessarily larger in the 'blue' PDA, compared to the 'red'. For example, if the 2 1 Ag -has a strong triplet-pair contribution, as some studies have suggested, one might even imagine that the excitation would spread more slowly, even though the monomers have stronger electronic coupling.

In terms of design rules for organic electronics our results highlight a number of key principles. Firstly, even in highly ordered systems introducing a small amounts of electronic disorder can potentially affect the transport behaviour as evidenced by measurements on PDA crystals of different thickness.

Additionally, the results suggest that not only delocalisation (via the backbone twist angle) is important in enhancing exciton diffusion but the symmetry of S1 should be considered when designing new conjugated polymers for exciton transport both in terms of annihilation and diffusion properties. Our results describing the interplay of electron-phonon coupling and J-coupling suggest that careful consideration of bonding motifs should be taken such that the contribution of both to transport can be optimised. Finally, although PDA has not found as widespread used in organic electronics as other semiconducting polymers our results with high diffusion coefficients call for a renewed interest in the material.

Methods

Sample preparation

'Blue' Polydiacetylene 3BCMU (3-methyl-n-butoxy-carbonylmethyl-urethane) diacetylene molecules were synthetized inhouse using the method previously outlined by Se et al. and references therein [START_REF] Kazunori | Urethane Substituted Polydiacetylene: Synthesis and Characterization of Poly[4,6-Decadiyn-1,10-Diol-Bis(n-Butoxy-Carbonyl-Methyl-Urethane)[END_REF] . The synthesis classically consisted of two steps: (i) oxidative coupling of 4-pentyn-1-ol (Hay's method) to produce the 4,6-decadiyn-1,10-diol and then, (ii) reaction of the diol with n-butylisocyanate acetate. Note that the source 4-pentyn-1-ol was not synthetized at the laboratory but purchased from Sigma-Aldrich (Merck). Ultrathin single crystals were grown between two coverslips using a melt-processing method.

The whole process was systematically carried out under a polarized optical microscope so as to be able to follow and control the sample elaboration: a very small amount of diacetylene powder is placed at one edge of the double-slides assembly; when heating above the melting temperature (~65°C) the liquid diacetylene fills the empty space by capillary action to form a thin liquid film between the two substrates. Rapid cooling leads to the formation of a highly polycrystalline film. The sample is then heated again to around the melting temperature until the melting of all the crystallites took place. When only a few crystal germs remain the sample is cooled again at a very slow cooling rate (typically < 0.1°C/mn) to induce the growth of large single monocrystalline domains from the germs. Typical polymer contents by weight are then in the 10 -4 -10 -3 range. The interchain separation (~100 nm; homogeneous distribution) is obtained from the absorption optical density (𝑂𝐷) and Beer Law, where 𝑂𝐷 = ∝ ×C ×D :.X where ∝ is taken as ~1 × 10 Z cm -1 20 for light polarized parallel to the long axis of the chains and 𝑙 (the thickness of the crystal) is ~500 nm. The spatial separation between chains is assumed to be significantly large enough that there are negligible inter-chain interactions. X-Ray diffraction [START_REF] Pandya | Optical Projection and Spatial Separation of Spin Entangled Triplet-Pairs from the S1 (2 1 Ag -) State of Pi-Conjugated Systems[END_REF] and microscopic optical transmission measurements (SI, S1) also confirm that chains remain linear and highly aligned over macroscopic length scales, again suggesting that inter-chain interactions can be ignored.

'Red' Polydiacetylene

To make multi-pulses microscopy possible by limiting scattering of the pump pulse ultra-thin crystalline monodomains of the ('red' PDA: 3Nf2) diacetylene were grown before being slightly polymerized.

Large size crystalline regions (~ mm 2 ) with sub-micronic thicknesses (~ 100 -200 nm) are obtained using a melt processing method [START_REF] Ribierre | A Solvent-Free and Vacuum-Free Melt-Processing Method to Fabricate Organic Semiconducting Layers with Large Crystal Size for Organic Electronic[END_REF] . A few mg of the purified diacetylene powder are melt (T ≥ 80 °C) and the liquid is injected, using capillarity action, in the space between two superimposed microscope coverslips. After rapid cooling the thin liquid film crystallization leads to the formation of a highly polycrystalline structure, being the assembly of microdomains. The sample is then heated again under a polarized optical microscope until almost all the melting takes place but preserving a few single crystal germs. At that point the sample is cooled again at a very slow cooling rate to induce the growth of hundreds of µm -mm scale domains from the germs until room temperature is reached. The (3Nf2) is a highly stable monomer that is characterized by both weak thermal-and weak UV photo-reactivity [START_REF] Al Choueiry | Twisted Polydiacetylene Quantum Wire: Influence of Conformation on Excitons in Polymeric Quasi-One-Dimensional Systems[END_REF] .

Polymer chains are thus generated by exposure to X-ray light in a diffractometer (Rigaku, Smartlab).

The polymer content is adjusted by trial and error and kept low enough (typically below 0.1 % in weight) to form a solid solution of isolated polydiacetylene chains inside their diacetylene host crystal.

