Fatima El Moussawi

Matthias Hofer

Damien Labat

Andy Cassez

Géraud Bouwmans

Siddharth Sivankutty

Rosa Cossart

Olivier Vanvincq

Esben Hervé Rigneault
email: herve.rigneault@fresnel.fr

Ravn Andresen
email: esben.andresen@univ-lille.fr

Esben Ravn Andresen

Tapered Multicore Fiber for Lensless Endoscopes

Keywords: Microscopy, Fluorescence, In-vivo imaging, Wave front shaping, Transmission matrix. 1

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Abstract

We present a novel fiber-optic component, a "tapered multi-core fiber (MCF)", designed for integration into ultra-miniaturized endoscopes for minimally invasive two-photon point-scanning imaging and to address the power delivery issue that has faced MCF based lensless endoscopes. With it we achieve experimentally a factor 6.0 increase in two-photon signal yield while keeping the ability to point-scan by the memory effect, and a factor 8.9 sacrificing the memory effect. To reach this optimal design we first develop and validate a fast numerical model capable of predicting the essential properties of an arbitrarily tapered MCF from its structural parameters. We then use this model to identify the tapered MCF design parameters that result in a chosen set of target properties (point-spread function, delivered power, presence or absence of memory effect). We fabricate the identified target designs by stack-and-draw and post-processing on a CO 2

Introduction

The term "lens-less endoscope" generally refers to an optical fiber based flexible imaging endoscope that reduces the diameter of the probe to the diameter of the fiber itself, typically 100-200 µm; and where light at the distal end (closest to the sample) is controlled by wave front shaping at the proximal side (opposite to the sample) [START_REF] Psaltis | Imaging with multimode fibers[END_REF][START_REF] Andresen | Ultrathin endoscopes based on multicore fibers and adaptive optics: a status review and perspectives[END_REF]. The lens-less endoscope is a promising ultra-miniaturized imaging tool which may enable minimally invasive and high-resolution observation of neuronal activity in-vivo inside deep brain areas [START_REF] Turtaev | High-fidelity multimode fibre-based endoscope for deep brain in vivo imaging[END_REF][START_REF] Vasquez-Lopez | Subcellular spatial resolution achieved for deep-brain imaging in vivo using a minimally invasive multimode fiber[END_REF][START_REF] Ohayon | Minimally invasive multimode optical fiber microendoscope for deep brain fluorescence imaging[END_REF]. The interest of this miniaturized endoscope stems from its ability to allow new functionalities because light source and detectors are remote as well as the light weight and flexibility of the optical fibers which constitute here the main part of the imaging system. These properties give the lens-less endoscope the ability to be fixed for instance onto a mouse's head allowing its free movement while studying neuronal activity. Also, it could reduce space constraints to allow fixing several probes simultaneously and thus the ability to study functional connectivity of neurons in two distant brain regions. The initial reports of lens-less endoscopy based on a bundle of single-mode fibers (SMFs) published in Ref. [START_REF] Thompson | Adaptive phase compensation for ultracompact laser scanning endomicroscopy[END_REF], and the first demonstrations of microscopic imaging of objects through multi-mode fiber which did not require any optical elements between the fiber and the object were published in Refs. 7, 8, 9. However, also multicore fiber (MCF) presents great potential as imaging wave guides due to its particular merits [START_REF] Andresen | Ultrathin endoscopes based on multicore fibers and adaptive optics: a status review and perspectives[END_REF]. In this paper, we focus on MCF because it is more readily compatible with two-photon imaging modalities [START_REF] Andresen | Twophoton lensless endoscope[END_REF][START_REF] Andresen | Ultrathin endoscopes based on multicore fibers and adaptive optics: a status review and perspectives[END_REF][START_REF] Kim | Adaptive multiphoton endomicroscope incorporating a polarization-maintaining multicore optical fibre[END_REF] widely used in bio-medical microscopy.

In a two-photon lens-less endoscope, the MCF must perform two tasks: (i) Imaging; and (ii) transport. For the imaging task the modes at the distal end face (closest to the sample) of the MCF must lend themselves to coherent combination to intense foci for point-scanning imaging-best results have been obtained with dense core layouts with down to 3.2 µm core separation as in Ref. 12; while for the transport task, the main section of the MCF must be able to transport an ultra-short pulse distributed over all cores without deforming it through dispersion or changes induced by twists and bends-for which a sparse core layout with around 12-15 µm core separation is better suited as in Refs. 13, 10, 14. In this paper we aim to realize a MCF comprising three segments comprising "injection", "transport" and "imaging" segments joined by conical tapers as sketched in Fig. 1. We will refer to this object with the shorthand "tapered MCF". Our aim is to reconcile the conflicting demands for both dense and sparse core layout in a tapered MCF in which the properties of injection, transport, and imaging segments are decoupled.

In this paper we report the design, fabrication, and application of a tapered MCF optimized for two-photon imaging with pulsed 920 nm excitation. The paper is organized as follows. In Sec. 3 we provide the main results of this work which we further discuss in Sec. 4. Detailed Methods are provided in Supplement 1 and referenced when appropriate. The presented Data is available as Dataset 1, Ref. 15. Specifically, in Sec. 3.1 we present a fast numerical model capable of predicting the essential properties of an arbitrarily tapered MCF from its structural parameters. In Sec. 3.2 we use this model to identify the design parameters that result in a chosen set of target properties. In Sec. 3.3 we report the fabrication and post-processing of the target designs as well as their optical characterization and validation. Finally, in Sec. 3.4 we demonstrate the application of the fabricated tapered MCFs in a two-photon lensless endoscope system and quantify the gains in imaging performance. (inj) and length L (inj) ; first conical segment of length L (conical,1) ; transport segment of diameter D (transport) and length L (transport) ; second conical segment of length L (conical,2) ; imaging segment of diameter D (im) and length L (im) .

D (inj) L (inj) L (conical,1) L (transport) L (conical,2) L (im) D (im) D (transport) (a) (b) (c) (d) L 2a

Results

Coupled-mode theory model

To assist in the identification of the optimal design parameters, we developed a code, based on coupled-mode theory (CMT) and perturbation theory, which we refer to as the "CMT model", and which is available as part of Dataset 1, Ref. 15. The CMT model considers the tapered MCF as a concatenation of uniform segments and requires the structural parameters of these segments. It returns the effective indices of the guided modes as well as its transmission matrix (TM) H which it calculates as the product of segment TMs: H = i H i (Suppl. Doc. Sec. A). The TM H is expressed in the basis of the fundamental modes of the N cores and is of dimension N × N (higher-order modes-if present-are simply left out). H alone allows to predict the cross-talk (XT, a measure of the energy exchanged among cores) in terms of the N × N "cross-talk matrix" (sometimes called the "power transfer matrix") X here defined as

X = |H| 2 λ (1)
where the average over λ is the average over a sufficiently large number of calculated TMs for different wavelengths (cf the experimental measurement with a filtered-about 10 nm spectral width-incoherent source); and the scalar "average total cross-talk" X ave

X ave = 1 N i ( j =i X ij ). (2)
In parallel with the CMT model, we developed a model based on the finiteelement beam propagation method (FE-BPM) [START_REF] Rahman | Finite element modeling methods for photonics[END_REF][START_REF] Koshiba | A wide-angle finite-element beam propagation method[END_REF] (Suppl. Doc. Sec. B) which we refer to as "the FE-BPM model". Unlike CMT, FE-BPM makes no prior assumptions on the modes of the MCF and so is generally the more accurate of the two. It is however much slower. Here, we have used the FE-BPM model as a "gold standard" against which we have compared the results of the CMT model. In Fig. 2(a) and 2(b) we present XT matrices for a N = 475 core MCF as predicted by the CMT model and the FE-BPM model respectively which look qualitatively and quantitatively similar. The XT preferentially occurs between neighbor cores. This is not easy to see from the Figure because the core numbering convention influences the appearance of the Figure . Typically, in the modelled case there is significant XT (X ij > -30 dB) between cores up to 5Λ apart. From this comparison we establish the accuracy of the CMT model up to average total cross-talk of at least X ave = -2 dB and down to effective indices 0.8e-3 above the cladding index.

The CMT model also returns the field of the fundamental core mode vs the radial coordinate; it can be interpolated on the cartesian "distal pixel basis" {x

(dist) j , y (dist) j }, 1 ≤ j ≤ N pix ; centering N copies on the core positions (x (core) j , y (core) j
), 1 ≤ j ≤ N ; each of these can be numerically propagated (assuming free-space propagation) to the far-field by a two-dimensional discrete Fourier transform, the resulting fields are expressed in the (angular) "far-field pixel basis" {f

(FF)
x,j , f (FF) y,j }, 1 ≤ j ≤ N pix ; and finally re-arranged to a mode-tofar-field pixel basis change matrix U of dimensions N × N pix . U together with H permit to calculate e (FF,pix) of dimensions N pix × 1 the field in the far-field of the MCF distal end face

e (FF,pix) = U † He (prox,mode) (3)
where e (prox,mode) is N × 1 and contains elements representing the complex modal field amplitudes injected into the proximal end of the MCF. It is from the vector e (FF,pix) that the properties of Strehl ratio (a measure of the intensity in the distal focus relative to the total intensity) and the memory effect (a measure of input-output angular correlations allowing distal point-scanning by proximal angular scanning) [START_REF] Freund | Memory effects in propagation of optical waves through disordered media[END_REF] can be inferred. e (FF,pix) can also be numerically propagated to any intermediate plane with an angular spectrum propagator if required. The elements of the N × 1 vector e (prox,mode,j0) are the optimal input modal amplitudes to obtain an optimal focus on far-field pixel j 0 and is found as the conjugate transpose of the j 0 'th row of U † H. e (FF,pix,j0) is the resulting complex field optimally focused on far-field pixel j 0 which is found by applying Eq. ( 3). The Strehl ratio S can then be defined as the ratio of the energy in the focus on far-field pixel j 0 to the total energy in the field, for example by plotting the norm-square of e (FF,pix,j0) versus {f

