Marie-Aline Martin-Drumel
email: marie-aline.martin@universite-paris-saclay.fr

Jessica P Porterfield

Manuel Goubet

Pierre Asselin

Robert Georges

Pascale Soulard

Matthew Nava

P Bryan Changala

Brant Billinghurst

Olivier Pirali

Michael C Mccarthy

Joshua H Baraban

Synchrotron-based High Resolution Far-infrared Spectroscopy of trans-butadiene

The high resolution far-infrared spectrum of trans-butadiene has been re-investigated by Fouriertransform spectroscopy at two synchrotron radiation facilities, SOLEIL and the Canadian Light Source, at temperatures ranging from 50 to 340 K. Beyond the well-studied bands, two new fundamental bands lying below 1100 cm -1 , ν 10 and ν 24 , have been assigned using a combination of cross-correlation (ASAP software) and Loomis-Wood type (LWWa software) diagrams. While the ν 24 analysis was rather straightforward, ν 10 exhibits obvious signs of a strong perturbation, presumably owing to interaction with the dark ν 9 + ν 12 state. Effective rotational constants have been derived for both the v 10 = 1 and v 24 = 1 states. Since only one weak, infrared active fundamental band (ν 23 ) of trans-butadiene remains to be observed at high resolution in the far-infrared, searches for the elusive gauche conformer can now be undertaken with considerably greater confidence in the dense ro-vibrational spectrum of the trans form.

Introduction

Much attention has surrounded 1,3-butadiene (C 4 H 6 ) because this conjugated diene is an ideal candidate to observe the effects of π-electron delocalization [START_REF] Craig | Equilibrium Structures for Butadiene and Ethylene: Compelling Evidence for π-Electron Delocalization in Butadiene[END_REF][START_REF] Wiberg | Butadiene and heterodienes revisited[END_REF], and because it is of central importance in a wide range of chemistry applications, from its archetypal role in the Diels-Alder reaction [START_REF] Diels | Synthesen in der hydroaromatischen Reihe[END_REF] to industrial rubber production [START_REF] Liao | Controlling guest conformation for efficient purification of butadiene[END_REF]. Characterizing the structural conformers and isomerization dynamics of this diene, however, has proven surprisingly challenging. 1,3-butadiene exists in two stable forms, the more stable planar trans, and the long-elusive gauche, lying 12 kJ/mol higher in energy [START_REF] Baraban | The molecular structure of gauche-1,3-butadiene: Experimental establishment of non-planarity[END_REF]. The latter was only conclusively detected in the gas phase recently by several authors of the present work, an effort which also yielded a nearly complete semi-experimental equilibrium structure (r SE e ) from a combination of pure rotational measurements of various isotopic species and high-level quantum chemical calculations [START_REF] Baraban | The molecular structure of gauche-1,3-butadiene: Experimental establishment of non-planarity[END_REF].

While electron diffraction studies provided insight into the structure of trans-butadiene as early as the late 1930's [START_REF] Schomaker | The electron diffraction investigation of the structure of benzene, pyridine, pyrazine, butadiene-1,3, cyclopentadiene, furan, pyrrole, and thiophene[END_REF][START_REF] Kveseth | Conformational Analysis. The Structure and Torsional Potential of 1,3-Butadiene as Studied by Gas Electron Diffraction[END_REF], an accurate determination of its equilibrium structure was only achieved in 2006 [START_REF] Craig | Equilibrium Structures for Butadiene and Ethylene: Compelling Evidence for π-Electron Delocalization in Butadiene[END_REF]. This centro-symmetric species possesses no permanent dipole moment, and consequently no pure rotational spectrum. Determination of the ground state constants thus relied on the assignments of rotational transitions in its dense vibrational bands, a challenging feat overcome in 2004 with the highresolution analysis of three infrared bands -namely ν 17 (3100 cm -1 ) [START_REF] Halonen | Structural issues in conjugated hydrocarbons: High-resolution infrared slit-jet spectroscopy of trans-1,3-butadiene[END_REF], ν 11 (908 cm -1 ) [START_REF] Craig | Analysis of the rotational structure in bands in the high-resolution infrared spectra of butadiene and butadiene-2,3-d 2 : Refinement in assignments of fundamentals[END_REF], and ν 20 (1596 cm -1 ) [START_REF] Craig | Analysis of the rotational structure in bands in the high-resolution infrared spectra of butadiene and butadiene-2,3-d 2 : Refinement in assignments of fundamentals[END_REF]. From extensive isotopic investigations, Craig and co-workers [START_REF] Craig | Equilibrium Structures for Butadiene and Ethylene: Compelling Evidence for π-Electron Delocalization in Butadiene[END_REF] soon afterwards derived a full semiexperimental structure from the ground state rotational Accepted article 1/10 constants of eight isotopologues: the normal [START_REF] Craig | Analysis of the rotational structure in bands in the high-resolution infrared spectra of butadiene and butadiene-2,3-d 2 : Refinement in assignments of fundamentals[END_REF], 1,1-d 2 [START_REF] Caminati | Microwave Fourier Transform Spectrum of s-trans-1,3-butadiene-1,1-d 2[END_REF], 2,3-d 2 [START_REF] Craig | Analysis of the rotational structure in bands in the high-resolution infrared spectra of butadiene and butadiene-2,3-d 2 : Refinement in assignments of fundamentals[END_REF], 1,4-trans,trans-d 2 [START_REF] Craig | Rotational analysis of bands in the high-resolution infrared spectra of Accepted article 9/10 the three species of butadiene-1,4-d 2 ; refinement of the assignments of the vibrational fundamentals[END_REF], 1,4-cis,cis-d 2 [START_REF] Craig | Rotational analysis of bands in the high-resolution infrared spectra of Accepted article 9/10 the three species of butadiene-1,4-d 2 ; refinement of the assignments of the vibrational fundamentals[END_REF], 1,4-cis,trans-d 2 [START_REF] Craig | Rotational analysis of bands in the high-resolution infrared spectra of Accepted article 9/10 the three species of butadiene-1,4-d 2 ; refinement of the assignments of the vibrational fundamentals[END_REF], 1-13 C 1 [START_REF] Craig | Rotational analysis of several bands in the high-resolution infrared spectrum of butadiene-1-13 c 1 : Assignment of vibrational fundamentals[END_REF], and 2,3-13 C 2 [START_REF] Craig | Analysis of rotational structure in the high-resolution infrared spectrum and assignment of vibrational fundamentals of butadiene-2[END_REF] species.

With the notable exception of butadiene-1,1-d 2 which is slightly polar and was thus investigated using microwave spectroscopy by Caminati et al. [START_REF] Caminati | Microwave Fourier Transform Spectrum of s-trans-1,3-butadiene-1,1-d 2[END_REF], the rotational constants of all other isotopologues were derived from highresolution ro-vibrational measurements. trans-butadiene belongs to the C 2h point group and possesses 12 IR-active modes, 4 of a u and 8 of b u symmetries. Among them, 6 modes lie below 1100 cm -1 , i.e. in the far-IR domain: ν 13 (162.42 cm -1 , a u ) [START_REF] Craig | An Investigation of the Rotamers of Butadiene by High-Resolution Infrared Spectroscopy[END_REF], ν 24 (∼ 299.1 cm -1 , b u ) [START_REF] Krasnoshchekov | Anharmonic Vibrational Analysis of the Infrared and Raman Gas-Phase Spectra of s-trans-and s-gauche-1,3-Butadiene[END_REF], ν 12 (524.57 cm -1 , a u ) [START_REF] Craig | An Investigation of the Rotamers of Butadiene by High-Resolution Infrared Spectroscopy[END_REF], ν 11 (908.07 cm -1 , a u ) [START_REF] Craig | Analysis of the rotational structure in bands in the high-resolution infrared spectra of butadiene and butadiene-2,3-d 2 : Refinement in assignments of fundamentals[END_REF], ν 23 (∼ 990.3 cm -1 , b u ) [START_REF] Krasnoshchekov | Anharmonic Vibrational Analysis of the Infrared and Raman Gas-Phase Spectra of s-trans-and s-gauche-1,3-Butadiene[END_REF], and ν 10 (∼ 1013.8 cm -1 , a u ) [START_REF] Krasnoshchekov | Anharmonic Vibrational Analysis of the Infrared and Raman Gas-Phase Spectra of s-trans-and s-gauche-1,3-Butadiene[END_REF]. Of these, only threeν 10 , ν 23 , and ν 24 -have not been observed at high resolution prior to this work, and thus their band centers (listed above) are probably accurate to at most a few wavenumbers based on low resolution gas phase investigation [START_REF] Krasnoshchekov | Anharmonic Vibrational Analysis of the Infrared and Raman Gas-Phase Spectra of s-trans-and s-gauche-1,3-Butadiene[END_REF]. In this article, we report the first ro-vibrational analysis of two of these, ν 10 (outof-plane C-H wagging [START_REF] Panchenko | Predictive abilities of scaled quantum mechanical molecular force fields: Application to 1,3-butadiene[END_REF]) and ν 24 (in-plane C--C-C deformation [START_REF] Panchenko | Predictive abilities of scaled quantum mechanical molecular force fields: Application to 1,3-butadiene[END_REF]). The spectra were recorded at two synchrotron radiation facilities, SOLEIL and the Canadian Light Source (CLS), during a search for the ro-vibrational spectrum of the gauche-butadiene.

Methods

Quantum chemical calculations

Vibrational frequencies and rotational constants were computed by second-order vibrational perturbation theory (VPT2) [START_REF] Mills | Vibration-Rotation Structure in Asymmetric-and Symmetric-Top Molecules[END_REF]. The force field constants were obtained using a hybrid approach, wherein a coupled-cluster treatment of electron correlation with singles, doubles, and perturbative triples and the frozen-core approximation [fc-CCSD(T)] was employed with the ANO1 basis set for the harmonic frequencies and the ANO0 basis set for the anharmonic constants [START_REF] Almlöf | General Contraction of Gaussian Basis Sets. I. Atomic Natural Orbitals for First-and Second-row Atoms[END_REF]. Calculations were performed using the CFOUR suite of electronic structure programs [START_REF] Stanton | Coupled-Cluster techniques for Computational Chemistry, a quantum-chemical program package[END_REF]. The expected far-IR spectrum of butadiene from literature experimental values, when available, or these calculations, otherwise, is simulated in Fig 1 . Tables S1 and S2 report the calculated band centers and intensities, and the calculated rotational constants, respectively.

A particularly noteworthy result of the VPT2 calculations is the predicted resonance between v 10 = 1 (a u symmetry) and v 9 = v 12 = 1 (a u ), whose zero-order energies lie less than 1 cm -1 apart, around 1034 cm -1 , according to the ANO1 harmonic frequencies. Using the GUINEA module of CFOUR with hybrid ANO1/ANO0 constants, the effective Hamiltonian between the v 10 = 1 and v 9 = v 12 = 1 states has been constructed and diagonalized. Once anharmonicity is taken into account, the states are found to be 22 cm -1 apart (with v 10 = 1 lower in energy, lying at 1012 cm -1 and v 9 = v 12 = 1 lying at 1034 cm -1 , see Table S1) with an overall (pure vibrational) coupling matrix element of 0.8 cm -1 .