Absorption spectroscopy

Polarised absorption spectroscopy of PDA crystals was performed using a home built setup with a white light source generated by focussing the fundamental of a Yb-based amplified system (PHAROS, Light Conversion) into a 4 mm YAG crystal. The resulting absorption (corrected for the sample substrate) was then collected by imaging with a Silicon photodiode array camera (Entwicklunsbüro Stresing; visible monochromator 550 nm blazed grating).

Femtosecond pump-probe spectroscopy

The fs-TA experiments were performed using a Yb-based amplified system (PHAROS, Light Conversion) providing 14.5 W at 1030 nm and 38 kHz repetition rate. The probe beam was generated by focusing a portion of the fundamental in a 4 mm YAG substrate and spanned from 520 nm to 1400 nm. The pump pulses were generated in home-built noncollinear optical parametric amplifiers (NOPAs). The NOPAs output (∼4 to 5 mW) was centred typically between 520 and 560 nm (FWHM ~65-80 nm), and pulses were compressed using a chirped mirror and wedge prism (Layerterc) combination to a temporal duration of ~9 fs. Compression was determined by second-harmonic generation frequency-resolved optical gating (SHG-FROG; upper limit) and further confirmed by reference measurements on acetonitrile where the 2200 cm -1 mode could be resolved. The probe white light was delayed using a computer-controlled piezoelectric translation stage (Physik Instrumente), and a sequence of probe pulses with and without pump was generated using a chopper wheel (Thorlabs) on the pump beam. The pump irradiance was set to a maximum of 38 μJ/cm 2 . After the sample, the probe pulse was split with a 950 nm dichroic mirror (Thorlabs). The visible part (520-950 nm) was then imaged with a Silicon photodiode array camera (Entwicklunsbüro Stresing; visible monochromator 550 nm blazed grating). The near infrared part was imaged using an InGaAs photodiode array camera (Sensors Unlimited; 1200 nm blazed grating). Measurements were carried out with a time step size of 4 fs out to 2 ps to minimize the exposure time of the sample to the beam. Unless otherwise stated, all measurements were carried out with the probe polarisation set parallel with respect to that of the pump

Figure 1 :

 1 Figure 1: Structural and optical characterisation of 'blue' and 'red' PDA. a. Chemical structure and packing side-chains of 'blue' (left) and 'red' (right) PDA. In 'blue' PDA the backbone is planar, whereas in 'red' PDA there is a 40° twist angle between adjacent monomer units in the chain. b. Absorption spectra of 'blue' and 'red' PDA. c-d. Selected spectral slices of pump-probe spectra of 'blue' (c) and 'red' PDA (d). The photoinduced absorption bands of the 2 1 Ag -and 1 1 Bu + states are marked. Shaded regions indicate probe wavelengths used in transient absorption microscopy experiments. The right panels show kinetics and associated exponential decay fits at indicated probe wavelengths.

Figure 2 :

 2 Figure 2: Femtosecond transient absorption microscopy results for 'blue' and 'red' PDA. a-b. MSD curves for 'blue' (a) and 'red' PDA (b), respectively. Solid line shows average curves whereas faint lines are from individual sample locations. Due to the faster decay in 'blue' PDA, the MSD was only recorded to 2.3 ps. c. Scatter-box plot of diffusion coefficients for 'blue' and 'red' PDA. Boxed area is 25-75% range, with horizontal line representing median and whiskers for interquartile range. Filled black rectangles represent the mean.

Figure 3 :

 3 Figure 3: Femtosecond transient absorption microscopy as a function of carrier density for 'blue' and 'red' PDA. a-b.In the case of 'blue' PDA (a) there is no dependence of the MSD on the carrier density, whereas in 'red' PDA (b) the diffusion coefficient increases at higher pump fluences. In this latter case exciton-exciton annihilation effects can be considered to be playing a role. n0 represents the number of (initial) excitations per nm of polymer chain.

Acknowledgements

We acknowledge financial support from the EPSRC and Winton Program for the Physics of Sustainability. R.P. additionally thanks the EPSRC for a Doctoral Prize Fellowship, Andrew Musser (Cornell University, USA) for useful initial discussions, and Lily Russell-Jones (London School of Economics) and Matthew Storer (London), for support during the preparation of this manuscript. R.P. also thanks Alexander J. Sneyd (Cambridge) for invaluable discussions on annihilation modelling.

A.M.A. acknowledges the support of the Engineering and Physical Sciences Research Council

(using a half-waveplate; Eksma) and along the PDA chains. The absorption spectrum of samples was measured after each pump-probe sweep to account for any sample degradation.