(FF) x,j , f (FF)
y,j } and integrating over the focus area and then over the whole field. The so-called "memory effect" is the effect by which the direction (phase ramp) of the output field changes by the same amount as the direction (phase ramp) of an input field [START_REF] Freund | Memory effects in propagation of optical waves through disordered media[END_REF]. In presence of the memory effect, it is therefore possible to displace a distal focus by simply adding a phase ramp to the proximal field. p(f

(0) x , f (0)
y ) is N ×1 and its elements are the [complex exponentials of] additional proximal modal phases required to move the focus in the far-field by (f

(0) x , f (0)
y ) using the memory effect:

p(f (0) x , f (0) y ) : p k (f (0) x , f (0) y ) = exp(i2πf (0) x x (core) k + i2πf (0) y y (core) k ) (4)
The memory effect can then be quantified as a two-dimensional curve M (f

(0) x , f (0)
y ) which is a ratio, the denominator containing the intensity of the field "optimally focused on far-field pixel j 1 " e (FF,pix,j1) ; and the numerator containing the intensity of the field "initially optimally focused on far-field pixel j 0 then shifted to far-field pixel j 1 using the memory effect":

e (FF,pix,j0→j1) = U † H[h (j0) • p(f (0) x , f (0) y )] (5)
where "•" signifies Hadamard product (element-by-element multiplication). M is constant = 1 for "full" memory effect, and = 0 everywhere except the origin for "no" memory effect. As outlined above, the quantities X ave , S, and M represent properties of tapered MCF that can be predicted by the CMT model from the tapered MCFs structural properties. Inversely, the CMT model can be used to identify the tapered MCF designs that result in a chosen set of properties.

Target properties and identification of target designs

The objective is to identify the design parameters cf Fig. 1(d) of a tapered MCF with triangular core layout with suitable properties for two-photon lensless imaging with a pulsed 920 nm excitation laser-the method would also work for any other core layout one has the capacity to fabricate.

As a first step, we want to minimize inter-core group delay dispersion in the transport segment cf Fig. 1(d) (center) which is debilitating for the shortpulsed excitation that we aim for. This dispersion arises from local variations among cores. We identify the rule-of-thumb that inter-core group delay dispersion decreases with increasing V -parameter. This holds for all the hypothesized physical origins that we considered ( [START_REF] Roper | Minimizing group index variations in a multicore endoscope fiber[END_REF][START_REF] Roper | Advances in multicore optical fibres for endoscopy[END_REF][START_REF] Andresen | Measurement and compensation of residual group delay in a multi-core fiber for lensless endoscopy[END_REF] and Suppl. Doc. Sec. D). We therefore choose to constrain the V -parameter to around 5.06, the second cutoff of a parabolic-index core, making the core bi-mode. To avoid exciting the higher-order mode in the transport segment, we decide to add an "injection" segment separated cf Fig. 1(d) (left) from the transport segment by a conical taper whose dimensions are scaled by the factor t (inj) = 0.6 resulting in V = 3.0 which is comfortably below the single-mode cutoff (V cutoff = 3.518). Next we want to fix the refractive index step, core size, and pitch in the transport segment with the aim of keeping X ave very low, say, below -25 dB in one metre of MCF. To this end, the refractive index step should generally be chosen as high as practically possible, ∆n = 30•10 -3 ; to conserve V = 5 the radius must then be chosen a = 2.5 µm. The only parameter left is the pitch Λ. We use the CMT model to predict that the chosen criterion on X ave is respected for Λ > 14 µm (predicted X ave = -62 dB per metre in the uniform segment, -26 dB in the injection segment with t (inj) = 0.6).

The injection segments evidently contributes to the properties of the tapered MCF, notably the cross-talk, so it is systematically included in the modelling for the remainder of the paper. At this point the number of cores N can be fixed. It may be chosen arbitrarily, generally "the more the better" as the number of resolvable image pixels is proportional to N . Other than that N only impacts on the outer diameter of the transport section, so it can be chosen so as to respect a chosen condition on the outer diameter. Here, we choose N = 349, resulting in an outer diameter D (transport) = 330 µm. Finally, and most importantly, we now want to choose the dimensions of the imaging segment cf Fig. 1(d) (right). All of its structural parameters are scaled by the factor t (im) relative to those of the transport segment. We are particularly interested in two values, a first one which maximizes the Strehl ratio under the condition that the "full" memory effect is conserved; and a second one which maximizes the Strehl ratio without any conditions on the memory effect. But in this case a new condition must be respected, that core guidance is maintained, i.e. the effective indices of all modes must remain comfortably above the cladding index so as not to be too close to the single-mode cutoff. In Fig. 3(d) we present the curves of memory effect for different t (im) ; in Fig. 3(e) the Strehl ratio; in Fig. 3(f) the effective indices; and in Fig. 3(g) the average XT as functions of t (im) as predicted by the CMT model (the rest of the parameters are as in Tab. 1). Based on the following discussion we identify t (im) = 0.6 (maximum Strehl ratio while retaining full memory effect) and 0.4 (maximum Strehl ratio while retaining core guidance) respectively.

Fig. 3(a)-3(c) present the predicted point-spread functions (PSF) at distance Z = Z 0 • t (im) from the distal end of the MCFs with Z 0 = 500 µm and t (im) = 1, 0.6, and 0.4 (per se they are far-field images scaled by x = f x Zλ, y = f y Zλ). It comprises a central spot surrounded by a dimmer periodic replicas for untapered MCF (t (im) = 1). However, it becomes more singly-peaked, very beneficial for image quality, as the taper ratio decreases while the focal spot size which is represented by the full-width at half-maximum (FWHM) of the PSF is fairly constant equal to 2.4 µm. From Fig. 3(f) which shows the predictions of effective indices it can be seen that at t (im) = 0.3 some of the indices drop below the cladding index (lower dashed line) making leaking to cladding modes inevitable. This is the reason the lowest value of t (im) retained as design parameter was 0.4 for which effective indices remain 2•10 -3 above the cladding index. Another important consequence of tapering, of primordial importance for two-photon imaging, is the dramatic increase in the Strehl ratio with decreasing taper ratio as shown in Fig. 3(e).

Fabrication and characterization of target designs

We fabricated a MCF to the specification requirement of the transport segment identified in Sec. 3.2 (∆n = 30•10 -3 ; Λ = 14 µm; a = 2.5 µm; N = 349) by the methods we used previously (Ref. [START_REF] Andresen | Toward endoscopes with no distal optics: video-rate scanning microscopy through a fiber bundle[END_REF] and Suppl. Doc. Sec. C).

Then we post-processed the uniform MCF to obtain the tapered MCF represented by Fig. 1(d) by tapering a short length at the terminals of the MCF using a CO 2 laser-based glass processing and splicing system (LZM-100 FU-JIKURA) (Suppl. Doc. Sec. E). The MCF is locally heated up with the CO 2 laser and simultaneously pulled lengthwise by stepper motors so as to obtain a fiber whose diameter varies longitudinally. This system provides extremely stable operation and allows control over a wide range of parameters to achieve the tapered MCFs precisely with the chosen design parameters Tab. 1.

We characterized the optical properties of the fabricated tapered MCF using a filtered super-continuum laser source (Suppl. Sec. F). We injected the focused laser beam into MCF cores in the injection segment successively, and imaged the emerging beam out of imaging segment on a camera. Consequently, we could measure the mode field diameter (MFD) of the visualized mode profile and confirm-for each t (im) -the agreement with the predictions of the CMT For the leftmost point the main contributor to X ave is the injection segment; for the remaining points both injection and imaging segments contribute, with the injection segment's contribution remaining constant. Unmentioned parameters are as in Tab. model. Additionally we were able to measure the XT between excited core and other cores and thus build up X in the core basis which is presented in Fig. 4(f) and which agrees reasonably with the predicted values in Fig. 3(g). To confirm that the tapered MCFs were effectively single-mode with the choice of injection segment parameters, we tested a long MCF (L (transport) > 1 m) with t (inj) = 0.6 and t (im) = 1. On the camera we visualized the intensity profile exiting one core and verified that it did not change while applying a perturbation on the fiber. This verifies that only the fundamental mode propagates through the transport segment despite it being bi-modal, thus the injection segment allows the selective excitation of only the fundamental mode of each core.