Butadiene sample preparation

In all of the experiments performed here, 1,3-butadiene was synthesized from thermal decomposition of sulfolene (C 4 H 6 SO 2 ) at 85-100 • C according to the reversible reaction:

C 4 H 6 SO 2 (s) ∆ ----C 4 H 6 (g) + SO 2 (g). (1)
Theoretical calculations of this reaction [START_REF] Goldstein | Density functional theory prediction of the relative energies and isotope effects for the concerted and stepwise mechanisms of the diels-alder reaction of butadiene and ethylene[END_REF] predict gaseous butadiene is initially produced in the gauche form through a cis transition state via a retro-Diels-Alder mechanism.

To prevent the reverse Diels-Alder reaction, and to obtain a high purity sample of butadiene, an SO 2 trap consisting of KOH pellets was used:

SO 2 (g) + KOH(s) --→ K 2 SO 3 (s) + H 2 O(l), (2)
where water released during reaction Eq. ( 2) was condensed in a cold trap maintained at about 0 • C. Butadiene passes through this trap, but was subsequently collected either by condensation in a vessel cooled with liquid nitrogen or was directly injected into the measurement cell. This purification procedure precludes the obtainment of nonthermal rotamer populations of butadiene (gauche:trans ratio) due to numerous collisions with the vessel walls.

Synchrotron-based FT-IR spectroscopy

The high resolution ro-vibrational spectrum of butadiene has been investigated using three complementary set-ups installed at two synchrotron facilities, SOLEIL and the Canadian Light Source. These set-ups allowed us to probe the sample at three different temperatures, namely 50, 273, and 343 K. In all cases, butadiene spectra were recorded using Fourier-transform spectroscopy (in the far-and mid-IR) using a Bruker IFS125HR spectrometer at its ultimate resolution of about 1 × 10 -3 cm -1 . When used as the light source, the synchrotron radiation was focused onto the entrance iris of the interferometer. In all cases, the interferometer was continuously evacuated to 10 -5 mbar to minimize absorption by atmospheric water. Each experiment is described in greater details in the following paragraphs.

Warm cell experiment

A synchrotron-based ro-vibrational spectrum of butadiene was recorded in the 400-1300 cm -1 region at the far-IR beamline of the CLS facility [START_REF] Mckellar | High resolution analysis of the ν 12 and ν 17 fundamental bands of acrolein, ch 2 chcho, in the 600 cm -1 region[END_REF]. About 40 µbar of sample was injected into a 2-m base-length multipass White-type cell aligned for a 72 m optical path length and maintained at a 343 K (70 • C) temperature [START_REF] Mckellar | High-resolution synchrotron far-infrared spectroscopy of acrolein: The vibrational levels below 700 cm -1[END_REF]. The cell was separated from the interferometer using KBr windows. 280 scans were collected at 0.00096 cm -1 resolution (according to Bruker definition) using a 80 kHz scanner velocity, a 1.15 mm entrance iris aperture, a KBr beamsplitter, and a Ge:Cu detector installed inside a QMC pulse tube cooled cryostat, and equipped with a KRS5 window. Simulation of the high resolution (at 0.001 cm -1 resolution and assuming a Doppler profile) far-IR spectrum of trans-butadiene (fundamental bands only) performed with the PGOPHER software using the experimental band centers and rotational constants when available, or those derived from the quantum chemical calculations (Table S1-S2). Computed intensities are plotted in arbitrary units. For the sake of clarity, the weak bands have been magnified by a factor 40 (ν 12 , ν 24 , in blue) or 1000 (ν 13 , in red). The two new bands investigated in this work (ν 10 and ν 24 ) have been highlighted in orange and the deformation associated with the two corresponding vibrations is represented by orange arrows on the molecular pictures.

The spectrum was calibrated using residual lines from HDO (around 1200 cm -1 ), HCN (around 715 cm -1 ), and CO 2 (around 670 cm -1 ) whose rest wavenumbers were taken from the HITRAN database [START_REF] Gordon | The hitran2016 molecular spectroscopic database[END_REF] and includes experimental data pertinent to this work [START_REF] Maki | Hightemperature infrared measurements in the region of the bending fundamental of h 12 c 14 n, h 12 c 15 n, and h 13 c 14 n[END_REF][START_REF] Zak | A room temperature co 2 line list with ab initio computed intensities[END_REF][START_REF] Horneman | High accurate peak positions for calibration purposes with the lowest fundamental bands ν 2 of N 2 O and CO 2[END_REF]. Residual lines of water, saturated in our experimental spectrum, were not considered in the calibration procedure. From the difference between the calibrated wavenumbers and the reference ones, accuracy of line positions for the entire spectrum should be better than ±2 × 10 -4 cm -1 (Fig. S1). Around 1100 cm -1 , where the ν 10 band of butadiene is predicted to lie, the Doppler full width at half maximum of the lines at 343 K is of about 2 × 10 -3 cm -1 ; thus, a wavenumber accuracy of about a tenth that value appears appropriate.

Room temperature experiment

A synchrotron-based ro-vibrational spectrum of butadiene was recorded in the 50-650 cm -1 region at the AILES beamline of the SOLEIL synchrotron facility [START_REF] Brubach | Performance of the ailes thz-infrared beamline on soleil for high resolution spectroscopy[END_REF]. A spectrum consisting of 220 co-added interferograms was recorded at a resolution of 0.00102 cm -1 (Bruker definition) using a 2.5-m base-length room temperature multipass White-type cell aligned for about 150 m optical path length [START_REF] Pirali | Rotationally resolved infrared spectroscopy of adamantane[END_REF] and filled with 180 µbar of sample. A 2 mm iris, a 6 µm thick mylar-silicon composite beamsplitter, and a liquid-helium Si bolometer were used for the measurements. The cell was isolated from the interferometer by two 50 µm thickness polypropylene windows.

Calibration was performed using residual H 2 O and CO 2 lines whose reference wavenumbers were taken from Refs. 29, 30, which in turn allowed line frequencies to be determined to better than ±1 × 10 -4 cm -1 below 500 cm -1 and ±2 × 10 -4 cm -1 above 500 cm -1 (Fig. S1).

Supersonic-jet experiment

Finally, a spectrum of butadiene in a supersonic jet expansion was recorded in the 850-1050 cm -1 region using the Jet-AILES apparatus at SOLEIL [START_REF] Cirtog | The (CH 2 ) 2 O-H 2 O hydrogen bonded complex. ab initio calculations and fourier transform infrared spectroscopy from neon matrix and a new supersonic jet experiment coupled to the infrared ailes beamline of synchrotron soleil[END_REF][START_REF] Asselin | Competition between interand intra-molecular hydrogen bonding: An infrared spectroscopic study of jet-cooled amino-ethanol and its dimer[END_REF]. A slit nozzle of 60 mm length and 130 µm width was used to expand the sample seeded in helium buffer gas, with respective flows of 1 and 20 standard liters per minutes (slm), into a large chamber that was continuously evacuated by a set of Roots pumps. The reservoir and chamber pressures were maintained at 260 and 0.8 mbar, respectively, and the chamber was separated from the interferometer and the detection compartment by two KBr windows. The high resolution spectrum is the result of the co-addition of 130 individual scans recorded at 0.00102 cm -1 resolution using a globar source, a 1.13 mm iris aperture, a KBr beamsplitter, and photovoltaic MCT detector. A low-pass optical filter was placed in front of the detector thereby limiting the upper wavenumber value of the measurements to 1050 cm -1 . Under our experimental conditions, the use of the synchrotron radiation instead of the globar source resulted in a lower signal-to-noise ratio. Additional spectra were recorded at lower resolution (0.01 cm -1 ) for different sample to buffer gas ratios (2 slm of butadiene and x slm of He, where x = 0, 5, 10, 20, 30) but otherwise identical experimental conditions. Since

Accepted article

3/10 accurate rotational constants of the ν 11 band of transbutadiene were available from prior work [START_REF] Craig | Analysis of the rotational structure in bands in the high-resolution infrared spectra of butadiene and butadiene-2,3-d 2 : Refinement in assignments of fundamentals[END_REF], simulations of the band profile (using the PGOPHER software [START_REF] Western | PGOPHER: A Program for Simulating Rotational, Vibrational and Electronic Spectra[END_REF]) allowed us to infer the rotational temperature of the spectra with different mixing ratios. A temperature of 30 K well reproduced the high resolution Jet-AILES spectrum, while temperatures of 150, 100, 70, 50, and 40 K were estimated to reproduce the lower resolution spectra with 0, 5, 10, 20, and 30 slm of He, respectively (Fig. S2).

In the relatively narrow range covered by the Jet-AILES spectrum, only two bands of trans-butadiene, ν 10 and ν 11 , are visible, which precludes a conventional frequency calibration using standard molecules. Instead, calibration was performed by comparing strong, isolated lines in both the well-calibrated warm spectrum recorded at CLS with those in this jet spectrum. Several drawbacks to this procedure should be noted: i) the wavenumber accuracy of the CLS transitions is at best 2 × 10 -4 cm -1 whereas accuracy as low as 1 × 10 -4 cm -1 should be possible in the cold Jet-AILES spectrum owing to the narrower lines and sparser spectrum that arises at colder rotational temperatures; and ii) because of the temperature difference between the two spectra, lines that are strong in the Jet-AILES spectrum may not be intense in the CLS spectrum, and vice versa. To increase our calibration reliability, we have selected as many lines as possible on the Jet-AILES spectrum that correspond to lines having a transmittance higher than 0.2 in the warm CLS spectrum. A conservative wavenumber accuracy of 5 × 10 -4 cm -1 is expected from this procedure (Fig. S1).