Femtosecond pump-probe microscopy

The femtosecond wide-field detected transient absorption microscope has been described in detail previously. [START_REF] Sung | Long-Range Ballistic Propagation of Carriers in Methylammonium Lead Iodide Perovskite Thin Films[END_REF][START_REF] Pandya | Femtosecond Transient Absorption Microscopy of Singlet Exciton Motion in Side-Chain Engineered Perylene-Diimide Thin Films[END_REF] Briefly, a Yb:KGW amplifier system (LightConversion, Pharos, 5 W, 180 fs, 1030 nm, 200 kHz) was used to seed two white-light stages for pump and probe generation. The pump white-light (3 mm Sapphire) was spectrally adjusted with a 650 nm short-pass filter (FESH650, Thorlabs), and compressed to 10 fs for all optical elements with two pairs of third-order compensated chirped mirrors and a wedge-prism pair (Layertec). Subsequently, the mode of the pump pulse is cleaned by a pinhole before being focused through the objective lens (NA = 1.1, oil immersion) to a spot size of ~340 nm (full-width-half-maximum). The probe white-light (3 mm YAG) was spectrally adjusted to 650 -900 nm in a home-build fused silica prism filter and compressed to 7 fs with a pair of third-order compensated chirped mirrors and a wedge-prism pair (Venteon) before being free-space focused onto the sample (20 micron Gaussian spot size full-width-half-maximum). The transmitted probe was imaged onto an emCCD (Rolera Thunder, Photometrics) at 55.5 nm/pixel as verified by a resolution target. The frame rate of the camera was set to 30 Hz with an integration time of 11 ms and pump off/on images were generated by a mechanical chopper at a frequency of 15 Hz. For the measurements, we adjusted the pump fluence to achieve initial concentration of carriers per unit length of polymer (n0) in the range: n0 = 0.1 -2 nm -1 for all samples.

Supporting Information

Optical microscopy of PDA, photothermal deflection spectroscopy, TAM modelling methods, pump fluence dependence, annihilation in 'blue' and 'red' PDA, theoretical modelling (EPSRC) for funding under grant EP/L015552/1. C.S. acknowledges financial support by the Royal Commission of the Exhibition of 1851.

Associated Content

The data associated with this manuscript is freely available at [url to be added in proof].


Antonios Christonasis
email: achristonasis@cc.uoi.gr

Damien Givry
email: damien.givry@univ-amu.fr

Can we quantify the stability of a teaching?

Keywords: Stability of teaching practices, Activity Theory, semiotic resources, video analysis

Our research proposes to show how we can quantify the stability of teaching, by refining a classical model of Activity Theory. Videotaped lessons of a model on energy chain, from two

French 5 th grade classes with the same teacher, were analysed in order to assess the percentage of stability as to: (i) realised subtasks and duration (ii) communicated themes and (iii)used semiotic resources. Using a self-designed scale, we show that these three-layered practices of the teacher ranged from stable to strongly stable.

INTRODUCTION

The stability of the practices of the teachers has already been underlined by [START_REF] Crahay | Contraintes de situations et interactions maître-élève, changer sa façon d'enseigner, est-ce possible?[END_REF], who indicated that teachers possibly modify their project during the preparation and not at the realisation of it. [START_REF] Berliner | Learning about and learning from expert teachers[END_REF] pointed out that the expert teachers automate the necessary repetitive operations, but they also make "in-flight" decisions. [START_REF] Hache | L'univers mathématique proposé par le professeur en classe[END_REF] searched and found stability in the discourse of the teacher, while Vandebrouck (2002) revealed invariances in the use of the blackboard by the same teacher in different classes. Pariès, Robert and Rogalski (2008) found that the main stabilities are linked with the precise management of the lesson's tasks and with the functions of the teachers' discourse.

Almost six decades ago, Rosenshine (1970) asked the question "A teacher who is effective or ineffective once is equally effective or ineffective a second time?" To answer this question, the constant effectiveness of a teacher (i.e. the positive impact on children's learning) should be linked with a stable characteristic, which may be the stability of the practices. Before the scientific community comes to a conclusion about this hypothesis, a way to measure the stability of the practices has to be developed, which is the main purpose of this paper. Based on the analysis of videotaped data of the same lesson being taught to two classes of 10 years old students (5 th grade), we propose a methodology to accomplish that. In order to do so, we refined a model of the Activity Theory, which combined with concepts linked to Social Semiotics worked as the theoretical framework.

THEORETICAL FRAMEWORK

Activity Theory

The organisational elements of the theory are: (1) the task, i.e. what the subject has to do, or "the aim to be reached under certain conditions" (Leontiev, 1978) and (2) the activity, i.e. what the subject really does.

Researches such as Rogalski ( 2008) make the distinction between the 'prescribed task' (i.e. the task according to the point of view of the person that prescribes it) and the 'effective task' (i.e. the subject's representation of what they have to do in the task). Since the prescribed task is redefined by the subject due to the representation, they have for it, we defined as realised task what the subject realised in relation to that task as such.

The classical model of Leontiev (1978) presents activity in a hierarchical system, in which activity is composed of actions, and these actions of operations. The actions of the teacher have a cognitive dimension, through the clarification of a proposed cognitive path, and a mediatory dimension, the actions of the teacher through communication (Pariès, Robert & Rogalski, 2008). By combining them, we could say that the teacher acts in order to communicate a cognitive path to the students, which is embodied in the themes communicated in the discourse.

As discussed by [START_REF] Jewitt | Teaching and learning: Beyond language[END_REF], the analysis of communication that does not take into account the full repertoire of the active semiotic resources is incapable of perceiving all the expressed meanings, and since, the operations are the means of the actions (see Leontiev, 1978), the semiotic resources constitute the operations of the communicative actions.