Two-photon imaging lensless endoscope

As a first step towards implementing the tapered MCFs in a lensless endoscope system we measured the imaging properties of the fabricated tapered MCFs. The measurements were done using a pulsed fs-laser shaped by a spatial-light modulator (SLM) for injection while measuring the resulting intensity distributions in a plane a distance Z from the distal end face. The SLM was programmed to generate the focus (Suppl. Doc. Sec. G). The experimentally measured properties are presented in Fig. 4 The rest of the observed trends are in line with the predictions, that is, increasing intensity in the central focus and decreasing intensity in the satellite peaks as t (im) is decreased. This trend is quantified in the measured Strehl ratio, presented in Fig. 4(e), from a value of 7 % at t (im) = 1, it increases to 15 % at t (im) = 0.6 and reaches 19 % at the smallest value of t (im) = 0.4. In terms of absolute power in focus these values correspond to 20, 43, and 54 mW. The experimentally measured memory effect curves are presented in Fig. 4(d). The taper ratios t (im) = 1, 0.6, and 0.4 were identified as key values in Sec. 3.2, the The formation of a two-photon fluorescence image in a lensless endoscopy system requires a single-point detector to acquire the two-photon fluorescence generated by the sample; and a raster scanning scheme where the focus is scanned across two dimensions. This can be realized in two different ways, either (i -"TM scan") the phases of the elements of a pre-measured TM can be converted to SLM masks which scan the focus across the sample when displayed sequentially on the SLM; or (ii -"memory effect (ME) scan") a focus initially on a center pixel can be displaced by the memory effect and scanned across the sample, this is done through the addition of a sequence of suitable phase ramps in a conjugate plane of the proximal MCF end face (Suppl. Sec. G). When possible, the ME scan method has the dual advantage of not requiring knowledge of a full TM; and the possibility to use fast galvo mirrors to generate the phase ramps (virtually unbound image acquisition time [START_REF] Andresen | Toward endoscopes with no distal optics: video-rate scanning microscopy through a fiber bundle[END_REF]). With the different tapered MCFs in our lensless endoscope system, we performed two-photon imaging with both the TM scan method and the ME scan method. For simplicity and pragmatism, here, we used the SLM for both (so we are unable to demonstrate the speed advantage). For these proof-of-principle demonstrations, we used yellow-green fluorescent beads as a sample whose emission spectrum is similar to that of yellow fluorescent proteins commonly expressed e.g. in mice neurons during in-vivo exploration of neuronal activity. The acquired two-photon images are presented in Fig. 5. The upper row Figs. 5(a)-5(c) show two-photon images acquired using the TM scan method with t (im) = 1, 0.6, and 0.4. The second row Figs. 5(d), 5(e) show two-photon images acquired using the ME scan method with t (im) = 1 and 0.6. The comparison to Figs. 5(a), 5(b) provides visual evidence that the memory effect is indeed retained at these values, as predicted. No image with the ME scan is shown for t (im) = 0.4 because the ME is absent. All images are on the same intensity scale which allows to appreciate the increase in two-photon signal with decreasing t (im) . It can be seen that the signal from single, isolated beads increases appreciably from Fig. 5(a) over 5(b) to 5(c). This two-photon signal increase is quantified in Fig. 5(f). The points represent a mean over two-photon signal measured on several single, isolated beads. The relative increases are a factor 6.0 and a factor 8.9 for t (im) = 0.6 and 0.4. The two-photon signal increase is expected to be proportional to the square of the Strehl ratio, represented by circles. And indeed the two curves are coincident to a high degree.

Discussion

Initially, we remark that for the experimental two-photon imaging, only 169 out of the N = 349 cores of the tapered MCFs were employed simultaneously. An experimental constraint limiting the number of cores that can be used simultaneously is the number of pixels on the SLM. A 800×600 pixel SLM can be divided into 169 40×40 pixel segments which must control not only the phase but also the focusing of light injected into a core (Suppl. Sec. G), and the latter is compromised as the number of pixels per segment decreases. So, while not a hard limit, 169 cores was the compromise we chose. Workarounds are to use one of the newer SLM generations with higher pixel count; or using the SLM in conjunction with a passive elements like a micro-lens array to take care of the focusing. The modelling was limited to the same number of cores (N = 169) in order to have directly comparable results.

Next, we remark that the CMT model has not correctly predicted the experimentally measured Strehl ratios [Fig. 3(e) compared to Fig. 4(e)]. For t (im) = 0.6 the prediction is S = 21 % versus the observed S = 15 %. This discrepancy can be accounted for by two factors not included in the model. (i) Polarization is not accounted for in the CMT model, but the fabricated tapered MCF has circular ie non-polarization-maintaining cores. In the experiment a polarizer selects one of the polarization components, each core thus contributes a random amplitude to the image on the camera. The effect can be modelled by multiplying each element of e (prox,mode,j0) by a random number between zero and one before applying Eq. ( 3), and the consequence is a drop of S down to 18 %. (ii) The model calculates the Strehl ratio in the far-field whereas the experiments were done in intermediate planes Z = 500, 300, and 200 µm. The predicted Strehl ratio should therefore be taken as upper bounds. If we numerically propagate the predicted far-field to the intermediate plane, this effect alone leads to a drop of S down to 16.5 %. Both effects together result in a drop of S to 12 %. These effects thus seem to fully explain the discrepancy. For t (im) = 0.4 the discrepancy is more flagrant, the prediction is S = 59 % versus the observed S = 19 %. Here; effects (i) and (ii) combined would only result in a drop of S down to 42 %. It is possible that at this small value of t (im) the effect of leakage from the cores into the cladding becomes important. This would signify that the CMT model is at the limit of or even beyond its regime of validity for t (im) = 0.4. Beyond this point one would have to resort to FE-BPM modelling to assure accurate results (using, for example, the open source FE-BPM tool in Ref. [START_REF] Veettikazhy | Bpm-matlab: an open-source optical propagation simulation tool in matlab[END_REF].

Inter-core group delay dispersion (GDD) [START_REF] Andresen | Measurement and compensation of residual group delay in a multi-core fiber for lensless endoscopy[END_REF] is also not accounted for in the CMT model. Non-zero inter-core group delay dispersion has the consequence of reducing the coherence between the fields emerging from different cores (since we are using ultra-short pulses) and hence the Strehl ratio. We have sampled inter-core GDD in some pairs of cores which indicate that the distribution of inter-core GDD is narrower than the duration (150 fs) of the ultra-short laser pulse. This is not expected to be a main source of the discrepancy in Strehl ratios and the above discussion corroborates this.

Had the inter-core GDD been larger, time-gated measurement of the TM could have been employed as a mitigation [START_REF] Conkey | Lensless two-photon imaging through a multicore fiber with coherence-gated digital phase conjugation[END_REF]. We take the occasion to emphasize that the reported tapered MCF with the long transport segment designed to exhibit low inter-core GDD has rendered it unnecessary to resort to the more complicated time-gated measurement. We must remark that the lensless endoscope system used to generate Figs. 4 and 5 employed a detector located at the distal end, after both MCF and sample, rather than at the proximal end as it would in a "true" endoscopic configuration, detecting fluorescence collected in the backward direction through the MCF. In a previous paper we did demonstrate how a double-clad MCF allows for efficient fluorescence collection and thus to work with a proximal detector only [START_REF] Andresen | Twophoton lensless endoscope[END_REF], where the double-cladding was a ring of air holes around the N cores. However we were unable to use the same kind of double cladding in the present work because the post-processing on the glass processing and splicing system would have collapsed the air holes. In the near term our aim is to realize versions of the tapered MCF presented above with double cladding. We envisage that coating the tapered MCFs with low-index polymer as a final step will achieve this effect.

One of the biggest challenges facing lensless endoscopes-be they multimode fiber-based or MCF-based-is the resilience to bending or lack thereof. Conformational changes of a multi-mode fiber or MCF generally alters its TM significantly, meaning that re-performing the initial calibration steps is mandatory after each conformational change, which becomes unfeasible even for slow conformational changes. The tapered MCFs presented above are also subject to this problem-even though we previously showed that bending a MCF results in a predictable phase ramp which only shifts the focus without degrading it [START_REF] Tsvirkun | Bending-induced inter-core group delays in multicore fibers[END_REF]. In a more recent paper [START_REF] Tsvirkun | Flexible lensless endoscope with a conformationally invariant multi-core fiber[END_REF] we reported the fabrication of a helically-twisted MCF and demonstrated that it is invariant to conformational changes in a fairly wide range of operational conditions when used in a lensless endoscope system. The fabrication method of the tapered MCFs outlined in the present paper takes as its starting point a standard (ie not helically twisted) MCF. But there would be no conceptual difficulties starting from a helically-twisted MCF instead. As such the pursuit of conformationally-invariant tapered MCFs also represents a near-term aim.

Conclusion

We have presented a novel tapered MCF for optimal two-photon lensless endoscopy by point-scanning imaging. Our key result is that optimized tapered MCF can increase two-photon yield by a factor 6.0 while still keeping the ability to point-scan by the memory effect, and up to a factor 8.9 while sacrificing the memory effect. These results are decoupled from the transport segment of the tapered MCF with low XT which can be metres in length without altering the results. We have outlined the entire procedure of conception, design, fabrication, and application in a lensless endoscopy system of this novel fiberoptic component. The procedure, namely the step involving the CMT model, is generic and applicable to other ranges of properties, other MCF core layouts e.g. a Fermat golden spiral layout [START_REF] Sivankutty | Nonlinear imaging through a fermat's golden spiral multicore fiber[END_REF], wavelength ranges, etc. We conclude that tailoring of the taper profile is a degree of freedom that can be efficiently exploited for ameliorating MCF based lensless endoscopes and we have shown here that it contributes to solving the major power delivery issue that MCF based lensless endoscopes [START_REF] Sivankutty | Miniature 120-beam coherent combiner with 3d-printed optics for multicore fiberbased endoscopy[END_REF]. More generally a tapered multi-core fiber is a valuable add-on into the lensless endoscope toolbox to design, optimize and fabricate ultra-miniaturized endoscope tailored to a broad range of applications.

Backmatter

FOR TABLE OF CONTENTS USE ONLY

Figure 6: "Tapered multi-core fiber for lensless endoscopes" by Fatima El Moussawi, Matthias Hofer, Damien Labat, Andy Cassez, Géraud Bouwmans, Siddharth Sivankutty, Rosa Cossart, Olivier Vanvincq, Hervé Rigneault, and Esben Ravn Andresen. We present a novel "tapered multi-core fiber" designed for use in ultraminiaturized minimally invasive two-photon point-scanning imaging lensless endoscopes. We demonstrate that a taper ie a progressive reduction of multi-core fiber diameter over a short distance can be designed to significantly ameliorate the point-spread function and hence the quality of acquired two-photon images without compromising other of the multi-core fiber's design properties.