Analysis procedure

Since several ro-vibrational bands of trans-butadiene have previously been assigned at high resolution, it is reasonable to assume that the vibrational ground state is very well-described, even though no pure rotational measurements of the main isotopologue are available. As a consequence, analysis of the newly-recorded bands appears well-suited for the Automated Spectral Analysis Approach (ASAP) developed by C. P. Endres and several of the authors of this paper [START_REF] Martin-Drumel | The soleil view on sulfur rich oxides: The s 2 o bending mode ν 2 at 380 cm -1 and its analysis using an automated spectral assignment procedure (asap)[END_REF][START_REF] Thorwirth | An asap treatment of vibrationally excited s 2 o: The ν 3 mode and the ν 3 + ν 2ν 2 hot band[END_REF]. Briefly, this procedure facilitates fast analysis of a ro-vibrational band when one of the two vibrational levels involved, the so-called reference state, is well-characterized. The second, or target state is thus the one from which spectroscopic information needs to be extracted. The method exploits the power of the Loomis-Wood approach for ro-vibrational assignments (see, e.g., Refs. [START_REF] Kisiel | Rotational spectrum of trans-trans diethyl ether in the ground and three excited vibrational states[END_REF][START_REF] Lodyga | Advanced graphical software for assignments of transitions in rovibrational spectra[END_REF] and the redundancy of the spectroscopic information in a ro-vibrational band with multiple selection rules (typically P-, Q-, R-branches and asymmetric splitting), i.e. when transitions originating from different rotational energy levels of the reference state reach the same rotational energy level of the target state. Since the energy levels of the reference state are well-determined, the error in the predicted ro-vibrational frequencies between the reference and target state are dominated by the error in the prediction of the target state. Since each set of transitions reaches the same rotational level, the frequencies of the sought-after experimental transitions all share the same offset relative to their prediction, with the value of that offset being the error in the predicted energy of the particular target rotational level. Assigning ro-vibrational transitions in such a set thus entails identifying lines lying at the same offset value from the prediction in each portion of the spectrum. ASAP simplifies this step by cross-correlating (multiplying) these portions of the spectrum. For spectra plotted in absorbance, the signal approaches zero between transitions, provided the spectrum is well-resolved. Thus, most frequency points will be multiplied by zero at least once except if a line is always present on each plot. This method allows for significant spectral simplification, as only a few points will stand out from the baseline, and by plotting successive J values in a Loomis-Wood fashion, one can assign series of lines. One consequence of the cross-correlation is that the actual rovibrational frequencies are no longer assigned; rather the energy of the sought after target state is directly extracted (predicted energy + assigned offset). Since it is no longer necessary to assign each transition for each selection rule, but only the target energy levels involved in the band, this procedure greatly speeds-up analysis time. The energy levels together with other available data are input to the SPFIT/SPCAT suite of programs [START_REF] Pickett | The fitting and prediction of vibration-rotation spectra with spin interactions[END_REF] to derive the rotational constants and band center of the target state.

The ASAP method has proven extremely helpful in assigning several bands of S 2 O, for which the rotational ground state was accurately known from extensive pure rotational investigations [START_REF] Martin-Drumel | The soleil view on sulfur rich oxides: The s 2 o bending mode ν 2 at 380 cm -1 and its analysis using an automated spectral assignment procedure (asap)[END_REF][START_REF] Thorwirth | An asap treatment of vibrationally excited s 2 o: The ν 3 mode and the ν 3 + ν 2ν 2 hot band[END_REF]. In the present case, however, since no such pure rotational data exists and since the ro-vibrational bands assigned in the literature were recorded at similar or lower resolution compared to the present work, once the target state rotational constants were derived from the ASAP analysis, a conventional assignment of each ro-vibrational transition was also undertaken. To do so, the LWWa software developed by W. Lodyga was used [START_REF] Lodyga | Advanced graphical software for assignments of transitions in rovibrational spectra[END_REF].

Results and discussion

Initial fit of the literature data

To obtain ground state constants that are as accurate as possible, we have refitted all of the ro-vibrational data previously published, i.e. in the ν 11 [START_REF] Craig | Analysis of the rotational structure in bands in the high-resolution infrared spectra of butadiene and butadiene-2,3-d 2 : Refinement in assignments of fundamentals[END_REF], ν 12 [START_REF] Craig | An Investigation of the Rotamers of Butadiene by High-Resolution Infrared Spectroscopy[END_REF], ν 13 [START_REF] Craig | An Investigation of the Rotamers of Butadiene by High-Resolution Infrared Spectroscopy[END_REF], ν 17 [START_REF] Halonen | Structural issues in conjugated hydrocarbons: High-resolution infrared slit-jet spectroscopy of trans-1,3-butadiene[END_REF], and ν 20 [START_REF] Craig | Analysis of the rotational structure in bands in the high-resolution infrared spectra of butadiene and butadiene-2,3-d 2 : Refinement in assignments of fundamentals[END_REF] bands. A Watson S-reduced Hamiltonian in the I r representation and the Pickett SPFIT program were used. Since A >> B ∼ C, we have chosen S-reduction although literature data were fitted using the A-reduction. Data from ν 12 and ν 13 [START_REF] Craig | An Investigation of the Rotamers of Butadiene by High-Resolution Infrared Spectroscopy[END_REF] were weighted according to their experimental uncertainty (0.0002 cm -1 ), while those in ν 11 and ν 20 [START_REF] Craig | Analysis of the rotational structure in bands in the high-resolution infrared spectra of butadiene and butadiene-2,3-d 2 : Refinement in assignments of fundamentals[END_REF], for which the estimated line accuracy was not stated, were weighted using the standard deviation obtained in the fit reported by the authors (0.0002 and 0.007 cm -1 , respectively). In the case of ν 17 , the authors [START_REF] Halonen | Structural issues in conjugated hydrocarbons: High-resolution infrared slit-jet spectroscopy of trans-1,3-butadiene[END_REF] noted that the expected wavenumber accuracy Accepted article 4/10 on each transition should be 10 times better than the standard deviation of their fit (0.003 cm -1 ) but perturbations, most likely from a rovibrational dark state, limited the fit convergence. Since in our study no attempt was made toward treating this perturbation, the value of this standard deviation was retained as the line uncertainty. In total, 7015 transitions (6929 different frequencies) are included in our fit with J = 0 -73 and K a = 0 -18. All quartic centrifugal distortion (CD) terms were varied in the ground state while those unfitted in the excited states were constrained to the ground state value. By using a 5σ rejection criterion (i.e. transitions were discarded if the observed-calculated frequency was five or more times the postulated uncertainty ), 16 lines from Refs. 9, 14 were rejected from the fit. In these instances, it appears that the inability to reproduce the literature data to within the experimental accuracy is due to perturbations rather than misassignments. We have therefore decided to keep these lines in the fit but to reduce their wavenumber accuracy by a factor 10 (see Table S4 for the list of transitions). The reduced standard deviation (unitless) of the combined fit is 1.16, ranging from 0.73 to 1.54 for individual bands.

To our knowledge, only ground state combination differences and some individual ro-vibrational bands were fitted together prior to this study. Since a combined effective fit has been performed here, the ground state rotational constants should be quite reliable.

The ν 24 vibrational band

The ν 24 band of trans-butadiene (a/b hybrid) is observed in the room temperature spectrum recorded at SOLEIL (Fig. 2).

Fig. 2.

Overview of the high-resolution spectrum of the ν 24 band of trans-butadiene observed at room temperature. Isolated lines arise from residual water vapor, and most of these are saturated. A portion of the Q-branch is also saturated.

ASAP assignments Using the ground state as the reference state and v 24 = 1 as the target state, ASAP searches were performed. The initial rotational constants for v 24 = 1 were obtained by scaling the VPT2 results by the experimentally-derived versus calculated constants in the ground state (Table S1-S2). The initial frequency for Because several series of lines are readily identifiable on the ASAP plots (Fig. 3), a preliminary assignment of the band could be performed almost immediately; it consists of 290 energy levels in v 24 = 1 with J = 4 -44, K a = 4 -14, measured to 0.0001 cm -1 accuracy. These energy levels were then added to the list of literature assignments, as defined in the previous section, and constants fit using the Pickett software. The resulting parameters are reported in Table 1. Even though we suspected that inclusion of higher-order CD parameters would help to reduce the standard deviation, at this juncture of the analysis the number of fitted parameters was limited because it appeared that the energies derived in the ASAP procedure were not as accurate as expected given our experimental resolution. Ostensibly this situation arises because of residual error in the ground (reference) state energies owing to the absence of extremely accurate pure rotational data. Indeed, after fitting, the ASAP series are still not uniformly aligned, and the distribution of residuals is larger than expected for a spectrum recorded at a resolution of 0.001 cm -1 . This may come from the fact that the data defining the ground state has at best an uncertainty twice what we expect from the ν 24 measurements (0.0002 vs. 0.0001 cm -1 ). Additionally, some Q-branch lines are saturated, which induces broadening of the ASAP lines. Because it is not possible to omit specific lines in the crosscorrelation procedure using the current version of the software, this effect may skew the ASAP plots. For both of these reasons, a subsequent line-by-line assignment of the ν 24 band has been performed using the LWWa software.

LWWa assignments Using the preliminary constants derived for v 24 = 1, assignments using the LWWa software are straightforward. Comparison of the LWWa diagrams with those from ASAP (Fig. 4) indicates that lines ap-Accepted article 5/10 pear better aligned (i.e. have a smaller dispersion) compared to the former plots. It should be noted, however, that the LWWa software represents a line as a triangle placed at its center frequency, with no consideration for the width of the actual line, while the ASAP plot displays a cross-correlated line profile. Nevertheless, the LWWa plots allow assignment over a wider range of quantum numbers without ambiguity. In total, 2840 transitions with J = 4 -87 and K a = 2 -18 have been assigned with a frequency uncertainty of 0.0001 cm -1 using this approach. The c-type ν 10 vibrational band of butadiene was observed both in the cold supersonic jet spectrum recorded at SOLEIL (Fig. 5) and in the warm one recorded at CLS. During the measurements with the supersonic jet, we were also able to observe the effect of rotational cooling on the band profile: from a standard c-type band with a sharp Q-branch at moderate temperature, the band evolves to a completely different profile with two Q-branch lobes at low temperature (Fig. 6).

ASAP assignments Following the same procedure as for the ν 24 band, assignments were first performed using the ASAP software. The ground state was used as the reference state, the initial constants of the target state (here v 10 = 1) were obtained by scaling the VPT2 calculation (Table S2), and the initial frequency of the band center was taken from the low-resolution gas-phase assignment of Ref. [START_REF] Krasnoshchekov | Anharmonic Vibrational Analysis of the Infrared and Raman Gas-Phase Spectra of s-trans-and s-gauche-1,3-Butadiene[END_REF].

As with the ν 24 analysis, several series of lines, slightly displaced from the prediction, were readily observed and assigned (Fig. 3): over 100 energy levels in total, with J = 1 -24 and K a = 0 -6. As before, the reduced standard deviation of the fit, assuming a 0.0005 cm -1 accuracy for the energy levels, is high (σ = 16 when fitting a single quartic CD term in addition to the rotational constants and band center; see Table 1). Owing to the limited range of the quantum numbers that are accessible at this temperature, no additional CD terms were varied at this stage of the analysis.

LWWa assignments Using the preliminary set of rotational constants and the LWWa software we were able to assign 927 lines with 682 different frequencies, including 131 lines with ∆K c = ±2, in the Jet-AILES spectrum. These transitions involve J = 1 -29 and K a = 0 -7.

Additionally, 1176 transitions (964 different frequencies) were also assigned in the warm CLS spectrum with J = 2 -49 and K a = 0 -7. More lines, likely involving higher J and K a values, are present in the warm spectrum, but it was not possible to assign them with confidence using our best model (see the Combined fit section).