Social Semiotics

The social semiotics deals with "the act of meaning making" (Thibault, 2004, p. 68) underlying that the communication is made up of the production and the interpretation of semiotic resources. Semiotic resources considered to be both actions and objects (Van Leeuwen, 2005), which are produced physiologically (with the vocal system, using muscles such as facial expressions and gestures) or technologically (ink and paper, software and hardware etc).

The meaning of any semiotic resource is inherently partial [START_REF] Airey | A disciplinary discourse perspective on university science learning: Achieving fluency in a critical constellation of modes[END_REF], since different semiotic resources emphasize different aspects of a concept, and it is not always specific, as there is a set of different meanings that any given semiotic resource can realise. According to [START_REF] Kress | Multimodality: A social semiotic approach to contemporary communication[END_REF], each semiotic resource has a meaning potential, different pieces of which, are activated in different contexts. As each individual resource, as well as their integration, can convey meaning in specific ways, the function of the semiotic resources should be deduced from the interplay of all the resources being used [START_REF] Givry | Toward a new conception of conceptions: Interplay of talk, gestures, and structures in the setting[END_REF]. Evaluating one of the above entities separately, the significance which results from their mutual interdependence, as a holistic meaning unit, is stripped down (Pozzer-Ardenghi & Roth, 2007).

As [START_REF] Givry | Ambiguities in representing the concept of energy: A semiotic approach[END_REF] explain, a typical semiotic approach in science teaching can focus on (a) acoustic signs (linguistic and paralinguistic) (b) kinesic signs (gestural sings, proxemics and mimic signs) and spatial signs (scenery and scenery objects). The gestural signs include gestures, (i.e. semiotic movement of hands and arms), and specifically forms which are called gesticulation: symbolic (descriptive), deictic (pointing) and ergotic (manipulation of objects) gestures. Writing and designing can be considered as (ergotic) gestures [START_REF] Flusser | Gestures[END_REF], the material result of manipulating an object that leaves a trace.

METHODOLOGICAL FRAMEWORK

Refined model

The following figure summarises the relevant elements used to study the teacher's activity. The purpose of the teaching is the gradual introduction of the energy chain model, which will take place through seven subtasks within a one-hour lesson to two classes of 5 th grade.

FIGURE 1

The refined model

Teaching content

The elements of the energy chain are the: (1) Object: which can emit light, heat or be in motion.

(2) Source of energy: a product which is used directly to produce energy. (3) Transformer: a factory or a living being that transforms a natural resource into a source of energy (4) Primary source (called Resource): a raw product collected in nature.

The associated geometric forms: a circle for the object, a rectangle for the source and resource of energy and a triangle for the transformer (Figure 2).

FIGURE 2

Energy chain model

The arrow was not used to represent the transfer of energy, but to illustrate "where does the energy come from", by going back to the chain, leading to the resource of energy.

The overview of the study

Our sample consists of a teacher, one class of 26 students (15 boys and 11 girls) and a second one of 22 students (13 boys and 9 girls). The teacher held a morning session from 9:00 to 10:00 with the first class and from 10:30 to 11:30 with the second. A video device (semi-fixed camera) was placed at the back of the classroom to monitor the teacher's movements and record his activity.

The overview of the analytic procedure

This section introduces the six-step process that was followed.

Step 1. Clarification of the subtasks carried out After getting acquainted with the data body, we separated the lesson into modules. Based on the description of the prescribed subtasks, we clarified what we called realised subtasks and their duration.

Step 2. Creating the text For each lesson, a transcript of the teacher's discourse was made -the oral speech was written down, while any other active resource was noted in brackets. The coding for the various semiotic resources presented in Table 1.

Step 3. Searching for themes (or thematic categories) and clarifying them The idea of identifying patterns within the data [START_REF] Braun | Using thematic analysis in Psychology[END_REF] was ideal as a starting point and helped us shape the main themes. We identified the "essence" of what a theme is about and which aspect of the data each theme describes. We considered a new theme when the teacher communicated another component of the same aspect or moved to a different aspect. Step 4. Review the themes/thematic categories Both researchers re-read the text and re-categorized the data. Comparing the work of the researchers, the rate of agreement [START_REF] Bakeman | Observing interaction: An introduction to sequential analysis[END_REF]) was over 80% the first time and over 90% the second time.

Step 5. Creating a combined table of analysis A table consisting of the themes that were communicated and the used semiotic resources, for each of them, was created. The coding for the comparison of the use of semiotic resources was:

Step 6. Stability rate In order to quantify stability in each analysed case, we adjusted the percentage agreement that is normally used to find interrater reliability [START_REF] Bakeman | Observing interaction: An introduction to sequential analysis[END_REF]. In our case, the rate of stability (S) is the percentage of the number of common elements divided by the sum of the number of common and uncommon elements, ie:

Finally, by adapting the idea of the six-point Likert scale, we have created the following table to help us describe the Stability rate (S).