Figure 1 :

 1 Figure 1: Sketch of tapered MCF (shown with reduced core count, for compactness). (a) Structure of MCF with triangular core layout. (b) Refractive index profile along the dashed line in (a). a, core diameter. Λ, pitch. n 1 Core refractive index. n 2 Cladding refractive index. (c) Example image of intensity emerging from the distal end facet of the tapered MCF showing the fundamental mode of seven cores. (d) Taper profile of the tapered MCF. From the left, injection segment of constant diameter D(inj) and length L (inj) ; first conical segment of length L (conical,1) ; transport segment of diameter D (transport) and length L (transport) ; second conical segment of length L (conical,2) ; imaging segment of diameter D (im) and length L (im) .

Figure 2 :

 2 Figure 2: XT matrix X of a uniform MCF segment computed by (a) CMT model; and (b) FE-BPM model at 800 nm. Core numbering convention, the center core is number 1, the six cores in the first hexagonal ring are 2-7, etc. Simulation parameters: L = 5 mm; N = 475; Λ = 7.5 µm; a = 1.4 µm.

4 Figure 3 :

 43 Figure 3: Properties of tapered MCF predicted by CMT model. (a)-(c) PSF for t (im) = 1.0, 0.6, and 0.4 in a plane Z = Z 0 • t (im) . Each image is normalized to its maximum intensity. Scale bar, 50 µm. Insets, 5× zoom on the central spot. (d) Memory effect curves. Horizontal axis normalized to half the distance between replica spots. (e) Strehl ratio. (f) Effective indices of supermodes. (g) Average XT.For the leftmost point the main contributor to X ave is the injection segment; for the remaining points both injection and imaging segments contribute, with the injection segment's contribution remaining constant. Unmentioned parameters are as in Tab.1. 8

  and organized in the same way as the Figure of predicted properties, Fig. 3. Figures 4(a)-4(c) show the PSFs with t (im) = 1, 0.6, and 0.4 and Z = Z 0 • t (im) = 500 µm, 300 µm, and 200 µm. The measured FWHMs of the PSFs are 4.2, 4.0, and 3.6 µm, slightly larger than predicted.

Figure 4 :

 4 Figure 4: Experimentally observed properties of the tapered MCFs with parameters given in Tab. 1; the parameters of the two tapered MCFs with t (im) = 0.8 and 0.5 not appearing in the Table are very similar. (a)-(c) PSF for t (im) = 1, 0.6, and 0.4 in a plane Z = Z 0 • t (im) = 500, 300, and 200 µm. Insets, 5× zoom on the central spot. Each image is normalized to its maximum intensity. Scale bar, 50 µm. (d) Memory effect curves. (e) Strehl ratio. (f) Average cross-talk.

Figure 5 :

 5 Figure 5: Two-photon excited fluorescence images of fluorescent beads with the lensless endoscope. (a)-(c) Point-scanning with an optimal focus on each pixel (TM scan) and taper ratios (a) t (im) = 1 and Z = 500 µm; (b) t (im) = 0.6 and Z = 300 µm; and (c) t (im) = 0.4 and Z = 200 µm. (d)-(e) Point-scanning with memory effect starting from an optimal focus on the central pixel (ME scan) and taper ratios (d) t (im) = 1; and (e) t (im) = 0.6. Spatial scales are identical for all the images. Intensities in (a) and (d) have been multiplied by 5 for visualization. Scale bar, 10 µm. (f) Evolution of two-photon signal with taper ratio. (dots) Observed two-photon signals; and (circles) the square of the observed Strehl ratios.

Funding

  Agence Nationale de la Recherche (ANR-11-IDEX-0001-02, ANR-20-CE19-0028, ANR-21-ESRS-0002 IDEC, ANR-21-ESRE-0003 CIRCUITPHOTONICS); IN-

Table 1 :

 1 [START_REF] Psaltis | Imaging with multimode fibers[END_REF]. The physical dimensions of the fabricated tapered MCF samples.

	Identifier	0	1	2
	D (inj) [µm]	196	194	196
	t (inj)	0.6	0.6	0.6
	L (inj) [m]	0.005 0.005 0.005
	L (conical,1) [m]	0.025 0.025 0.025
	D (transport) [µm]	330	330	330
	L (transport) [m]	0.23	0.21	0.24
	L (conical,2) [m]	0	0.05	0.05
	D (im) [µm]	330	199	135
	t (im)	1	0.6	0.4
	L (im) [m]	0	0.005 0.005

Acknowledgments

Parts of this work were developed at IRCICA (USR CNRS 3380, https://ircica.univlille.fr/) using FiberTech Lille facilities (https://fibertech.univ-lille.fr/en/). This work was supported by the French Ministry of Higher Education and Research, the "Hauts de France" Regional Council, the European Regional Development fund (ERDF) through the CPER "Photonics for Society".

Disclosures

The authors declare no conflicts of interest.

Data availability The following data underlying the results presented in this paper are available in Dataset 1, Ref. 15: Matlab script of the CMT model described in Sec. 3.1 and Suppl. Sec. A. The data presented in Fig. 2. The data presented in Fig. 3. The data presented in Fig. 4. The data presented in Fig. 5. All other data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplementary Information Supporting Information: A. Coupled-mode theory model; B. Model based on finite-element beam propagation method; C. Methods: Fiber drawing; D. Intercore group-delay dispersion; E. Methods: Fiber post-processing (tapering); F. Methods: Optical properties of fabricated tapered multi-core fibers; G. Methods: Lensless endoscope.


M G Corbel

A Oueslati

K Khirouni
email: kamel.khirouni@fsg.rnu.tn

M Gargouri

A Rousseau

J.-F Bardeau

Structural phase transitions in [(CH 3 ) 4 P] 3 M 2 Cl 9 (M = Bi and Sb) hybrid compounds revealed by temperature-controlled Raman spectroscopy and X-ray diffraction

Two phosphonium chlorobismuthate (III) and chloroantimonate (III) crystals have been prepared and their thermal behaviors have been studied by thermal analyses, temperature-controlled Raman spectroscopy and temperature-controlled X-ray diffraction. Refinements of the diffraction pattern, carried out by the Le Bail method, confirm the high purity of both samples. Raman spectroscopy revealed a decomposition of the com-pounds above 250 • C, later confirmed by energy dispersive X-ray (EDX) analysis and X-ray powder diffraction (XRPD). In addition, temperature-controlled Raman spectroscopy and X-ray powder diffraction allowed us to reveal the existence of transitions in the 25-150 • C temperature range, which had previously gone completely unnoticed despite numerous published studies. These transitions are related to reorientations of the tetramethyl phosphonium cations [(CH 3 ) 4 P] + and rotations of [M 2 Cl 9 ] 3-(M = Bi or Sb) dimers in the structure. The rear-rangements of the cationic and anionic entities then change the thermal expansion coefficients measured by diffraction along the trigonal crystallographic axes of the structure.

Introduction

Among a large number of hybrid compounds discovered so far, R a M b X (3b+a) (R: cations = ((C n H 2n+1 ) 4 Y where Y= P, N)); M = Sb, Bi and X = Cl, Br, I) compounds has received considerable attention due to its interesting physical (ferroelastic, ferroelectric, piezoelectric) properties [1][START_REF] Rao | Stabilization of [BiCl 6 ] 3and[END_REF][START_REF] Jakubas | Structure and ferroelectric properties of (C 3 N 2 H 5 ) 5 Bi 2 Cl 11[END_REF][4][5]. These properties are mainly related to the dynamical reorientation of the alkyl cations chains and/or to displacements of the M (III) ions within the anionic networks [1,[START_REF] Zarychta | Phase transitions mechanism and distortion of SbCl 6 3octahedra in Bis(n-butylammonium) Pentachloroantimonate(III) (C 4 H 9 NH 3 ) 2[END_REF][7][START_REF] Szklarz | Organic-inorganic hybrid crystals, (2,4,6-CH 3 PyH) 3 Sb 2 Cl 9 and (2,4,6-CH 3 PyH) 3 Bi 2 Cl 9 . Crystal structure characterization and tunneling of CH3 groups studied by 1 H NMR and neutron spectroscopy[END_REF].

The compound with the poly-anionic units [M 2 X 9 ] 3-forming two dimensional networks and the one with bioctahedral [M 2 X 11 ] 5-units in which two [MX 6 ] 3-octahedral are linked together by a bridging halogen atom [START_REF] Jakubas | Structure and ferroelectric properties of (C 3 N 2 H 5 ) 5 Bi 2 Cl 11[END_REF]. Halogenobismuthates ions (Bi (III)) and antimonates ions (Sb (III)) based on elements of groups 15 and 14 (column V), respectively, are well known for their ns 2 electronic lone pair which can be stereochemically active or inactive [START_REF] Nedil'ko | Synthesis of barium cuprate by secondary induction heating and its electrical properties[END_REF][START_REF] Tmar Trabelsi | Synthesis, characterization and ionic conductivity of Ca 8-x Sr x Bi 2 (PO 4 ) 6 O 2 for x = {3, 4, 5[END_REF]. Consequently, the interest of this family of compounds therefore comes not only from its structural diversity but also from its potential applications in the development of low-cost electronic devices (thin-film transistor, organic-inorganic light-emitting diode) and in terms of conversion of energy in solar cells [START_REF] Sobczyk | Self-assembly of Sb(III) and Bi(III) halocoordinated octahedra in salts of organic cations. Structure, properties and phase transitions[END_REF][START_REF] Kagan | Organic-inorganic hybrid materials as semiconducting channels in thin-film field-effect transistors[END_REF][START_REF] Toledano | La ferroélasticité[END_REF].