Other bands

In the FIR spectra we recorded, limited sensitivity prevented detection of any combination bands. Searches for ν 23 (lying around 977 cm -1 ) were also unsuccessful. This band is expected to be extremely weak (see Fig. 1 and Table S1) and increasing the pressure in hopes of detecting it resulted in full optical saturation of ν 10 and ν 11 in the room temperature and warm spectra.

Combined effective fit

A fit has been performed using the Pickett software for all the literature data and the newly assigned transitions. Energy levels assigned at the ASAP step were not included, nor were the assignments of the ν 10 band from the warm CLS spectrum (only the cold Jet-AILES data were considered for this band). The best-fit constants and relevant fit parameters for the ν 10 and ν 24 bands are summarized in Table 1, while the complete set of constants (all states in the fit) is given in Table S3. For ν 24 , inclusion of two higher-order (sextic and octic: H K and L K ) CD terms, not needed for the ground state, are required to reproduce all assigned transitions to their experimental accuracy (resulting in a reduced standard deviation of 1.1). Taken together with the larger value of D K in v 24 = 1 compared to that in v = 0, this behavior likely indicates v 24 = 1 is somewhat less "rigid" than the ground state.

The situation for the ν 10 band is significantly more complicated; the band shows obvious signs of a (strong) perturbation. To reproduce as many transitions from the cold spectrum as possible, three sextic (H K , H JK , and H K J ) CD terms are required. These parameters, however, as well as the quartic CD terms, have magnitudes that are much larger than expected (e.g., D JK is 38 times larger than the ground state value and H K is 50 times larger than the corresponding value in v 24 = 1). Despite their inclusion, 163 transitions (120 different frequencies) were still rejected from the fit using a 5σ criterion. These transitions are almost exclusively high J values in each K a branch, meaning that the highest values of J included in the fit decrease with K a . For completeness these transitions are reported in the Supporting Information. In light of the difficulties encountered in the cold Jet-AILES spectrum, which samples a limited range of rotational excitation, no attempt was made to fit the warm CLS spectrum. Based on the available data it appears that a strong perturbation occurs for K a ≥ 4, likely from the nearly degenerate dark state ν 9 + ν 12 , if the VPT2 calculations can be trusted.

Treatment of the perturbation in ν 10 has been attempted. From our quantum chemical calculations, the v 9 = v 12 = 1 level is predicted to lie at 1034 cm -1 , and the interaction between this state and v 10 = 1 is predicted to be Fermi and/or c-type Coriolis, the former with a coupling value of order 0.8 cm -1 . The same calculation also predicts v 10 = 1 at 1012.2 cm -1 (about 1 cm -1 away from the experimental value) and scaled rotational constants that are consistently very close to the experimental values for the other bands investigated in this work, i.e. about 0.2 % for B and C, and 5% for the A constant (see Tables 1 andS2). Nevertheless, no satisfactory fit was obtained for the combined fit, regardless of the magnitude of the coupling parameter. At present, we believe this fitting failure is no accident, but rather due to a coincidence between K a = 4 of v 10 = 1 and the first K a values in v 9 = v 12 = 1, based on effective Hamiltonian predictions (Fig. 7). This is consistent with the large residuals associated with the K a = 4 value in the ν 10 band. The main obstacle is the lack of accurate data on the dark v 9 = v 12 = 1 state: no pure rotational data are available and there is no evidence of the ν 9 + ν 12 band in the experimental spectra, presumably because its calculated intensity is more than two orders of magnitude lower than that of ν 10 .

While unsatisfactory in some respects, the effective fit reported here improves our understanding of the rovibrational structure of trans-butadiene. Concerning the band centers, for instance, the value determined in this work for the ν 24 band is 2 cm -1 lower than the previous value, derived from low-resolution spectroscopy reported in Ref. [START_REF] Krasnoshchekov | Anharmonic Vibrational Analysis of the Infrared and Raman Gas-Phase Spectra of s-trans-and s-gauche-1,3-Butadiene[END_REF]. A much closer agreement is obtained for the ν 10 band (0.5 cm -1 ) as a consequence of this band being a c-type, which is characterized by a sharp Q-branch at room temperature. The rotational constants in the ground state are also extremely reliable, since they have now been derived from measurements of roughly 10,000 lines.

Accepted article

7/10

[ • ] [ • ] 53.51(82) d 1 ×10 9 -3.5988(53) [ • ] [ • ] -3.5927(71) [ • ] [ • ] [ • ]

Conclusions

While attempting to record the ro-vibrational spectrum of gauche-butadiene, we have re-investigated the far-infrared spectrum of its more stable trans conformer. Two new bands, ν 24 and ν 10 , have been observed and assigned, meaning that only one IR active band below 1100 cm -1 , namely ν 23 , has evaded detection at high resolution. The ASAP software was used to initially assign the spectra of both bands in a few hours. While the assignment was straightforward, the fit has proven more challenging since the ν 10 band appears strongly perturbed, most likely by the dark v 9 = v 12 = 1 state. In absence of any experimental information for this state (rotational constants or band center), no reasonable treatment of the perturbation could be achieved. Further investigation of the species under rotationally cooled conditions may prove useful to determine the ν 9 + ν 12 band centers, and ideally rotational constants, assuming a sufficient signal-to-noise ratio can be reached. Such measurements may also enable the detection of the ν 23 band. Nevertheless, now that the far-IR ro-vibrational spectrum of the most abundant conformer of butadiene is nearly complete, renewed attempts to assign the gauche spectrum can be undertaken with greater confidence.

Supporting Information. Calibration plots (Figure S1); Experimental Jet-AILES spectra and comparison with simulations (Figure S2); Calculated band centers and intensities (Tables S1); Predicted rotational constants (Table S2); Full list of derived rotational constants (Table S3); Literature transitions for which the estimated uncertainty was increased in the present fit (Table S4). Complete list of observed transitions and fit files (in text format).

  y / a.u.

Fig. 1 .

 1 Fig.1. Simulation of the high resolution (at 0.001 cm -1 resolution and assuming a Doppler profile) far-IR spectrum of trans-butadiene (fundamental bands only) performed with the PGOPHER software using the experimental band centers and rotational constants when available, or those derived from the quantum chemical calculations (TableS1-S2). Computed intensities are plotted in arbitrary units. For the sake of clarity, the weak bands have been magnified by a factor 40 (ν 12 , ν 24 , in blue) or 1000 (ν 13 , in red). The two new bands investigated in this work (ν 10 and ν 24 ) have been highlighted in orange and the deformation associated with the two corresponding vibrations is represented by orange arrows on the molecular pictures.

1 Fig. 3 .

 13 Fig. 3. Initial ASAP views of the ν 24 band (left panel, for one of the two K a = 7 asymmetric branches) and the ν 10 band (right panel for a K a = 1 branch). Predicted energies are represented by the black vertical line (0 cm -1 offset) and visible series of lines are highlighted in blue.

Fig. 4 .

 4 Fig. 4. Initial LWWa view of the ν 24 band for one of the two K a = 8 branches in comparison with the corresponding final ASAP plot (i.e., following the initial fit).

3. 3 .Fig. 5 .

 35 Fig. 5. Overview of the high resolution, rotationally cold spectrum recorded using the Jet-AILES apparatus: both the ν 10 and ν 11 bands of trans-butadiene are clearly visible.

Fig. 6 .

 6 Fig.6. Evolution of ν 10 band profile as a function of rotational temperature showing an expanded view of the Q-branch (gray insert). Spectra were all recorded at medium resolution (0.01 cm -1 ) but with different sample:He ratios. Rotational temperatures determined from simulations of the ν 11 band profile are indicated on the far right (see text in the Methods section and Fig.S2for further details).

d 2 ×10 9 -Fig. 7 .

 97 Fig. 7. Rotational energy level diagram in v 10 = 1 (best fit, black dots) and v 9 = v 12 = 1 (best prediction, red triangles). The term 1 /2(B + C)J(J + 1) is subtracted from the energy. K a values are noted at the head of each series.

Table 1 .

 1 Rotational constants of butadiene in the ground state, v 24 = 1, and v 10 = 1 after initial assignments using the ASAP software and the final fit in comparison with predicted values. Numbers in parenthesis are one standard deviation in units of the last quoted digit. Brackets indicate parameters fixed to the ground state values.

	D J ×10 6 D JK ×10 6 D K ×10 6	C	B	A	E	Constant
	7.2929(11) [ • ]	-0.23874(16) [ • ]	0.0287444(89) [ • ]	0.133694116(62) 0.1338	0.147885664(59) 0.1478	1.39038285(40) 1.3455	299.1	Final Pred. a	v = 0
	19.4396(32) 21.9587(59)	-0.22862(60) -0.24369(17)	0.027890(37) 0.0278283(90)	0.13364383(222) 0.133616011(65)	0.14793757(207) 0.147960763(63)	1.43140808(57) 1.43163076(55)	297.0970719(258) 297.0923451(62)	Ini. (ASAP) Final	v 24 = 1
		[ • ] -16.3923(253)	[ • ] [ • ]	0.1337 0.13381633(72)	0.1469 0.14945258(72)	1.3950 1.3861088(66)	1013.8 1013.371993(93)	Pred. a Ini. (ASAP)	v 10 = 1
		-9.191(57)	0.0470(12)	0.13382711(68)	0.14952571(74)	1.386966(15)	1013.368967(66)	Final

Acknowledgement

Part of the research described in this paper was performed at the Canadian Light Source, a national research facility of the University of Saskatchewan, which is supported by the Canada Foundation for Innovation (CFI), the Natural Sciences and Engineering Research Council (NSERC), the National Research Council (NRC), the Canadian Institutes of Health Research (CIHR), the Government of Saskatchewan, and the University of Saskatchewan. We acknowledge the SOLEIL facility for provision of synchrotron radiation and we would like to thank the AILES@SOLEIL and Far-IR@CLS beamlines staff for their assistance. This work has been performed under the SOLEIL proposal 20171389 and CLS proposal 27G08965. We thank Mohammed Bali for assistance in the recording of the SOLEIL jet spectrum. MCM and JPP acknowledge support from NASA grant 80NSSC18K0396 and NSF grant CHE-1566266.


Member, IEEE Olivier D Bernal

Senior Member, IEEE Usman Zabit

Francis Jayat

Senior Member, IEEE Thierry Bosch

ADVERTISING OR PROMOTIONAL PURPOSES, CREATING NEW COLLECTIVE WORKS FOR RESALE OR REDISTRIBUTION TO SERVERS OR LISTS, OR REUSE OF ANY COPYRIGHTED COMPONENTS OF THIS WORK IN OTHER WORKS 1 Sub-λ/2 Displacement Sensor with Nanometric Precision based on Optical Feedback Interferometry used as a Non-Uniform Event-Based Sampling System

Keywords: Optical feedback interferometry, self-mixing, displacement measurement, non-uniform sampling

In this paper, a method based on the inherent event-based sampling capability of the laser optical feedback interferometry (OFI) is proposed to recover sub-λ/2 displacement with a nanometric precision. The proposed method operates in open-loop configuration and relies on OFI's fringe detection, thereby improving its robustness and ease of use. The measured white noise power spectral density is less than 100 pm/ √ Hz with a corner noise frequency of approximately 80 Hz for a laser diode emitting wavelength λ0 of 785 nm placed at 30 cm of the target.