TABLE 3 The category of comparison and its coding

RESULTS

Teacher Activity

The realised subtasks are strongly stable (100%)

The table 5 shows that the progress of both lessons was the same, with the same subtasks to succeed one another. In that way, we can see that the teacher managed the same 9 subtasks following exactly the same order in both lessons. Furthermore, the pedagogical form of work was exactly the same for each subtask: the pupils work on their place and were questioned either individually, by name or not, or collectively. The teacher introduces the elements of the energy chain by discussing the related themes with the children. The students work in groups of four the exercises referring to the design of the energy chain model before moving to the introduction of a new element of the chain.

The teacher taught exactly the same subtasks (in the same order) with the same pedagogical form of work.

Since the teacher realised nine subtasks but there were seven subtasks imposed to him (see Figure 1) we could say that there is a gap between what he had to do and what he really did. To quantify this gap, there is 78% (78% = 7/7+2) stability or 22% instability between the prescribed and the realised subtasks. The duration of the subtasks is stable (78%)

TABLE 5 Subtasks managed the by teacher

The first lesson lasted 52 minutes and 30 seconds while the second one 55 minutes and 40 seconds. The percentage of stability for the overall duration of the lessons is strongly stable (94%). Most pertinently the percentage of stability of the duration by counting the common and uncommon duration per subtask is stable (78%).

More particularly, we can see that the duration of: (a) four subtasks (n° 1, 2, 5, and 9) are strongly stable (83% ≤ S <100%), (b) two subtasks (n° 6 and 8) are stable (67% ≤ S <83%) and (c) three subtasks (n° 3, 4 and 7) are slightly stable (50% ≤ S <67%).

Communicated themes

The number of themes that the teacher communicated is strongly stable (84%) Most categories of themes were found in both lessons. The table 6 includes the number of the common and uncommon categories of themes.

Semiotic Resources

The use of semiotic resources is stable (72%) when communicating the common themes Comparing the semiotic resources between the two lessons for the aforementioned 60 thematic communications, we created the following table (Table 9). In the two out of six tasks, the use of semiotic resources was slightly stable, with the teacher using the same semiotic resources in half of the cases. In one task the teacher used the same semiotic resources in 77% of the cases. The rest of the tasks are considered strongly stable, since the thematic communications were executed with the same way.

DISCUSSION

At the same time, while conducting a different study, in which we studied how differentiations between the material that two groups of students worked on influenced their understanding of the energy chain model, we had to be certain that all the other factors were controlled, and therefore the teacher's practices. The teacher had been given the same prescribed task/subtasks, but is that sufficient in an experimental research? In order to answer this question, we tried to quantify the stability of the practices, with this three-level analysis of video data being our suggestion.

Bearing in mind that the stability measure we adopted has a certain range (>50% -100%), it is the next step for every subsequent research to check the effect of the (in)stability on students' learning, since a 80% stability means a 20% of instability at the same time.

Taking this discussion one step further, if the stability of practises of the teachers will be found to be a characteristic of their activity and concurrently a linkage with students' performance, we could consider an ineffective teacher to be ineffective for all their professional life. The research on stability of teachers' activity over the years is proposed, in order to have a more holistic view over this topic and to further understand the factors which influence it.

From a didactics point of view, the way the lesson was analysed could work vice versa as a way to organise a teaching, starting with the subtasks and their duration, proceeding to the necessary themes for communication, to finish by selecting the most appropriate semiotic resources for every occasion.

Since this is a case study, this kind of analysis was performed only in one teacher, so in order to validate this method, it should be tested on a bigger sample and various courses.

  theme (1) communicated with the use of the same semiotic resource(s) (1) When the same theme (1) communicated with the use of different semiotic resource(s)

  of the Energy Chain with three elements(objectsource-transformer) 5.Introduction of the Energy Chain with three elements(object-sourcetransformer) 5.Introduction of the Energy Chain with three elements(object-sourcetransformer) 5. Pupils' Exercise 2: Realisation of the energy chain of three elements in various cases 6.Pupils' Exercise 2: Realisation of the energy chain of three elements in various cases 6.Pupils' Exercise 2: Realisation of the energy chain of three elements in various cases -7.Examples of Activity 2 7.Examples of Activity 2 6. Introduction of the Energy Chain with four elements(objectsource-transformer-resource) 8.Introduction of the Energy Chain with four elements(object-sourcetransformer-resource) 8.Introduction of the Energy Chain with four elements(object-sourcetransformer-resource) 7. Pupils' Exercise 3: Realisation of the energy chain of four elements in various cases 9.Pupils' Exercise 3: Realisation of the energy chain of four elements in various cases 9.Pupils' Exercise 3: Realisation of the energy chain of four elements in various cases Total of common and uncommon tasks 9 and 0Percentage of stability between the realised subtasks 100 % = 9/(9+0)

TABLE 1

 1 Coding of Semiotic Resources

	Semiotic resources	Coding
	Speech	s
	Deictic Gestures	d.g
	Symbolic Gestures	s.g
	Ergotic Gestures	e.g
	Scenic Objects	s.o
	Scenery -Projection	sc.pr
	Scenery-Paper elements on the black	sc.el
	board
	Scenery-Black Board	scen
	Writing	eg.w
	Drawing	eg.dr