It should be mentioned that there is a particular interest in the inorganic halide compounds R 3 M 2 X 9 , since the discovery of their pyroelectric properties [1]. The study of hybrid compounds based on tetra-alkylphosphonium cations of the general formula [(C n H 2n+1 ) 4 P], (where n = 1, 2, 3, 4) revealed several structural phase transitions caused by reorientation dynamics of the cationic groups. According to P. P. Shi et al. [14], [(CH 3 ) 4 P]FeCl 4 and [(CH 3 ) 4 P]FeBr 4 compounds show a paraelectric-ferroelectric phase transitions, above room temperature (at 362 and 353 K respectively), accompanied by notable dielectric anomalies, reversible polarizations and nonlinear optical (NLO) responses. These transitions can be attributed to the rotations of the cations [(CH 3 ) 4 P] + , which is confirmed by the calculations of the potential energies of rotation.

Among these materials, the compounds [(CH 3 ) 4 P] 3 Bi 2 Cl 9 (PCB) and [(CH 3 ) 4 P] 3 Sb 2 Cl 9 (PCA) exhibit a low temperature paraelesticferroelastic phase transition at -122 and -138 • C, respectively [1]. Raman and infrared studies suggested that these transitions are of the order-disorder type (the vibrational states of cations groups change markedly through the phase transitions) [1,4,[START_REF] Zarychta | Phase transitions mechanism and distortion of SbCl 6 3octahedra in Bis(n-butylammonium) Pentachloroantimonate(III) (C 4 H 9 NH 3 ) 2[END_REF][7][START_REF] Szklarz | Organic-inorganic hybrid crystals, (2,4,6-CH 3 PyH) 3 Sb 2 Cl 9 and (2,4,6-CH 3 PyH) 3 Bi 2 Cl 9 . Crystal structure characterization and tunneling of CH3 groups studied by 1 H NMR and neutron spectroscopy[END_REF]. Moreover, these PCA and PCB compounds show a high temperature phase transition at 277 and 261 • C, respectively. The dilatometric and calorimetric studies suggest that these transitions are 'displacive' in nature without any mechanism being identified by Wojtas et al. [1,4,5].

Given the interest in better understanding the phase transition mechanisms of [(CH 3 ) 4 P] 3 Bi 2 Cl 9 (PCB) and [(CH 3 ) 4 P] 3 Sb 2 Cl 9 (PCA) compounds, including the origin of the high temperature phase transition, we studied the structural modifications of these materials by complementary approaches by using temperature-controlled Raman spectroscopy and X-ray diffraction. PCB and PCA compounds will be synthesized by a slow evaporation method at room temperature before studying their thermal behaviours by thermal analyses (TGA-DTA and DSC), temperature-controlled Raman spectroscopy and temperaturecontrolled X-ray powder diffraction.

Experimental details

The synthesis of the [(CH 3 ) 4 P] 3 Bi 2 Cl 9 (PCB) and [(CH 3 ) 4 P] 3 Sb 2 Cl 9 (PCA) compounds were carried out using a preparation procedure reported elsewhere [START_REF] Elgahami | Crystal structure, Hirshfeld surface analysis, Thermal study and Conduction mechanism of [(C 4 H 9 ) 4 P] 3 Bi 2 Cl 9 compound[END_REF][START_REF] Megha | Study of alternating current conduction mechanism in polypyrrolemagnesium ferrite hybrid nanocomposite through correlated barrier hopping model[END_REF][START_REF] Ben Gzaiel | Using Raman spectroscopy to understand the origin of the phase transitions observed in [(C 3 H 7 ) 4 N] 2 Zn 2 Cl 6 compound[END_REF]. The bismuthate chloride and antimonate chloride were prepared by dissolving Bi 2 O 3 or Sb 2 O 3 in concentrated hydrochloric acid (HCl, 37%). Then, a stoichiometric quantity of tetramethylphosphonium chloride (purity 98%) was added to these two solutions. The corresponding sequences of chemical reactions are:

M 2 O 3 + 6 HCl→2 MCl 3 + 3H 2 O
(1)

3[(CH 3 ) 4 PCl] + 2MCl 3 →[(CH 3 ) 4 P] 3 M 2 Cl 9 (2)
where M = Sb or Bi. The solution is left open to the air at room temperature for 5 days, after which bright and transparent crystals begin to appear. Crystals of millimetric size are then isolated, washed with absolute ethanol and kept under standard atmospheric conditions before characterizing them. The phase purity of the powder was verified by recording X-ray powder diffraction (XRPD) pattern at room temperature on a PANalytical θ/θ Bragg-Brentano Empyrean diffractometer (CuKα 1+2 radiations) equipped with the PIXcel 1D detector. XRPD pattern was collected at room temperature in the [5 • -100 • ] scattering angle range, with a 0.0131 • step size, for a total acquisition time of 7 h 30. Thermal stability in air of the as-prepared powder was studied by temperature-controlled X-ray powder diffraction. XRPD patterns were recorded on the same diffractometer between 25 and 130 • C (heating rate of 10 • C/min, temperature stabilisation for 20 min, cooling rate of 60 • C/min, and air flow of 40 mL/min) by using an XRK 900 Anton Paar reactor chamber. The sample was deposited on the sieve (pore size Φ = 0.2 mm) of the open sample holder cup, both made of glass ceramic Macor®, thus allowing air to flow through the sample. For each temperature, the XRPD pattern was collected in the [5 • -55 • ] scattering angle range, with a 0.0131 • step size, for a total acquisition time of 120 min. The refinements of the XRPD pattern were carried out by the Le Bail method [START_REF] Bail | Ab-initio structure determination of LiSbWO 6 by X-ray powder diffraction[END_REF] of the Fullprof program [START_REF] Rodríguez-Carvajal | Recent advances in magnetic structure determination by neutron powder diffraction[END_REF].

The differential scanning calorimetry (DSC) measurement was performed using PerkinElmer DSC-7 instrument rate of 5 • C mn -1 . It was carried out at two successive heating and cooling runs over a temperature range of 25-300 • C in order to identify the precise phase transition temperature.

The thermogravimetric and differential thermal analysis were simultaneously performed on raw powders of PCA and PCB with a TGA/ DTA Q600 SDT TA Instruments apparatus (Pt crucibles, α-Al 2 O 3 as a reference) under air flow (100 ml/min) from the room temperature (RT) to 270 • C range (heating/cooling rate of 5 • C/min).

A first series of Raman measurements was carried out on PCB and PCA crystals in the wavenumber range 10-3500 cm -1 at 25 • C and heated to 250 • C, then a second series on the temperature range 25-130 • C. The spectra were recorded with a HORIBA Jobin-Yvon T64000 Raman spectrometer equipped with a single monochromator (600 gratings per mm) coupled to a nitrogen cooled front-illuminated charge-coupled device (CCD) detector. The spectra were collected in backscattering geometry under a BX41 Olympus microscope equipped with aMSPlan50x (N.A. 0.55) objective focusing the incident wavelength radiation at 514.5 nm of an argon-krypton ion laser (Innova, Coherent, France). During the experiment, the laser power was less than 2 mW on the sample and the Raman spectra were systematically recorded twice with an integration time of between 40 and 60s. Acquisition and basic analysis of spectra (determination of peak positions and full width at half maximum (FWHM) of the Raman bands using Gaussian fitting functions) have been performed using the Lab-SpecV5.25 (Jobin-Yvon, Horiba Group, Kyoto, Japan) software.

Confocal Raman mapping 100 × 100 μm 2 area (2D map step size of around 5 μm) was performed at room temperature on a crystal of PCA and PCB heated at 270 • C for 72h. The spectra were recorded using a WITec Alpha 300R confocal Raman spectrometer (WITec GmbH, Ulm Germany) equipped with both a RayShield Coupler and a 600 gmm -1 grating blazed at 500 nm and a thermically Peltier-cooled Si-based CCD front-illuminated detector (Andor, Oxford Instrument, Belfast, Ireland). Raman scattered signals were collected under a microscope equipped with a Zeiss EC Epiplan-Apochromat Dic 50x (numerical aperture of 0.95) focusing the 532 nm line of a solid-state sapphire laser (Coherent INC., Santa Clara, USA) with a laser power of 2mW on the sample and integration times set at 2s. The spectral analysis was performed using the WITec Project FIVE plus software (version 5.248, WITec GmbH, Germany) and in particular the True Component Analysis (TCA). This dedicated tool of the WITec Project FIVE plus software allows both identifying pixels of a map with similar spectral features and providing these spectral characteristics (i.e. similar chemical response) in an intensity distribution image. Let us note that before analysis, spectral raw data were baseline corrected and pre-processed by removing the cosmic rays.

The surface morphology of PCB and PCA powder sample was observed by a field emission scanning electron microscope (SEM) using a JEOL JSM-6510LV microanalysis. An OXFORD Energy Dispersive X-ray Spectrometer (EDS) coupled to the microscope was also used to verify the presence of chemical elements in the crystals. Before analysis, small crystals were deposited on a carbon tape and a thin film of gold was sprayed on them.