I. INTRODUCTION

O PTICAL feedback interferometry (OFI), also referred to as self-mixing (SM) effect in laser diodes (LD) [START_REF] Donati | Developing self-mixing interferometry for instrumentation and measurements[END_REF], [START_REF] Taimre | Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing[END_REF], [START_REF] Mezzapesa | High-resolution monitoring of the hole depth during ultrafast laser ablation drilling by diode laser self-mixing interferometry[END_REF] has been widely investigated for the last decades as it results in a self-aligned and cost effective sensing system. The resolution of a stationary OFI based displacement sensor depends on the employed signal processing techniques. Displacement measurement with a basic resolution of halfwavelength (λ 0 /2) can be easily achieved with an OFI sensor under moderate optical feedback regime by fringe counting [START_REF] Donati | Developing self-mixing interferometry for instrumentation and measurements[END_REF]. The basic resolution can be improved by locking the laser phase to half-wavelength [START_REF] Giuliani | Self-mixing laser diode vibrometer[END_REF] or by fringe duplication [START_REF] Wei | Obtaining scalable fringe precision in self-mixing interference using an even-power fast algorithm[END_REF], [START_REF] Jiang | Multiple self-mixing interferometry algorithm based on phase modulation for vibration measurement[END_REF] or by utilizing phase unwrapping techniques. Different phase unwrapping techniques (based on time-domain OFI signal processing) have been proposed in literature [START_REF] Bes | Displacement measurements using a self-mixing laser diode under moderate feedback[END_REF], [START_REF] Bernal | Study of laser feedback phase under self-mixing leading to improved phase unwrapping for vibration sensing[END_REF], [START_REF] Fan | Improving the measurement performance for a self-mixing interferometry-based displacement sensing system[END_REF], [START_REF] Arriaga | Real-time algorithm for versatile displacement sensors based on self-mixing interferometry[END_REF], [START_REF] Merlo | Reconstruction of displacement waveforms with a single-channel laser-diode feedback interferometer[END_REF], [START_REF] Ehtesham | Analysis and implementation of a direct phase unwrapping method for displacement measurement using self-mixing interferometry[END_REF], [START_REF] Zabit | Spectral processing of self-mixing interferometric signal phase for improved vibration sensing under weak-and moderatefeedback regime[END_REF] providing accuracy from λ 0 /8 to λ 0 /60. For accuracy exceeding λ 0 /40, these methods [START_REF] Bernal | Study of laser feedback phase under self-mixing leading to improved phase unwrapping for vibration sensing[END_REF], [START_REF] Fan | Improving the measurement performance for a self-mixing interferometry-based displacement sensing system[END_REF] require elaborate time-domain SM signal segmentations as well as estimations of key OFI parameters, such as optical feedback coupling parameter C. Except for the fringe-locking method [START_REF] Giuliani | Self-mixing laser diode vibrometer[END_REF], [START_REF] Melchionni | Development of a design tool for closed-loop digital vibrometer[END_REF], [START_REF] Magnani | Selfmixing digital closed-loop vibrometer for high accuracy vibration measurements[END_REF], to the best of our knowledge, none of the previously mentioned methods exploiting the modulation of optical output power by OFI (also referred to as amplitude modulation (AM) channel) have achieved precision down to the nanometer yet. More recently, new methods exploiting the frequency modulation (FM) channel of OFI instead of the AM channel have shown that it is possible to further significantly improve the vibrometer performances by at least two decades [START_REF] Donati | Self-mixing interferometer with a laser diode: Unveiling the fm channel and its advantages respect to the am channel[END_REF]. However, to achieve these performances, it is required to convert FM-to-AM via narrow band optical frequency filter such as gas cell [START_REF] Contreras | Edge filter enhanced self-mixing interferometry[END_REF] or Mach Zehnder [START_REF] Norgia | Exploiting the fm-signal in a laser-diode smi by means of a mach-zehnder filter[END_REF], which might hinder the simplicity of OFI systems.

Subsequently, using the AM SM signal alone, we do propose here a new open-loop approach that allows to recover sub-λ 0 /2 displacement with nanometric precision without requiring any closed-loop scheme for an LD of wavelength λ 0 . This approach retains the inherent simplicity of OFI as the required hardware consists only in amplifying and acquiring the SM signal contrary to [START_REF] Azcona | A nanometric displacement measurement system using differential optical feedback interferometry[END_REF]. Further, as opposed to [START_REF] Giuliani | Self-mixing laser diode vibrometer[END_REF], in open-loop configuration, the distance D 0 from the LD to the target at rest neither directly affects the measurement precision nor the dynamic range. D 0 can affect the quality of the SM signals in terms of noise and C. However, only the change in the external cavity length corresponding to the target vibration will be embedded in the OFI signal. On the contrary, in closed-loop configuration, the loop compensates any target displacement by applying the required wavelength variation which is inversely proportional to D 0 [START_REF] Giuliani | Self-mixing laser diode vibrometer[END_REF], in order to maintain the interferometer phase constant. In addition, in this configuration, the achievable dynamic range is limited neither by the tunability range of the LD wavelength λ 0 nor by the finite value of the architecture's open-loop gain.

Here, in the moderate feedback regime (1 < C < 4.6), we propose to perceive SM interferometers as an inherent nonuniform sampling system with its own embedded phase levelcrossing detector. Non-uniform sampling (NUS) approaches are often used in applications for which the retrieved information is sparse. Based on the non-uniform sampling theory, we show that it is possible to reconstruct the target displacement based on fringe detection only. Consequently, phase unwrapping techniques are not necessary to achieve nanometric precision. In addition, to recover sub-λ/2 displacements for which a maximal of only one level crossing can be detected, we propose to add a phase dither Φ d , either obtained by vibrating the sensor itself or by modulating the LD driving current, to the SM phase so that both the number of crossed levels as well as the rate of level crossings can be increased. We will also show that using the NUS theory, estimating C is not necessary when using dithering.

Note that dithering techniques were already employed in OFI to achieve high displacement resolution [START_REF] Giordano | Phaseresolved terahertz self-detection near-field microscopy[END_REF], [START_REF] Xia | Sinusoidal phase-modulating self-mixing interferometer with nanometer resolution and improved measurement velocity range[END_REF] when combined to lock-in techniques. In [START_REF] Giordano | Phaseresolved terahertz self-detection near-field microscopy[END_REF], OFI is employed to design a simple and compact scattering-type scanning nearfield optical microscopy. However, the quantum cascade laser was operated at low temperature ( ≈ 15 K) in the very weak feedback regime (C < 0.1) and the dithering signal provided by a resonant vibrating tip was employed to allow amplitude and phase information retrieval on the medium optical response under test by applying lock-in techniques. In [START_REF] Xia | Sinusoidal phase-modulating self-mixing interferometer with nanometer resolution and improved measurement velocity range[END_REF], the dithering signal is required for displacements both larger and smaller than λ/2. It is obtained using an electrooptic modulator operating at 20 kHz up to 0.2 MHz while the LD is working in weak feedback regime. The instantaneous interference phase change related to the target displacement is retrieved by calculating the arc-tangent of the ratio between the zeroth-order and first-order harmonic amplitudes of the SM signal. These amplitudes are also obtained using lockin techniques with the dithering signal and its corresponding phase quadrature signal.

In the following section II, we present the non-uniform sampling theory applied on SM signals. We show how by applying a dithering signal, sub-λ/2 displacements can also be recovered. Then, in section III, different experimental testbenches are described and results are analyzed to assess the system performances. Finally, effects of speckle on the achieved performances are discussed in section IV.

II. PROPOSED METHOD

In this section, a short background on OFI as well as on uniform and non-uniform sampling is first provided in order to introduce the proposed interpretation of OFI as an inherent non-uniform sampling system.

A. OFI overview

In OFI, a portion of the laser beam can be back-scattered from target placed at a distance D 0 from the laser (moving with displacement D(t)) and can thus re-enter the active laser cavity (Fig. 1). This causes a mixing of generated and phase-shifted back-scattered beams. This "self-mixing" causes fluctuation in the optical output power (OOP) of the laser, denoted as P (t), given by [START_REF] Donati | Developing self-mixing interferometry for instrumentation and measurements[END_REF]:

S(t) Sample Amplitude S(t) Sample t 2 t k t 1 Q 1 Q 2 Q k Q n Q k Q n Q 2 Q 1 t n Time Time
P (t) = P 0 (1 + m cos (Φ F (t))) (1)
where P 0 is the emitted optical power under free-running conditions, m is the modulation index and Φ F (t) is the laser output phase in the presence of feedback. Φ F (t) is related to the laser output phase without feedback Φ 0 (t) = 4πD (t) /λ 0 by:

Φ 0 (t) = Φ F (t) + C sin (Φ F (t) + arctan α) (2)
where α is referred to as the linewidth enhancement factor [START_REF] Donati | Developing self-mixing interferometry for instrumentation and measurements[END_REF], [START_REF] Taimre | Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing[END_REF]. Depending on C , the laser can operate into different regimes. SM sensing is generally performed under weak feedback regime (C < 1), moderate feedback regime (1 < C < 4.6), or strong feeback regime ( C > 4.6). However, moderate feedback regime (1 < C < 4.6) is usually preferred as the apparently simple saw-tooth shaped SM fringes belonging to such a regime [START_REF] Bernal | Classification of laser self-mixing interferometric signal under moderate feedback[END_REF] intrinsically provide motion direction indication and require simplified SM fringe detection processing [START_REF] Magnani | Self-mixing vibrometer with real-time digital signal elaboration[END_REF].