TABLE 2

 2 Configuration of themes

	Aspect	Component	Example
	Element of the model	1. object	object
		2.source3.transformer
		4.resource 5. link
	Symbolisation of elements	1. circle 2. rectangle 3.	circle
		triangle 4. arrow to the
		right
	Element's position at the model	first, second, third,	beginning of the
		fourth, between	chain
	Element's association with material	-	car
	entities
	Elements' relationship	-	need for gasoline
	Element's function	-	movement
	Everyday life example	-	-

TABLE 4

 4 Characterisation of the stability rate

	Stability Rate	Characterisation
	0% ≤ S< 17%	Strongly Instable
	17% ≤ S < 33%	Instable
	33%≤ S < 50%	Slightly Instable
	50% ≤ S < 67%	Slightly Stable
	67% ≤ S < 83%	Stable
	83% ≤ S ≤100%	Strongly Stable

TABLE 8

 8 Common and uncommon thematic communications per subtask

	Realised subtask Lesson 1 Lesson 2	Common Uncommon	Percentage of stability
	Subtask n°1	2	2	2	0	100%
	Subtask n°2	23	27	23	4	85%
	Subtask n°3	2	11	2	9	18%
	Subtask n°4	22	28	22	6	79%
	Subtask n°5	6	10	6	4	60%
	Subtask n°6	5	21	5	16	24%
	Total thematic communications	60	39
	Percentage of Stability		61%=60/(60+39)

TABLE 9

 9 Comparison of used semiotic resources

	Realised subtask	Common semiotic resources (1.1)	Uncommon semiotic resources (1.0)	Percentage of stability per task
	Subtask n°1	1	1	50% (=1/2)
	Subtask n°2	12	11	52% (=12/23)
	Subtask n°3	2	0	100% (=2/2)
	Subtask n°4	17	5	77% (=17/22)
	Subtask n°5	6	0	100% (=6/6)
	Subtask n°6	5	0	100% (=5/5)
	Total	43	17
	Percentage of Stability		72% = 43 /(43+17)

The common categories of themes were twenty-one, while the uncommon were four. This means that the number of the thematic categories communicated by teacher is strongly stable between both lessons (84%).

The thematic communications per subtask are slightly stable (61%)

As it can easily be understood the 21 common thematic categories were communicated various times in the progress of each lesson, sometimes in the exact same subtask sometimes not. Specifically, there were 60 times where the same themes were communicated in the same subtask and 30 times that they were not, which leads, according to our ranking, to a slight stability (61 %) between the two lessons regarding the thematic communications per subtask.

From the one hand there are two subtasks strongly stable, one stable and one with slight stability, from the other hand there are two subtasks which can be characterised as unstable. Leontiev, A. N. (1978). Activity, consciousness, and personality. Englewood Cliffs, NJ: Prentice-Hall. Pariès, Μ., Robert, A., & Rogalski, J. (2008). Analyses de séances en classe et stabilité des pratiques d'enseignants de mathématiques expérimentés du Second degré. Educational Studies in Mathematics, 68(1), 55-80. Pozzer-Ardenghi, L., & Roth, W.-M. (2007). On performing concepts in science lectures. Science Education, 91(1), 96-114. Rogalski, J. (2008). Théorie de l'activité et cadres développementaux pour l'analyse liées des pratiques des enseignants et des apprentissages des élèves. In F. Vandebrouck (Éd.), La classe de mathématiques : Activités d'élèves et pratiques des enseignants (pp. 237-280)


B Kibler

C Billet
email: cyril.billet@univ-fcomte.fr

P.-A Lacourt

R Ferriere
email: richard.ferriere@univ-fcomte.fr

Member, IEEE J M Dudley
email: john.dudley@univ-fcomte.fr

All-Fiber Source of 20-fs Pulses at 1550 nm Using Two-Stage Linear-Nonlinear Compression of Parabolic Similaritons

Keywords: Optical fiber amplifiers, solitons, ultrafast optics

Parabolic pulses from a similariton amplifier at 1550 nm are compressed using a two-stage fiber compressor employing linear compression in photonic bandgap fiber and soliton effect compression in highly nonlinear fiber. Precise optimization of the compressor design using frequency resolved optical gating allows the generation of minimally chirped pulses as short as 20 fs. This duration corresponds to only four optical cycles of the underlying electric field at this wavelength.

I. INTRODUCTION

M OTIVATED by numerous emerging applications in ultra- fast photonics, the development of fiber-based sources of femtosecond pulses around 1550 nm is a subject of much current research [START_REF] Wada | Femtosecond all-optical devices for ultrafast communication and signal processing[END_REF]. A widespread fiber-based technique used for femtosecond pulse generation is soliton-effect compression, which exploits the initial nonlinear spectral broadening and temporal compression phase of higher order soliton propagation in the anomalous dispersion regime of an optical fiber [START_REF] Mitschke | Ultrashort pulses from the soliton laser[END_REF]. In the context of generating pulses consisting of only a small number of optical cycles, however, achieving high pulse quality with this technique has required the use of multiple nonlinear stages and/or specially profiled dispersion decreasing fiber [START_REF] Matsui | Generation of 20-fs optical pulses from a gain-switched laser diode by a four-stage soliton compression technique[END_REF], [START_REF] Tsuchiya | Sub-100 fs SDPF optical soliton compressor for diode laser pulses[END_REF]. The use of simpler single-stage soliton compression setups has typically yielded multiple pulse pedestals, or longer pulses in the range of 35-40 fs [START_REF] Mitschke | Ultrashort pulses from the soliton laser[END_REF], [START_REF] Nicholson | High-power single mode all-fiber source of femtosecond pulses at 1550 nm and its use in supercontinuum generation[END_REF], [START_REF] Takayanagi | Generation of high-power femtosecond pulse and octave-spanning ultrabroad supercontinuum using all-fiber system[END_REF].