Results and discussion

X-ray powder diffraction analysis

A portion of the millimetre-sized crystals have been crushed into micrometric powder in order to verify, by X-ray powder diffraction (XRPD), the homogeneity of the two compounds. A refinement of the XRPD pattern (for each sample), collected at room temperature, was carried out by the Le Bail method using the trigonal cell parameters and the P31c (n • 159) space group reported in Ref. [1]. Fig. 1 shows the observed, calculated, and difference diffraction patterns for PCB and PCA. All Bragg peaks were successfully indexed and satisfactorily fitted using this crystallographic data, thus confirming the high purity of both samples. The conventional reliability factors of the refinement are R p = 6.47%, R ωp = 6.77%, R exp = 1.94% for PCB and R p = 8.09%, R ωp = 7.81%, R exp = 2.68% for PCA. The trigonal cell parameters (a = 9.5030 (4) Å, c = 22.3276(9) Å for PCB and a = 9.4658(5) Å, c = 22.20(1) Å for PCA) are in good agreement with those determined from XRD data collected on a single crystal of PCB and PCA in the literature [1].

In the crystal structure of both PCB and PCA compounds, each tetramethyl phosphonium [(CH 3 ) 4 P] + cation has one of its P-C bonds oriented along the threefold axis of the crystal structure (Fig. 2). Two [MCl 6 ] 3-octahedral (M = Bi or Sb) share one of their triangular faces to form isolated dimers of [M 2 Cl 9 ] 3-(Fig. 2). Each organic cation is surrounded by three dimers arranged around the threefold axis (Fig. 2). With such an arrangement, three carbon atoms of the [(CH 3 ) 4 P] + cation interact directly with the chlorine atoms shared between two [MCl 6 ] 3- octahedra (each carbon thus has two first neighbor chlorine atoms at distances d 2 and d 3 , respectively), while the fourth carbon atom located on the threefold axis only interacts with the chlorine atoms of one of the triangular faces left free (all atoms at the same distance d 1 ), as indicated by the grey dashed lines in Fig. 2. Thereby, the methyl groups of each tetramethyl phosphonium [(CH 3 ) 4 P] + cation have two different chemical environments.

Thermal analysis

The DSC curves of PCB and PCA compounds recorded upon heating and cooling at 5 • Cmin -1 are displayed in Fig. 3 (a) and (b), respectively. For each compound, one endothermic peak and one exothermic peak, both of small intensity, are detected on heating and cooling, respectively (temperatures read on the top of the peak: 269/258 • C for PCB and 256/ 254 • C for PCA). These phenomena are completely reproducible since these peaks are also detected on the second heating/cooling run. The associated entropy value for PCB and PCA are ΔS = 2.09 Jmol -1 K -1 and ΔS = 1.89 Jmol -1 K -1 , respectively.

Fig. 4 displays the TGA and DTA curves of the PCB and PCA samples recorded upon heating and cooling. For both compounds, no weight loss is measured from RT to 270 • C indicating that the sample is presumably thermally stable in this temperature range. No endothermic and exothermic peaks appear on the DTA curve upon heating and cooling, reflecting the lower sensitivity of DTA, compared to DSC, to detect such weak signals. So, the nature of the peaks that appears on the DSC curve on heating and cooling cycles is still unclear. Wojtas et al. [1] postulated that this high temperature phase is displacive in nature but gave no experimental evidence for this. In order to identify the nature of this high temperature transition, Raman spectra were recorded in the wavenumber range 10-3500 cm -1 over the temperature range 180-250 • C. XRPD pattern and scanning electron microscope (SEM) were carried out after heating the sample up to 270 • C.

Thermal behavior above 250 • C 3.3.1. Raman analysis

The Raman spectra of PCB and PCA crystals at 25 and heated at 250 • C are shown in Fig. 5. At room temperature, the assignment of the vibrational modes (Table 1) is based on the comparison with similar compounds in literature [4,[START_REF] Nakamoto | Infrared and Raman Spectra of Inorganic and Coordination Compounds[END_REF][START_REF] Varma | Infrared and Raman spectroscopic investigations of methylammonium haloantimonates(III), [N(CH 3 ) 4n H n ], Sb 2 X 9 (n = 0-3, X = Cl or Br), through their phase transitions[END_REF][START_REF] Kruger | Tri-und TetraMethylammoniumchloroantimonate(III)[END_REF][START_REF] Wojtaś | Vibrational study of structural phase transitions in[END_REF]. The bands observed in the 50-400 cm -1 wavenumber range can be attributed to the vibrational modes of [M 2 Cl 9 ] 3-anions (Table S1) and the bands above 400 cm -1 are related to the vibrational modes of cationic [(CH 3 ) 4 P] + cations. After heating the samples up to 250 • C additional broad bands near 1600 cm -1 appear; these bands are commonly attributed to the stretching modes of graphitic carbon. At 250 • C, a decrease in the intensity of the peaks of the stretching modes of (CH 3 ) is also observed for the PCB and the almost total disappearance of these bands for the PCA compound. The presence of graphitic carbon clearly shows that a decomposition process must have taken place on the surface of PCB and PCA crystals when they are heated above 250 • C. To identify the decomposition products, we then decided to characterize each compound after a long heating process above 250 • C by Raman mapping, SEM-EDX analysis and X-ray powder diffraction.

Raman mapping

Fig. 6 shows a Raman mapping based on 400 spectra recorded at ambient temperature in 100 × 100 μm 2 area of PCB crystal surface after heating 72h at 270 • C. Although a large part of the material was modified (easily visible under an optical microscope by its change in surface state), a specific area was chosen to perform the Raman mapping because it remained a small area not modified by temperature. A True Component Analysis (TCA) was used to identify spectral components that are most prevalent in the dataset. The average spectra, which can be extracted from the components, are then helpful to identify and localize different molecular structures. The blue-coded spectral signature corresponds to the PCB crystal not affected by heating (Fig. 6b). The redcoded spectral signature is different from that of the PCB crystal (Fig. 6c). The intense bands at 142 and 199 cm -1 can be attributed respectively to A 1g internal Bi-Cl stretching mode and E g internal Bi-Cl stretching mode of BiOCl compound (with space group D 7 4h ) [START_REF] Shamaila | WO 3 /BiOCl, a novel heterojunction as visible light photocatalyst[END_REF][START_REF] Sarkar | Faceted growth of morphologically tuned of BiOCl[END_REF], which is in total agreement with diffraction results (rectangle in Fig. 8a). This attribution is also consistent with the presence of a very weak intensity band at 394 cm -1 , due to the motion of oxygen atoms. The presence of weak bands at 107, 232 and 276 cm -1 (assigned to PCB crystal not affected by heating) in the red-coded spectrum confirms, as shown by X-ray diffraction (Fig. 8), that the thermal decomposition is not complete. Let us note that the absence of other vibrational modes on the spectrum does not allow the identification of other bismuth oxychlorides of different stoichiometry or bismuth oxides (Bi 2 O 3 or Bi 2 O 5 ).

A Raman mapping was also carried out on the PCA crystal surface after heating 72h at 270 • C (Fig. 7). Concomitantly, the CH stretching bands decrease very strongly and three bands at 194, 402 and 455 cm - appear which can be attributed to the bending and stretching vibration of the Sb-O bonds in Sb 2 O 4 [START_REF] Mestl | Sb 2 O 3 /Sb 2 O 4 in reducing/oxidizing environments: an in situ Raman spectroscopy study[END_REF], compound indeed identified by diffraction (rectangle in Fig. 8b).

Considering the decomposition of the compounds above 250 • C, it may be surprising that no weight loss is detected on the thermogravimetric curves displayed in Fig. 4. Nevertheless, due to the heating rate of 5 • C/min and the absence of an isothermal plateau at 270 • C during the thermogravimetric analyses, it can the considered that the time during which the samples are heated above 250 • C is too short to cause their total decomposition and therefore measure the associated weight losses.

Accordingly, the ageing study of PCA and PCB crystals carried out at 270 • C for 3h or 72h clearly demonstrates that the endothermic peak detected by DSC for each compound is associated with its decomposition rather than with a displacive structural phase transition, as postulated by Wojtas et al. [1].

SEM-EDX analysis

During Raman experiments, we observed that the surface goes from a reflective state to a much more diffusive state from 250 • C. In order to understand this surface phenomenon, one single crystal of each compound was heated at 270 • C for 3h and then studied by a scanning electron microscope (SEM), coupled with the energy dispersive X-ray (EDX). Figs. S1a andS1b show the SEM images and EDX elemental mappings and EDX spectra of PCA and PCB crystal before and after heating.

The SEM micrograph observed before heating reveals smooth surfaces with small fragments of other crystals. After heating, crystal surfaces are rougher with grey and black areas (Fig. S1a andS1b). From the elemental analyses, we notice that the percentage oxygen increases for both crystals after heating. Although the percentage of carbon is skewed due to the use of carbon tape to stick the sample to the holder, carbon also covers the crystal surface. The carbon at the crystal surface must probably be in an amorphous form (black carbon) since it is not visible on the XRPD pattern (see hereafter). A significant percentage of oxygen is also present in the samples thus leading to a reduction in the Bi/Cl and Sb/Cl ratios of the PCB and PCA compounds, respectively. SEM and EDX results confirm the partial decomposition of PCB and PCA and show that a surface oxidation also occurs up on heating to 270 • C in air. Because this oxidation can lead to either oxychlorides or pure oxides, X-ray powder diffraction was used to identify the products formed.