B. Uniform and non-uniform sampling principles

A discrete-time representation of a band-limited signal S(t) is usually obtained using an analog-to-digital converter (ADC) which uniformly samples S(t) at a frequency f s =1/T s and quantizes it to the nearest quantization level (Q 1 . . . Q n ) [START_REF] Sayiner | A level-crossing sampling scheme for a/d conversion[END_REF], [START_REF] Vezyrtzis | Processing of signals using level-crossing sampling[END_REF]. ADCs thus represent the signal in time-amplitude ordered pairs [kT s , Q k ] with Q k the nearest quantization level at kT s (see Fig. 2 a)). As shown in Fig. 2 a), the quantization process induces an estimation error of the correct amplitude of S(t). To reduce the quantization error and improve the dynamic range without increasing the number of quantization levels, oversampling techniques can be used in order to distribute the quantization error power over a larger bandwidth than the signal of interest [START_REF] Sayiner | A level-crossing sampling scheme for a/d conversion[END_REF]. While the quantization process of the previously described ADC's scheme is clock driven, ADCs can be designed to be event driven instead. These ADCs are often referred as level crossing ADCs which record the time instants t k at which S(t) crosses any of its quantization levels

Q k resulting in time-level pairs [t k , Q k ] (see Fig. 2 b
)) [START_REF] Sayiner | A level-crossing sampling scheme for a/d conversion[END_REF], [START_REF] Vezyrtzis | Processing of signals using level-crossing sampling[END_REF]. The time instants t k of the level-crossings are then quantized into Q (t k ) using a clock that must provide both a high resolution and precision. As a result, if the average sampling rate of the input exceeds twice the input signal bandwidth, it is possible to retrieve the original band-limited signal S(t) from the non-uniformly spaced in time samples

[Q (t k ) , Q k ] [26].
In addition, in order to both improve the dynamic range of the level-crossing ADCs without increasing the number of level-crossings and fulfill the requirements on the average sampling rate of level crossings, adding a dithering signal to the input signal can be employed to ensure that low-amplitude (lower than the quantization levels) or slowly varying signals are sampled and converted accurately [START_REF] Jungwirth | Improved sayiner level crossing adc[END_REF], [START_REF] Wang | A level-crossing analog-to-digital converter with triangular dither[END_REF]. This dithering principle can be illustrated by Fig. 3 where a 10 Hz 0.9 V sinusoidal input S(t) is applied at the input of a level-crossing level ADC. Fig. 3 a) shows that it crosses only one of the threshold levels uniformly-spaced by 1V, which results in a poor analog-to-digital conversion. However, if a 50 Hz 3 V sinusoidal dithering signal is added to S(t), a lot of samples are generated (Fig. 3 b)). Consequently, S(t) can be reconstructed using spline interpolation for instance after having removed the dithering signal with a small remaining reconstruction error. The level-crossing ADC principle can be summarized by the block diagram shown in Fig. 4.

C. OFI as a non-uniform sampling system

Based on the previous description of level-crossing ADCs, in the moderate feedback regime, based on (1) and ( 2), we propose to perceive SM interferometers as an inherent nonuniform sampling system with its own embedded phase levelcrossing detector. The information on D (t) is completely enclosed within the phase Φ F [START_REF] Taimre | Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing[END_REF]. By monitoring the OOP discontinuities, a phase domain level crossing every 2π can thus be obtained (Fig. 5). Further, contrary to classical levelcrossing analog-to-digital converters (ADCs), the number of phase quantization levels (PQL) Φ 0 (k) is ideally infinite due to the folding or wrapping occurring every 2π introduced by the cosine in the OOP (see [START_REF] Donati | Developing self-mixing interferometry for instrumentation and measurements[END_REF] and [START_REF] Plantier | Behavioral model of a self-mixing laser diode sensor[END_REF]). However, as shown in Fig. 5, these phase quantization levels Φ 0 (k) are slightly different (by an amount denoted ∆Φ) for an increasing and decreasing Φ 0 phase. These PQLs can thus be referred to as Φ 0 R and Φ 0 F when Φ 0 is increasing or decreasing respectively. They are completely defined by (2) with Φ F as given in [START_REF] Plantier | Behavioral model of a self-mixing laser diode sensor[END_REF] whenever Φ F has infinite slopes:

Φ F,R = kπ -arctan (α) + arccos -1 C (3)
Φ F,F = (k + 2) π -arctan (α) -arccos -1 C ( 4
)
where k is an even integer. It will be later shown that this ∆Φ does not cause any issue for the proposed approach and can

∆Φ = 2 arccos -1 C + C 2 -1 -π (5)
The SM phase level-crossing detector outputs time-phase pairs [t n , Φ n ]. For each pair, t n corresponds to the time instant when Φ 0 (t) crosses one PQL which can be either Φ 0 R (k) or Φ 0 F (k). Since these pairs [t n , Φ n ] must be recorded, these time instants are quantized Q (t n ) with a time resolution of 1/f s (where f s is the sampling frequency of the data acquisition system) to generate non-uniform samples (NUS)

[Q (t n ) , Φ (t n )].
As shown in [START_REF] Marvasti | Nonuniform sampling: theory and practice[END_REF], the continuous time input signal can be reconstructed from these samples if the quantization sampling rate of the input signal exceeds twice the input signal bandwidth. In addition, to be further processed, these [Q (t n ) , Φ (t n )] sets are usually fed to an interpolator to generate a uniformly sampled rate output signal. However, for sub-λ 0 /2 displacements, none or only one level crossing (depending on the initial phase value of Φ F ) can be detected thereby leading to a poor displacement reconstruction.

D. Dithering and sub-λ 0 /2 displacement recovery

Here, in a manner similar to approaches used in NUS ADCs, a phase dither Φ d can be added to the phase Φ 0 so that both the number of crossed levels as well as the rate of level crossings can be increased. The resulting equivalent displacement D Σ (t) is obtained by summing the target displacement D (t) to the equivalent dither displacement D d (t). Φ d can be implemented either by directly vibrating the laser itself with D d (t) or indirectly by modulating the laser driving current, thereby modulating λ 0 . Here, a cosine vibration dithering has been chosen with an amplitude A d and a frequency f d . To achieve an accurate displacement reconstruction, the average nonuniform sampling rate should be at least twice the D d (t) bandwidth according to the Nyquist criterion. Consequently, f d can be chosen to be greater than the bandwidth of interest and A d should then be chosen greater than λ 0 /2 to induce at least two level crossings, thereby guaranteeing the Nyquist criterion. Note that this criterion can also be satisfied with lower f d values associated with correspondingly higher A d values. However, it makes it more difficult to retrieve the target displacement as the dithering signal would then be within the bandwidth of interest. As a result, here f d is chosen greater than the displacement bandwidth of interest so that it is not required to precisely know A d (it should only be > λ 0 /2).

If the Nyquist criterion is satisfied then the band-limited displacement D(t) can be reconstructed from the non-uniform samples (NUSs). As previously mentioned, to achieve this reconstruction, interpolators are required to convert these NUSs into uniform samples. Different interpolators can be employed to fulfill this task such as: splines, polynomials [START_REF] Marvasti | Nonuniform sampling: theory and practice[END_REF], sinc-based polynomials [START_REF] Vezyrtzis | Processing of signals using level-crossing sampling[END_REF]... In addition, if the SM signals are acquired using an ADC, then phase unwrapping algorithms correspond to the perfect interpolator. Here, the Consecutive Samples based Unwrapping (CSU) approach that consists in adding the normalized SM signal to the staircase approximation as interpolator since it is an attractive alternative with respect to reduced computational requirements [START_REF] Ehtesham | Analysis and implementation of a direct phase unwrapping method for displacement measurement using self-mixing interferometry[END_REF].

Φ stair of Φ 0 based on [Q (t n ) , Φ (t n )], is used R A =A d / λ 0 R f =f s /f d
The achieved reconstruction performance depends directly on the number of effective quantization levels (NEQLs) N [START_REF] Sayiner | A level-crossing sampling scheme for a/d conversion[END_REF], and on the accuracy of both the quantization levels δΦ and of f s . Based on [START_REF] Sayiner | A level-crossing sampling scheme for a/d conversion[END_REF], [START_REF] Wang | A level-crossing analog-to-digital converter with triangular dither[END_REF], the combined root mean square (rms) displacement quantization error rms per sample due to f s and δΦ can be expressed as:

rms = 1 2π λ 0 2 δΦ 2 3 + 16π 9
A d f d λ 0 f s 2 (6)
(6) and Fig. 6 clearly show that a high ratio R f of the sampling frequency f s to the dither signal frequency f d is required to achieve low displacement errors while the ratio R A = A d /λ 0 can be kept small. Even if a higher value of A d results in a higher number of level crossings, a higher value of A d necessitates higher f s value to keep decreasing rms [START_REF] Sayiner | A level-crossing sampling scheme for a/d conversion[END_REF], [START_REF] Wang | A level-crossing analog-to-digital converter with triangular dither[END_REF]. Further, for high A d , the SM signal might suffer from speckle effects [START_REF] Zabit | Self-mixing laser sensor for large displacements: Signal recovery in the presence of speckle[END_REF]. Here A d is thus chosen to be equal to a few λ 0 only since even with a low number of quantization levels a high resolution can still be achieved [START_REF] Sayiner | A level-crossing sampling scheme for a/d conversion[END_REF]. As a result, the accuracy of the system is mainly dependent on δΦ for which different noise sources can be identified [START_REF] Giuliani | Self-mixing laser diode vibrometer[END_REF]: the LD linewidth [START_REF] Giuliani | Laser diode linewidth measurement by means of self-mixing interferometry[END_REF], the mechanical noise of the experimental setup, the LD driving current noise and the temperature noise that can both affect λ 0 . Fig. 7 shows the effect of NEQLs on the displacement reconstruction error using the CSU. As expected, this error increases with C [START_REF] Ehtesham | Analysis and implementation of a direct phase unwrapping method for displacement measurement using self-mixing interferometry[END_REF] and decreases with increasing dithering amplitudes [START_REF] Sayiner | A level-crossing sampling scheme for a/d conversion[END_REF].

E. Impact of the Optical Feedback Factor C on the reconstructed displacement

As previously mentioned, it is important to also note that as shown in Fig. 5, SM interferometers cannot be considered as ideal NUS systems since their PQL Φ 0 (k) are slightly different (by an amount denoted ∆Φ) for an increasing and decreasing Φ 0 phase (see ( 8)). It is thus clear that even if Φ stair is usually used as a rough estimation of Φ 0 , Φ stair does not take into account ∆Φ and hence inherently contains some errors. Consequently, it strongly suggests that not taking into account ∆Φ might result in reduced performances if C is not considered. To estimate this error, it is interesting to look at this issue from a different perspective. Instead of considering the PQLs Φ 0 R and Φ 0 F to be different, they can be supposed to be equal if a virtual square displacement of amplitude λ 0 ∆Φ/8π is added on top of D d at f d . Fig 9 shows the simulated results obtained for a SM signal corresponding to a 4 µm sinusoidal displacement for C = 2 with its corresponding erroneous Φ stair , its ideal Φ stair and the reconstructed displacement errors using Spline interpolation in both cases. The maximum error max ≈ 42 nm is thus in accordance with the predicted error λ 0 ∆Φ/8π. Fig. 10 shows the influence of the C value on the displacement error. Using a spline interpolator and the non-uniform sampling theory, the RMS error of the reconstructed displacement rms is 2.2 nm and 40.7 nm with (by adding or subtracting ∆Φ/2 to Φ 0 R or Φ 0 F respectively if C is known [START_REF] Bes | Displacement measurements using a self-mixing laser diode under moderate feedback[END_REF], [START_REF] Fan | Improving the measurement performance for a self-mixing interferometry-based displacement sensing system[END_REF], [START_REF] Bernal | Study of laser feedback phase under self-mixing leading to improved phase unwrapping for vibration sensing[END_REF], [START_REF] Bernal | Robust method of stabilization of optical feedback regime by using adaptive optics for a self-mixing micro-interferometer laser displacement sensor[END_REF]) and without C taken into account respectively.