A recent technology that has attracted particular interest for compression applications is based on the generation of similariton pulses in fiber amplifiers with normal dispersion. In particular, similariton amplifiers based on both doped fiber and Raman gain have been shown to lead to the self-similar generation of parabolic output pulses possessing a strictly linear chirp that facilitates pulse compression to the femtosecond regime [START_REF] Tamura | Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers[END_REF]- [START_REF] Billet | Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm[END_REF]. An additional important feature of such similariton pulses is that they possess improved resistance to input pulse noise because they are asymptotic attractors toward which any arbitrary input pulse of given energy will converge irrespective of its specific pulse profile or duration [START_REF] Finot | Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers[END_REF]. In this letter, we take advantage of the novel properties of similariton amplifiers to develop an all-fiber source of 20-fs pulses at 1550 nm based on a simple two-stage compression setup, and using only short lengths of commercially available specialty fibers. Specifically, we generate linearly chirped parabolic pulses from an erbium-doped fiber (EDF) amplifier followed by a length of normal dispersion highly nonlinear fiber (HNLF), and subsequently compress these pulses using two independent propagation steps. The first stage uses air-core photonic bandgap fiber (PBGF) to linearly compress the parabolic pulses to 240-fs duration, and this is followed by a second stage using nonlinear soliton-effect compression to generate 20-fs pulses in the few-cycle regime. An important feature of our setup is the careful optimization of the nonlinear compression through numerical simulations using frequency-resolved optical gating (FROG) measurements of the input pulses to the compressor as initial conditions [START_REF] Dudley | Complete characterization of ultrashort pulse sources at 1550 nm[END_REF].

II. EXPERIMENTAL SETUP AND RESULTS

Fig. 1 shows the experimental setup. A 19.8-MHz repetition rate passively mode-locked fiber laser (Pritel FFL) generates 1.4-ps [full-width at half-maximum (FWHM)] linearly polarized hyperbolic secant pulses of 100-pJ energy. FROG measurements confirmed that the pulses were minimally chirped. These pulses are then injected into an EDF similariton amplifier and a segment of HNLF to undergo self-similar amplification and propagation respectively.

The similariton amplifier is based on 10 m of commercially available La-Al-codoped EDF designed for -band operation (OFS R37003) with large normal dispersion ps/nm km and nonlinearity coefficient W km . The amplifier is pumped codirectionally by a 1480-nm continuous-wave (CW) Raman laser (Keopsys), with 170 mW of pump power yielding 14.5-dB gain. Under these conditions, the average output power from the amplifier was 65 mW, and FROG measurements confirmed that the amplifier operated in the similariton regime generating high-quality parabolic pulses with linear chirp [START_REF] Billet | Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm[END_REF]. The pulse duration and spectral bandwidth (both FWHM) at the amplifier output were 5.4 ps and 17 nm, respectively. These pulses were then injected into a 10-m segment of dispersion-flattened HNLF (OFS Specialty Fiber) to undergo further self-similar spectral and temporal broadening [START_REF] Limpert | High power femtosecond Yb-doped fiber amplifier[END_REF]. At 1550 nm, the fiber normal dispersion is ps/nm km, the dispersion slope is 0.002 ps/nm km, and the nonlinearity coefficient W km . FROG measurements verified that this additional propagation step in the HNLF was also self-similar, with the output pulses retaining their parabolic profile and linear chirp even while undergoing significant temporal and spectral broadening [START_REF] Anderson | Wave-breaking-free pulses in nonlinear-optical fibers[END_REF]. The similariton pulses at the HNLF output had 380-W peak power and temporal and spectral FWHM of 6.1 ps and 34 nm, respectively.

The first (linear) compression stage used hollow-core PBGF (Crystal Fiber/BlazePhotonics HC-1550-02) possessing large anomalous dispersion ps/nm km around 1550 nm. The nonlinearity of this hollow core fiber is negligible at our power levels. Based on the FROG measurements of the similariton intensity and chirp, the length of PBGF required could be precisely determined from numerical simulations, avoiding the need for expensive trial and error. Optimal compression used a fiber length of 3.5 m, and yielded 240-fs compressed pulses with minimal phase structure. Fig. 2 shows the pulse spectrum and the intensity and phase of these linearly compressed pulses obtained from the FROG measurements. We note that although depolarization effects in the short lengths of EDF and HNLF used in our setup were negligible, precise control of the input polarization prior to injection in the PBGF was required for optimal compression. These 240-fs compressed pulses were then injected into the second (soliton effect) compression stage. This was based on dispersion-flattened HNLF (OFS Specialty Fiber) with anomalous dispersion ps/nm km, dispersion slope of 0.02 ps/nm km, and nonlinearity coefficient W km . Soliton effect compression requires that the fiber length be precisely tailored to the input pulse characteristics, and a number of analytic guidelines for this purpose have been developed [START_REF]Applications of Nonlinear Fiber Optics[END_REF]. However, because higher order soliton propagation is an intrinsically highly nonlinear process, even small intensity and chirp substructure on the input pulse can dramatically impact on the quality of the compressed pulse obtained. Moreover, the presence of any noise on the input pulse can induce pulse break through soliton fission, and lead to severe pulse-to-pulse fluctuations in the compressed pulse characteristics.