X-ray powder diffraction

In order to study the structural modifications of the PCB and PCA compounds above 250 • C, XRPD patterns were collected at room temperature after heating the powders at 270 • C under air for 72 h (Fig. 8). When compared with the XRPD pattern of the raw PCB sample, the long heating caused many peaks to appear alongside those attributed to the PCB phase (Fig. 8a), only some of which can be attributed to BiOCl (10, 26, 33 and 34 • , JCPDS card n • 98-007-4502 [START_REF] Keramidas | The crystal structure of BiOCl[END_REF]). The PCB sample partially decomposes during the prolonged heating to 270 • C. In Fig. 8b, the thermal decomposition of PCA is, in contrast, complete since only broad diffraction peaks of Sb 2 O 4 (JCPDS card n • 98-000-4109) is detected on the XRPD pattern collected on the powder heated at 270 • C for 72 h.

Thermal behavior in the range 30-130 • C

In order to probe the thermal behavior of both compounds in the temperature range 30-130 spectroscopy and XRPD were used.

Raman spectroscopy analysis

Raman spectroscopy is one of the powerful tools to study hybrid compounds and reveal phase transitions caused by reorientations of organic entities in the crystal structure [START_REF] Khalfa | Synthesis, structural and electrical characterization of a new organic inorganic bromide: [(C 3 H 7 ) 4 N] 2 CoB 4[END_REF][START_REF] Trigui | Ferroelectric properties and Raman spectroscopy of the [(C 4 H 9 ) 4 N] 3 Bi 2 Cl 9 compound[END_REF][START_REF] Trigui | Raman study of order-disorder phase transition in [(C 3 H 7 ) 4 N] 3 Bi 3 Cl 12 compound[END_REF][START_REF] Moreira | Raman-spectroscopic evaluation of the longrange order in Ba(B 1/3 ′ B 2/3 ′′ )O 3 ceramics[END_REF][START_REF] Ben Brahim | Organic-inorganic interactions revealed by Raman spectroscopy during reversible phase transitions in semiconducting [(C 2 H 5 ) 4 N]FeCl 4[END_REF]. Raman spectra of PCB and PCA compounds, collected in the temperature range 30-130 • C, are presented in Figs. 9 and 10, respectively. For more clarity, only selected spectra were shown in the wavenumber ranges 50-350 cm -1 , 550-1550 cm -1 , 2750-3050 cm -1 . A careful analysis of these spectra makes it possible to observe small variations in terms of positions and half-widths of bands in the temperature range 30-130 • C. Fig. S2 shows an example of deconvolution of the spectrum of PCA compound recorded at 30 • C in the 50-400, 600-1500 and 2750-3150 cm -1 spectral ranges.

PCB compound.

In order to verify whether a phase transition (not detected by thermal analysis) is connected to a change in the dynamical state of the anions and the cations, we presented in Fig. 11 the temperature dependence of the position and the full width at half maximum (FWHM) for the most intense vibrational bands of PCB compound.

The vibration modes of cationic [(CH 3 ) 4 P] + parts are sensitive indicators to phase transitions. As previously mentioned in the XRPD analysis, PCB compound exhibits two different chemical environments for the methyl group of the tetramethyl phosphonium cation [(CH 3 ) 4 P] + (Fig. 2). This results in a splitting of the symmetric and asymmetric (CH 3 ) stretching vibration modes (ν s and ν as , respectively) with relative integrated intensities in the ratio 1 to 3, as expected. So, the more intense bands at 2919 and 2994 cm -1 are assigned respectively to the symmetric and asymmetric stretching vibrations of methyl group in interaction with the equatorial chlorine atoms at distances d 2 and d 3 (Fig. 2) and the bands located at 2911 and 2982 cm -1 to the symmetric and asymmetric stretching vibrations of methyl group [START_REF] Nakamoto | Infrared and Raman Spectra of Inorganic and Coordination Compounds[END_REF] in interaction with the terminal chlorine atoms at distance d 1 (Fig. 2). In the following, we studied the behavior of all these vibration modes with temperature.

The bands at 2911, 2919, 2982 and 2994 cm -1 , shift about 1 cm -1 after 80 • C (Fig. 11c) but the evolution of their widths shows a change in slope between 70 and 80 • C, suggesting local changes in the interactions of hydrogen with chloride anions after 70 • C (Fig. 11f). As a consequence, the stretching vibration band of C-P bonds within the tetrahedron C 4 P starts broadening from 70 • C (Fig. 11e). The bands at 1412 and 1415 cm -1 , associated to CH 3 bending vibration [START_REF] Nakamoto | Infrared and Raman Spectra of Inorganic and Coordination Compounds[END_REF], move at 80 • C towards lower wavenumber with a broadening of the band of 1 cm -1 (Fig. 11b) similarly to what can be observed for the symmetric ν s (C 4 P) and asymmetric ν as (C 4 P) stretching vibrations (Fig. 11 e). The phase transition at around 70-80 • C would result from a local reorientation of the organic [(CH 3 ) 4 P] + cation.

Chlorobismuthate [Bi 2 Cl 9 ] 3-anions are also involved in the phase transition as [(CH 3 ) 4 P] + cations since the Raman bands assigned to the vibration of Bi-Cl bonds (Table 1) also change in position and width at around 80 • C. So, for chlorobismuthate [Bi 2 Cl 9 ] 3-anions, the bands observed at 282 and 240 cm -1 can be assigned to stretching vibrations of terminal Bi-Cl bonds respectively. The band at 182 and 116 cm -1 are attributed to the stretching vibration of equatorial Bi-Cl bonds and to the bending vibration mode of Bi-Cl, respectively (Table S1). All these bands are shifted toward lower wavenumber by 1 cm -1 (Fig. 11 a) with a broadening of 2 cm -1 (Fig. 11 d) at 80 • C. At the phase transition, the reorientation of the organic [(CH 3 ) 4 P] + cations is accompanied by a rotation of the [Bi 2 Cl 9 ] 3-anions, the former starting 10 • C before the latter as mentioned above. As the movements of the anions and cations are local and progressive, they do not require a large energy input to occur. This explains the absence of an endothermic peak on the DSC curve (Fig. 3) which is characteristic of a second order type transition. Such a behavior has been also observed in the two other chlorobismuthate (CH 3 NH 3 ) 3 Bi 2 CI 9 and (C 5 H 5 NH) 6 Bi 4 Cl 18 [START_REF] Belkyal | X-ray diffraction and Raman scattering in (CH 3 NH 3 ) 3 Bi 2 Cl 9 (MACB) single crystals[END_REF][START_REF] Tarasiewicz | Raman studies of the anionic sublattice vibrations in (C 5 H 5 NH) 6 Bi 4 Cl 18[END_REF].

3.4.1.2. PCA compound. Fig. 12 shows the temperature dependence of the position and the full width at half maximum (FWHM) for the most intense vibrational bands of PCA compound. As previously mentioned for PCB compound, the vibration modes of (CH 3 ) are sensitive indicators to phase transition (Fig. 12a andd). The strongest bands at 2914 and 2990 cm -1 , assigned respectively to the symmetric and asymmetric stretching vibrations of methyl group in interaction with the equatorial chlorine atoms, shift in position at 57, 80 and 95 • C concomitantly with the variation of their width. The positions and the widths of the bands associated with the bending vibration modes δ (CH 3 ) also changes significantly at 57 and 95 • C (Fig. 12b ande). As with PCB, any change in the way the methyl groups vibrate naturally affects the C-P bonds within the C 4 P tetrahedron whose the vibration bands are at 649 and 780 cm -1 . The difference is that the PCA compound undergoes three successive transitions at 57, 80 and 95 • C, compared to only one for PCB at 70-80 • C.

The bands observed at 314 and 287 cm -1 are assigned to stretching vibrations of terminal Sb-Cl bonds and the bands at 241 and 126 cm -1 to the stretching vibration modes of equatorial Sb-Cl bonds and bending vibration mode of Sb-Cl, respectively (Table S1). All these bands change abruptly in position at 57, 80 and 95 • C (Fig. 12c) and broaden (Fig. 12f) however a more pronounced broadening is observed for the band at 126 cm -1 which starts from 40 • C until the first transition at 57 • C. This behavior is certainly correlated with the structural changes responsible for the broadening of the symmetric and asymmetric stretching vibrations of the methyl group interacting with the terminal chlorine atoms, which incidentally both begin at 30 • C. All these results confirm that the Thus, for PCA and PCB compounds, we have shown that CH 3 undergoes position and width modifications during phase transitions. In order to complete these Raman analyses, we wanted to verify if the integrated intensities of these vibration bands, and in particular the symmetric and asymmetric stretching vibrations of methyl group in interaction with the terminal chlorine atoms, could be used to highlight the phase transitions. So, we calculated the ratio between the integrated intensity of the symmetric over asymmetric stretching vibrations of methyl (meaning at 2905 cm -1 and 2977 cm -1 , respectively). Intensity variations are clearly observed around 57, 80 and 92 • C (Fig. S3), suggesting that this method could be applied to other hybrid materials when it is difficult to highlight transitions involving organic components. Finally, let us note that a similar calculation was carried out for the PCB compound and that only one intensity variation at 80 • C was found (Fig. S3).