In presence of a dithering signal, the change of direction occurs at the dithering frequency f d , which is outside of the signal bandwidth of interest. Consequently, ∆Φ has a direct impact on the retrieved amplitude of the dithering signal D d but none on the desired displacement D (t). Fig. 11 shows the reconstructed displacement spectrum with and without a 2 µm 1 kHz dithering signal for a 4 µm 90 Hz sinusoidal displacement using Spline interpolation. It clearly shows that unwanted harmonics generated by ∆Φ are efficiently removed by the dithering and appeared at frequency higher than f d . This is also clearly shown in Fig. 12: while Fig. 12 d) illustrates that the C induced error on the reconstructed signal (blue curve) is similar to the one obtained previously without dithering signal (Fig. 9), Fig. 12 e) demonstrates that by removing all the signal frequencies ≥ f d , the remaining error is similar to the one obtained when C is taken into account. In addition, the RMS error is 2.8 nm and 18 nm with and without dithering signal respectively over a 1 kHz bandwidth, which is similar to the 2.2 nm RMS error obtained when C is taken into account.

III. RESULTS

A SM test bench (see Fig. 13) was developed to assess the performances of the proposed approach through two main test procedures. The aim of the first one is to verify the principle of dithering. In this case, the target generates both the dithering displacement and the sub-λ 0 /2 displacement while the laser diode remains stationary. In the second case, as a proof of concept, the target only generates the subλ 0 /2 displacement to be recovered while the laser diode is vibrated at one of the mechanical resonance frequencies of the system by a shaker to generate the dithering displacement. The LD, driven by a constant injection current of 30 mA, is a Hitachi HL7851G emitting at λ 0 =785 nm. The system benefits from the autofocus based on the liquid lens ARTIC 39N0 from Varioptic [START_REF] Bernal | Robust method of stabilization of optical feedback regime by using adaptive optics for a self-mixing micro-interferometer laser displacement sensor[END_REF]. A piezoelectric transducer (PZT) from Physik Instrumente (P753.2CD) is used as a target positioned at 30 cm from the LD. It is equipped with an internal capacitive feedback position sensor with a 0.2 nm resolution and 2 nm repeatability. The data is acquired by a NI USB 6251 data acquisition system operating at 10 6 sample/s with a 16 bit resolution.

Prior to any measurements, rms can be estimated through the system phase noise measurement based on [START_REF] Giuliani | Laser diode linewidth measurement by means of self-mixing interferometry[END_REF] and [START_REF] Jiang | Multiple self-mixing interferometry algorithm based on phase modulation for vibration measurement[END_REF]. As a result, δΦ is estimated to be approximately 0.136 rad corresponding to rms ≈ 4.9 nm.

Here, the data is firstly processed using the CSU method in order to reconstruct a first estimate of D Σ . Then, an FFT analysis is performed to filter out all the signals out of the bandwidth of interest (f ≥ f d ). Figure 14 shows the measured amplitude of the displacement extracted from the SM signal acquired during 1 s in order to retrieve the sub-λ 0 /2 displacement of a target vibrating at 20 Hz while the dithering vibration is set at A d = 2.5 µm and f d = 90 Hz. As a consequence, R f is approximately 11 k and R A is 1.6 corresponding to 13 NEQLs approximately. For each displacement amplitude, a set of 10 measurements is performed. Note that spline interpolation can also be used instead of CSU. However, the obtained performances are slightly better for CSU. Figure 15 shows a typical spectrum of the reconstructed displacement signal together with the dithering signal. The proposed method's performances for 3 different dithering amplitudes and D ∈ [2.5 nm; 500 nm] are summarized in Table I in terms of the absolute mean error | | , and standard deviation σ. It is interesting to note that the best result is not obtained for the Note that the proposed approach can be also applied to displacements larger than λ 0 /2 (Fig. 16) as even more PQLs would then be generated. The measured white noise power spectral density is less than 100 pm/ √ Hz with a corner frequency f c related to the 1/f noise estimated to be approximately 80 Hz (Fig. 16). Finally, the proposed approach can also recover other kind of vibrations. For instance, Fig. 17 shows the reconstructed displacement of a damped harmonic vibration at 55 Hz compared to the PZT.

IV. DISCUSSION AND PERSPECTIVES

For large displacements, C might suffer slow drifts due to speckle. Subsequently, the PQL will also drift accordingly. Contrary to the precedent analysis on the impact of C on the proposed method performances, this will then result in reconstruction error since the phase difference between two consecutive PQL of Φ 0 R and Φ 0 F are no more equal to 2π. To correctly recover the displacement, knowing C real-time value is thus necessary. In another study, it will be shown that using our NUS method such estimation can be obtained contrary to all existing methods that only provide an average value of C.

However, note that except for the fringe locking method [START_REF] Giuliani | Self-mixing laser diode vibrometer[END_REF], all the other SM unwrapping based methods will suffer from the same issues. Nonetheless, due to the inherent simplicity of the proposed approach and its ability to deal with sub-λ 0 /2 displacement (for which speckle is usually negligible), the proposed algorithm is still of great interest. In addition, Fig. 18 shows the evolution of the total rms error rms as a function of the C modulation due to speckle (C = C 0 + C m sin (2πf t) where C 0 = 2 and f is equal to the displacement frequency) and of the displacement amplitude D. Based on (6), without any C modulation, the estimated rms is approximately 3.3 nm. The results obtained in the presence of speckle show that the achievable performances are comparable to this figure for C m < 0.3.

As previously mentioned in section II, if C is not taken into account, it generates virtual square displacement, the amplitude of which is related to (2). It can thus be possible to both detect and estimate it via the reconstructed displacement spectrum. In addition, the high dithering frequency compared to the slow varying C value might give the opportunity to observe the evolution of C with time, which might allow fine C correction and real-time C adjustment via adaptive optics [START_REF] Bernal | Robust method of stabilization of optical feedback regime by using adaptive optics for a self-mixing micro-interferometer laser displacement sensor[END_REF].

V. CONCLUSION

The proposed sensor can measure sub-λ 0 /2 displacement and achieve a high precision with a noise power spectral density ≤ 100 pm/ √ Hz while based on both a relatively simple set-up and processing method compared to [START_REF] Azcona | A nanometric displacement measurement system using differential optical feedback interferometry[END_REF]. We propose to interpret OFI as a non-uniform event based sampling system where the quantization levels are clearly defined by the signal discontinuities in the moderate optical feedback regime. It was shown that the CSU ( [START_REF] Ehtesham | Analysis and implementation of a direct phase unwrapping method for displacement measurement using self-mixing interferometry[END_REF]) can be used as an efficient interpolator. However, to design an even more efficient system, it might be interesting to choose another kind of mathematical interpolators since no analog-to-digital conversion would then be required. With the non-uniform sampling approach, the instants when quantization levels are crossed contain all the information. It is thus possible to envision the development of a system based only on fringe detection using an analog front-end similar to [START_REF] Siddiqui | All analog processing of speckle affected self-mixing interferometric signals[END_REF] and then process the acquired data. As a result, the amount of acquired data to be processed can be much less than that of a classical approach based on an ADC front-end, resulting in simpler and cheaper embedded monitoring systems. Based on this non-uniform event based proposed interpretation, a dithering signal was added here to successfully recover sub-λ 0 /2 displacements as well as higher amplitude ones. Up to now, the system's main limitation is the bandwidth (here 190 Hz corresponds to the maximal mechanical resonant frequency of the shaker-laser system) directly related to the dithering vibration frequency due to the use of a bulky system. This can be greatly improved by designing a very light weight laser head with an embedded PZT. Modulating the laser driving current might also be another promising way to achieve higher bandwidth.

  O. D. Bernal, F. Jayat and T. Bosch are with the LAAS-CNRS, University of Toulouse, INP-ENSEEIHT, 31000 France e-mail: olivier.bernal@toulouseinp.fr. U. Zabit is with National University of Sciences and Technology, NUST, Islamabad, 44000, Pakistan. DOI:10.1109/JSEN.2020.2970599

Fig. 1 .

 1 Fig. 1. Self-Mixing displacement sensor set-up with a piezoelectric transducer (PZT) used as a target. A dithering signal can be added either via the laser drive current I 0 + I d (t) or via a shaker A d (t).

Fig. 2 .

 2 Fig. 2. Uniform sampling and quantization of S(t) a) and non-uniform sampling of S(t) in a level-crossing ADC b).The samples are shown as red crosses.

Fig. 3 .

 3 Fig.3. Simulated level-crossing ADC: a) a 0.9 V sinusoidal input S(t) at 10 Hz crosses only the 0V-threshold level, b) after adding a 3 V 50 Hz dithering signal to S(t) many more samples are generated, c) reconstructed input signal using spline interpolation after having subtracted the dithering signal and d) the remaining reconstruction error.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Block diagram of a typical level-crossing ADC

Fig. 6 .

 6 Fig. 6. Normalized Displacement RMS error to λ 0 induced by the time instants quantization through R f = fs/f d and the dither signal amplitude through R A = A d /λ 0

Fig. 7 .Fig. 8 .

 78 Fig. 7. Absolute displacement error retrieved with the CSU interpolating method vs different C values and the target displacement amplitudes D for different dithering amplitude A d

Fig. 9 .

 9 Fig.9. Simulated SM signal (b) obtained for a 4µm displacement at 90 Hz with C=2 (a) with a laser diode operating with a λ= 785 nm. Reconstructed Φ stair (c) by discontinuity detection with (red) and without (blue) C value taken into account. Obtained reconstructed displacement error induced with (red) and without (blue) C taken into account while using the non-uniform sampling approach with a spline interpolator.

Fig. 10 .

 10 Fig.10. Simulated displacement error (blue line) and estimated error λ 0 ∆Φ/8π (red dashed line) induced when C is not taken into account in Φ stair obtained for a 4µm 90 Hz displacement and spline interpolation.

Fig. 11 .

 11 Fig. 11. Simulated reconstructed displacement frequency spectrum with (red dashed line) and without (black line) a dithering signal of 2 µm at f d =1 kHz for C=2, a target displacement of a 4 µm and 90 Hz displacement and spline interpolation.

Fig. 12 .

 12 Fig.[START_REF] Ehtesham | Analysis and implementation of a direct phase unwrapping method for displacement measurement using self-mixing interferometry[END_REF]. Simulated SM signal (b) obtained for a 4µm displacement at 90 Hz with C=2 (a) with a dithering signal of 2 µm at f d =1 kHz for a laser diode operating with a λ= 785 nm. Reconstructed Φ stair (c) by discontinuity detection with (red) and without (blue) C value taken into account. Finally d), obtained reconstructed displacement error induced with (red) and without (blue) C taken into account while using the non-uniform sampling approach with a spline interpolator and e) obtained from d) curves by removing all frequencies f > f d .