In this regard, however, recent studies of soliton fission in the context of supercontinuum generation have provided improved insight into the way soliton fission-related instabilities can be minimized [START_REF] Dudley | Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers[END_REF], [START_REF] Foster | Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires[END_REF]. Two factors that are crucial are 1) the use of a low noise source to minimise the seeding of soliton fission by modulation instability and 2) the limitation of the soliton compressor fiber to a length where the injected pulse attains its maximum bandwidth, just prior to the point where soliton fission is generally induced. In our setup, these conditions are met 1) by taking advantage of the high quality of the compressed similariton pulses, and 2) by precisely determining the optimal fiber length through numerical simulations of the soliton compression process using the measured input pulse intensity and chirp as the initial conditions.

Because the input and output coupling between the PBGF and HNLF used nonoptimal simple physical contact, the 240-fs pulses from the PBGF excited only a moderate soliton order of (1.2-kW peak power) when injected into the anomalous dispersion HNLF. For these injection parameters, and using the measured intensity and phase profile of the 240-fs pulses, numerical simulations predict a minimum achievable pulse duration of 20 fs for an optimal HNLF length of 0.5 m. These simulation results were confirmed experimentally, and Fig. 3 shows the broadened spectrum and the pulse intensity and phase obtained from the FROG measurements at the output of the anomalous dispersion HNLF. To illustrate the accuracy of our numerical design procedure, the dashed line in Fig. 3(b) shows the expected output compressed pulse profile based on the numerical simulations of the soliton compression process described above. It is clear that there is very good agreement.

Although some residual energy ( 13%) remains in low amplitude substructure in the wings, the central compressed pulse lobe has a peak power of 14 kW and duration of 20 fs, corresponding to only four optical cycles under the FWHM. The residual pedestal arises because the nonlinear chirp across the propagating higher order soliton is dominated by self-phase modulation, and thus only the central part is compressed by anomalous group velocity dispersion. Energy in the wings remains uncompressed and appears as low amplitude substructure [START_REF]Applications of Nonlinear Fiber Optics[END_REF]. In addition, effects such as higher order dispersion can also play an important role in modifying the pedestal structure [START_REF] Chan | Short pulse generation by higher order soliton-effect compression: Effects of optical fiber characteristics[END_REF].

It is also instructive to consider a numerical reconstruction of the electric field waveform amplitude across the pulse centre, obtained by multiplying an ideal sinusoidal carrier by the retrieved complex pulse envelope [START_REF] Taft | Measurement of 10-fs laser pulses[END_REF]. This is shown in Fig. 3(c). Although such a reconstruction is somewhat artificial because the absolute carrier phase cannot be determined from FROG measurements alone, it does highlight clearly the chirp-free and few-cycle nature of the compressed pulses. Note in this regard that the absolute phase zero in Fig. 3(c) is set to the peak of the envelope.

III. CONCLUSION

The pulse amplification and compression system described here allows the generation of few-cycle pulses around 1550 nm using a compact experimental setup, with a total length of only 24 m of commercially available optical fiber. Although we have obtained pulses with duration as short as 20 fs, we have checked using numerical simulations that shorter pulses 15-fs duration should be obtainable with improved PBGF coupling efficiency and the corresponding use of a shorter soliton compressor HNLF segment. These results represent what we believe to be an exciting new phase in the development of similariton-related technology and applications, and we expect such sources to find wide use in ultrafast photonics.

Fig. 1 .

 1 Fig. 1. Experimental setup. The wavelength-division multiplexer (WDM) is used to separate the 1480-nm pump from the 1550-nm signal pulses at the output of the EDF. All the other symbols have their usual meaning.

Fig. 2 .

 2 Fig. 2. Results after linear compression of similariton pulses in PBGF. (a) Measured spectrum, and (b) intensity (left axis) and phase (right axis) from FROG measurements. The intensity FWHM in (b) is 240 fs.

Fig. 3 .

 3 Fig. 3. Results after soliton-effect compression in HNLF showing (a) spectrum and (b) intensity (solid line, left axis) and phase (solid line, right axis) from FROG measurements. The intensity FWHM in (b) is 20 fs. The dashed line shows the optimally compressed intensity profile predicted from numerical simulations. The inset in (c) shows an illustrative reconstruction of the electric field over the central peak in order to highlight the few-cycle and chirp-free nature of the compressed pulse.

ACKNOWLEDGMENT

We thank the University of Bath for providing the photonic bandgap fiber used in this work.