X-ray diffraction analysis

In addition to studies by Raman spectroscopy, temperaturecontrolled X-Ray powder diffraction can provide information about any changes in the crystal structure of PCB and PCA compounds. XRPD patterns collected in air on the both powder PCB and PCA, heated at different temperatures in the range RT to 130 • C, are superimposed in determined from the refinement of a XRPD pattern at each temperature were used as starting values for the refinement of the pattern at the next temperature. All XRPD patterns can be satisfactorily fitted in this way, thus confirming that the trigonal cell of the both crystals structure of PCB and PCA is preserved in the whole temperature range investigated. The thermal evolutions of the cell parameters a and c of PCB and PCA are displayed in Fig. 14. Fig. 14a shows two thermal domains with linear dependences of each cell parameters of (PCB), corresponding to a small increase in thermal expansion coefficient (TEC) above ≈ 75 • C. The thermal expansions coefficients along the a crystallographic axis is 4.64 × 10 -4 • C -1 and 5.87 × 10 -4 • C -1 in the range RT-75 • C and 75-130 • C, respectively. The thermal expansion coefficients along the c crystallographic axis are 1.62 × 10 -3 • C -1 and 1.84 × 10 -3 • C -1 in the range RT-75 • C and 75-130 • C, respectively. Fig. 14b shows two thermal domains with linear dependences of each cell parameters of (PCA), corresponding to increase in thermal expansion coefficient above ≈ 65, 85 and 95 • C. Whatever the thermal domains considered, the thermal expansion along the [001] direction is much larger in magnitude than the one in the basal (a,b) plane. Both changes in TEC concomitantly occur with the discontinuities in the thermal evolution of the vibrational modes associated with the tetramethyl phosphonium [(CH 3 ) 4 P] + cations. This suggests that the discontinuities in the thermal evolution of the vibrational modes and changes in TEC around 80 • C for PCB and 65, 85 and 95 • C for PCA, are due to smaller orientations of the tetramethyl phosphonium cations [(CH 3 ) 4 P] + and rotations of [M 2 Cl 9 ] 3-dimers in the crystal structure. This explains why these new phase transitions are of second order.

When six-fold coordinated to anions, trivalent antimony is slightly smaller in size that trivalent bismuth [START_REF] Shannon | Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides[END_REF]. This difference in size explains why the unit cell parameters a and c for PCA are also smaller than that for PCB (Fig. 14). Because of this reduction in cell volume, we believe that reorientations of the tetramethyl phosphonium cations [(CH 3 ) 4 P] + and rotations of [Sb 2 Cl 9 ] 3-dimers in the structure of PCA are not as easy as in the structure of PCB. This could explain that these movements take place in three successive steps and not in a single step as in PCB.

Conclusion

In this paper, the [(CH 3 ) 4 P] 3 Bi 2 Cl 9 (PCB) and [(CH 3 ) 4 P] 3 Sb 2 Cl 9 (PCA) compounds have been synthesized by slow evaporation method at room temperature. The structural refinement confirms that both compounds crystallize in the trigonal system with a space group of P31c. Raman and XRD studies reveal that the structure of PCB and PCA crystals start to decompose above 250 • C. The SEM image and the EDX analysis show the inhomogeneity of both crystals surface and the increases of proportion of carbon and oxygen after heating (above 270 • C). The analysis of the positions and widths of the vibration bands of PCB and PCA compounds, as a function of the heating temperature, allowed us to highlight specific temperatures which can be assigned to phase transitions based on a dynamical reorientation of the organic [(CH 3 ) 4 P] + cations with a rotation of [M 2 Cl 9 ] 3-anions (M = Bi or Sb). The behavior of the thermal evolutions, which shows a slope change of the cell parameters a and c of both compounds confirmed the presence of the new phase transitions detected by Raman spectroscopy. PCB exhibits only one phase transition at 80 • C while 3 successive phase transitions at about 60, 80 and 95 • C are detected for PCA. The number of transitions may be related to the free volume available for reorientations of the tetramethyl phosphonium cations [(CH 3 ) 4 P] + and rotations of [M 2 Cl 9 ] 3- dimers in the unit cell, this free volume being smaller in PCA than in PCB.

Fig. 1 .

 1 Fig. 1. Comparison of the observed diffraction patterns (red dots) with the patterns calculated by the Le Bail method (black lines) for [(CH 3 ) 4 P] 3 Bi 2 Cl 9 (PCB) (a) and [(CH 3 ) 4 P] 3 Sb 2 Cl 9 (PCA) (b) compounds. Vertical markers give Bragg peak positions of the space group P 31c (No.159).

Fig. 2 .

 2 Fig. 2. Structural representation of [(CH 3 ) 4 P] 3 Bi 2 Cl 9 (PCB) and [(CH 3 ) 4 P] 3 Sb 2 Cl 9 (PCA) compounds.

Fig. 3 .

 3 Fig. 3. DSC curve of [(CH 3 ) 4 P] 3 Bi 2 Cl 9 (PCB) (a) and [(CH 3 ) 4 P] 3 Sb 2 Cl 9 (PCA) (b) compounds recorded during heating and cooling.

Fig. 4 .

 4 Fig. 4. Thermogravimetric (red) and differential thermal (blue) curves recorded during heating and cooling of [(CH 3 ) 4 P] 3 Bi 2 Cl 9 (PCB) (a) and [(CH 3 ) 4 P] 3 Sb 2 Cl (PCA) (b) powders in air.

Fig. 5 .

 5 Fig. 5. Raman spectra of [(CH 3 ) 4 P] 3 Bi 2 Cl 9 (PCB) (a) and [(CH 3 ) 4 P] 3 Sb 2 Cl 9 (PCA) (b) compounds at 25 and 250 • C.

Fig. 6 .

 6 Fig. 6. (a) and (b) Raman mapping recorded at ambient temperature in 100 × 100 μm 2 area of [(CH 3 ) 4 P] 3 Bi 2 Cl 9 (PCB) crystal surface heated 72h at 270 • C. (c) Raman spectra of (PCB) after heating to 270 • C.

Fig. 7 .

 7 Fig. 7. Raman spectra of [(CH 3 ) 4 P] 3 Sb 2 Cl 9 (PCA) powder, heated at 270 • C.

Fig. 8 .

 8 Fig. 8. X-ray diffraction diagram of raw powder of [(CH 3 ) 4 P] 3 Bi 2 Cl 9 (PCB) (a) and [(CH 3 ) 4 P] 3 Sb 2 Cl 9 (PCA) (b) compounds heated 72h at 270 • C.

Fig. 9 .

 9 Fig. 9. Temperature evolution of the Raman spectra of [(CH 3 ) 4 P] 3 Bi 2 Cl 9 (PCB) compound in the temperature range 30 • C-130 • C for selected spectral ranges [50-350 cm -1 ], [500-1550 cm -1 ] and [2750-3050 cm -1 ].

Fig. 10 .

 10 Fig. 10. Temperature evolution ofthe Raman spectra of [(CH 3 ) 4 P] 3 Sb 2 Cl 9 (PCA) compound in the temperature range 30 • C-130 • C for selected spectral ranges [50-350 cm -1 ], [500-1550 cm -1 ] and [2750-3050 cm -1 ].

Fig. 11 .

 11 Fig. 11. Temperature evolution of Raman wavenumbers of [(CH 3 ) 4 P] 3 Bi 2 Cl 9 (PCB) in the frequency region assigned to (a) the stretching modes of [Bi 2 Cl 9 ] 3-anions, (b) the asymmetric bending modes of (CH 3 ) and the stretching modes of (C 4 P) and (c) the stretching modes of (CH 3 ). Temperature evolution of full width at half maximum (FWHM) of the Raman bands in the frequency region assigned to (d) the stretching modes of [Bi 2 Cl 9 ] 3-anions, (e) the asymmetric bending modes of (CH 3 ) and the stretching modes of (C 4 P) and (f) the stretching vibration modes of (CH 3 ).

Fig. 13 .

 13 No change in the shape of diffraction peaks or disappearance/ appearance of new ones is noticed on the XRPD patterns collected above 75-80 • C. If the XRPD patterns are comparable in whole temperature range RT -130 • C, slight and progressive shifts of all diffraction peaks towards low scattering 2θ angles occur as the temperature increases. These shifts are characteristic of a positive thermal expansion of the unit cell parameters. The refinement of each XRPD pattern was carried out by the Le Bail method. For the refinement of the XRPD pattern collected at RT, the trigonal cell parameters, previously determined in section 3.1, were used as starting values. Successively, the trigonal cell parameters

Fig. 12 .

 12 Fig. 12. Temperature evolution of Raman wavenumbers of [(CH 3 ) 4 P] 3 Sb 2 Cl 9 (PCA) in the frequency region assigned to (a) the stretching modes of [Sb 2 Cl 9 ] 3-anions, (b) the asymmetric bending modes of (CH 3 ) and the stretching modes of (C 4 P) and (c) the stretching modes of (CH 3 ). Temperature evolution full width at half maximum (FWHM) of the Raman bands in the frequency region assigned to (d) the stretching modes of [Sb 2 Cl 9 ] 3-anions, (e) the asymmetric bending modes of (CH 3 ) and the stretching modes of (C 4 P) and (f) the stretching modes of (CH 3 ).

Fig. 13 .

 13 Fig. 13. XRPD patterns collected on the [(CH 3 ) 4 P] 3 Bi 2 Cl 9 (a) and [(CH 3 ) 4 P] 3 Sb 2 Cl 9 (b) powders upon heating in air from RT to 130 • C.

Fig. 14 .

 14 Fig. 14. Thermal evolutions of the trigonal cell parameters a and c determined from the refinement by the Le Bail method of the XRPD patterns of the [(CH 3 ) 4 P] 3 Bi 2 Cl 9 (PCB) (a) and [(CH 3 ) 4 P] 3 Sb 2 Cl 9 (PCA) (b) compounds.

Table 1

 1 Band assignm ent of Ram an spectra of [(CH 3 ) 4 P] 3 Bi 2 Cl 9 (PCB) and [(CH 3 ) 4 P] 3 Sb 2 Cl 9 (PCA).

Acknowledgment

This work was funded by the Tunisian Ministry of Higher Education and Scientific Research and French Ministry of Higher Education and Research through funds accorded to the implied research Labs.

Data availability

Data will be made available on request.

CRediT authorship contribution statement

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi. org/10.1016/j.jpcs.2023.111227.