Fig. 13 .Fig. 14 .

 1314 Fig. 13. The sub-λ/2 SM displacement sensor test bench. Piezoelectric transducer (PZT) containing internal capacitive feedback sensor with 2 nm resolution is used to generate target displacement D(t). Mechanical shaker provides the dithering vibration D d (t) to the SM sensor.

Fig. 15 .Fig. 16 .

 1516 Fig. 15. FFT of the reconstructed displacement obtained with the spline interpolation for a 20 Hz target vibration with an 2.5 nm (black curve) and 25 nm (blue curve) amplitude and a 1.2 µm dithering amplitude obtained by the shaker at 50 Hz.

Fig. 17 .

 17 Fig. 17. Reconstructed damped harmonic displacement (red) for A d =0.8µm and f d =190 Hz compared to PZT displacement (blue) (a) and (b) FFT of the reconstructed measured (blue) and simulated (red) displacement both using the CSU.

9 Fig. 18 .

 918 Fig. 18. Simulated total RMS error rms over the full bandwidth for A d = 2 µm at 200 Hz for different PZT displacement amplitudes at f = 45 Hz and varying C value described here as C = C 0 + Cm sin (2πf t).

  c 2020 IEEE. PERSONAL USE OF THIS MATERIAL IS PERMITTED. PERMISSION FROM IEEE MUST BE OBTAINED FOR ALL OTHER USERS, INCLUDING REPRINTING/ REPUBLISHING THIS MATERIAL FOR ADVERTISING OR PROMOTIONAL PURPOSES, CREATING NEW COLLECTIVE WORKS FOR RESALE OR REDISTRIBUTION TO SERVERS OR LISTS, OR REUSE OF ANY COPYRIGHTED COMPONENTS OF THIS WORK IN OTHER WORKS

4

be expressed as a function of C:

TABLE I

 I

	Measured Displacement Reconstruction Performances vs Dithering
	Amplitude A d or NEQLs using the CSU interpolator with 1 Hz
		bandwidth FFT analysis
	A d (µm) 2.5 *	NEQLs Mean| | (nm) 13 1.9	Mean σ(nm) 0.85
	4.5 *	22	2.2	1.4
	1.2 * *	6	1.7	0.96
	* PZT vibrating at 90 Hz
	** Shaker vibrating at 50 Hz
	highest NEQLs of 22 but for 13. This might be explained by
	the presence of a higher speckle level for the NEQLs of 22
	that can introduce C variations, thereby inducing some more
	phase errors.

ACKNOWLEDGMENT

The authors would like to thank Agence Nationale de la Recherche (ANR) : LabCom CapIRO in cooperation with ACOEM, Thierry MAZOYER


Sylvain Tourancheau
email: sylvain.tourancheau@univ-nantes.fr

Patrick Le Callet

Dominique Barba

IMPACT OF THE RESOLUTION ON THE DIFFERENCE OF PERCEPTUAL VIDEO QUALITY BETWEEN CRT AND LCD

Keywords: Quality control, Liquid crystal displays, Cathode-ray tubes, Image resolution, HDTV

The incoming of high-definition new visual experience at home has boosted new display technologies such as liquid crystal displays (LCD), plasma and projectors. These technologies enable the increase of screen size necessary to feel a cinema-like experience. However, they introduce some new visual shortcomings not present on mature CRT technology.

In this paper, subjective tests are described which highlight a difference of perceptual video quality between CRT and liquid crystal displays. Moreover, it's observed that this loss of quality on LCD is more important with high resolution sequences than with standard resolution ones. This influence of the resolution is particularly explainable for the LCD motion blur defect.

INTRODUCTION

High-definition television (HDTV) broadcasting systems will soon substitute standard television (SDTV). With the incoming of this new visual experience in terms of pictures resolution, new display technologies have grown. They enable the increase of screen size necessary to sense immersion, impact and immediacy as in a movie theater [START_REF] Fujio | Future broadcasting and high definition television[END_REF]. However, these new display technologies, such as liquid crystal displays (LCD) and plasma, introduce some new visual shortcomings [START_REF] Itu | Report on results of comparative subjective picture quality assessment test between CRT and LCD[END_REF] and make compression distortions more visible than on CRT [START_REF]Maximizing the quality of SDTV in the flat-panel environment[END_REF].

It has been shown that larger pictures resolution becomes a drawback when the level of coding distortions increases: observers then prefer standard definition as this reduces the visual impact of the distortions [START_REF] Péchard | From SD to HD television: effects of H.264 distortions versus display size on quality of experience[END_REF]. It could be interesting to know if the same behaviour appears with display-dependent visual defects.

In this paper, subjective quality assessment tests are described in Section 2. They're performed for both HDTV and SDTV sequences, on both CRT and LCD. A difference of perceptual video quality between CRT and LCD is highlighted in Section 3. Moreover, results show that this loss of quality on LCD is larger with high resolution sequences. Section 4 is a discussion of these results.

SUBJECTIVE QUALITY ASSESSMENT

In this part, subjective quality assessment tests are described. The perceptual quality of HDTV and SDTV sequences is assessed on both CRT and LCD.

Material

Four ten-second long 1080i HDTV sequences from SVT research have been used (namely New Mobile and Calendar, Parkrun, Shields and Stockholm). Each reference (uncompressed) sequence has been distorted with H.264 compression standard using JM reference software. Seven bit-rates (not necessary the same for each sequence) have been chosen to cover the entire quality range.

SDTV sequences are computed from these HD sequences through a half-band filtering followed by a downsampling by a factor of 2 (both along horizontal and vertical directions). This processing is performed on each field of the interlaced HD sequence. Resulting 540i sequences are an approximation of actual SDTV whereof format is 576i, with the advantage that it doesn't necessitate any interpolation. As with HD sequences, SD videos have been encoded using the H.264 JM reference software, with the same parameters. Six bit-rates (not necessary the same for each sequence) have been chosen to cover the entire quality range.

Tests have been performed in a specific showroom. Lighting conditions and display parameters have been measured and adjusted according to BT.500-11 and BT.710-4 ITU recommendations. Two HDTV displays have been used: a JVC DT-V 1910CG and a Philips T370 HW01 which both can display 1080i format. Viewing distance was set to 3H for HD sequences and 6H for SD sequences (where H is the height of the screen), according to recommendations. Tests have been led in four parts: HD sequences on a CRT, HD sequences on a LCD, SD sequences on a CRT and SD sequences on a LCD.

Observers

Observers were mostly male students in their mid twenties. All were familiar with standard television and cinema but not with HDTV. Every candidate has been first checked for color blindness with Ishihara test and for acuity with Monoyer's plates. People with at least one error in Ishihara's test or less than 9/10 in Monoyer's test have been rejected. Between 20 and 25 people took part in each of the four parts tests, but there was not necessary the same people from a part to another.

Protocol

The assessment method required here should allow observers to precisely construct their judgment. As very little quality differences must be detected, the method must force the quality discrimination. A well known stable method for this purpose is the SAMVIQ method [START_REF] Ebu | SAMVIQ -Subjective assessment methodology for video quality[END_REF], developed by France Telecom R&D and standardised by the European Broadcasting Union (EBU). Observers compare seven distorted sequences (for HD content, six for SD) and one hidden reference both between them and with the explicit reference. Notation scale is continuous, each score can take a value between 0 and 100.

SAMVIQ is a multi stimuli continuous quality scale (MSCQS) protocol. It provides a precise and reliable [START_REF] Blin | New quality evaluation method suited to multimedia context: Samviq[END_REF] measure of the subjective video quality which can be compared directly to the reference. It is important to note that this reference may or may not be the original video signal. As the observers can directly compare the impaired sequences among themselves and against the reference, they can grade them accordingly. This feature permits a high degree of resolution in the grades given to the system. Moreover, observers have a random access to the sequences, which permits to choose exactly the sequence they want to assess. This allows them to precisely build their assessment opinion. This is particularly interesting in this context where very little quality differences have to be identified.

The consistency of the individual scores is evaluated after the tests have been completed by all the subjects. It is done by applying a suitable "rejection" technique. This is a process in which all scores from a particular subject are omitted from the analysis of data. Following the application of the rejection process, 15 valid subjects should be retained at minimum. Same quality scores are shown for the SD versions of sequence Shields in Figure 2. In the case of standard resolution materials, the difference between the two displays is not so conspicuous. CRT scores are a little higher than LCD ones, but with regard to confidence intervals it's not statistically reliable. In order to have a global view of these results, quality MOS obtained on LCD devices have been plotted as a function of quality MOS obtained on CRT displays in Figure 3. At the top (Figure 3a), MOS of the eight HD versions of the four sequences are plotted. At the bottom (Figure 3b), MOS of the seven SD versions of the four sequences are plotted.

RESULTS AND OBSERVATIONS

Quality mean opinion scores (MOS) are presented in

HD quality MOS are globally shifted towards the area where CRT MOS are higher than LCD ones. For SD materials, it's less blatant, points are nearer the frontier than for HD materials. Moreover, confidence intervals are widely overlapping this limit.

DISCUSSION

Visual shortcomings of liquid crystal displays

Subjective tests have been conducted with HDTV sequences in order to compare quality picture between CRT and LCD [START_REF] Itu | Report on results of comparative subjective picture quality assessment test between CRT and LCD[END_REF]. It has resulted that, as a whole, majority of the observers felt that the picture quality on LCD was lower than the one on CRT. Many defects have been counted by viewers. Despite of recent improvements, motion blur remains still annoying for moving pictures with significant movements. The luminance at black level is higher for LCD monitor, so that black areas look glossy or lighter than on a CRT. Difference in reproduced colours has been also observed between CRT and LCD, particularly with flesh colours. Concerning the overall impression, observers generally notice that there is no depth-feel in images displayed on LCD. CRT produces natural feelings and textures while on LCD images are displayed too sharply, leading to unnatural perspective.

LCD motion blur has been widely studied in recent works [START_REF] Michiel | The temporal MTF of displays and related video signal processing[END_REF][START_REF] Hao Pan | LCD motion blur modeling and analysis[END_REF][START_REF] Feng | LCD motion-blur analysis, perception, and reduction using synchronized backlight flashing[END_REF]]. It's mainly caused by the hold-type LCD's displaying method: the light intensity is maintained on the screen for the duration of the frame, whereas on CRT light intensity is a pulse which fades over the frame duration. The main difference happens when the eyes of the observer are tracking a moving object on the LCD screen: for a given frame, the picture is sustained on the screen while the eyes are still moving slightly anticipating the movement of the object. Edges of this object are displaced on the retina resulting in a blur [START_REF] Kurita | Moving picture quality improvement for hold-type AM-LCDs[END_REF].

Impact of LCD motion blur

Recent study has shown that the difference of perceptual quality between CRT and LCD devices for moving pictures could be roughly predicted from the quantity of movements in the sequence. The width W (in pixels) of motion blur that appears on the edges of a moving object is proportional to its velocity V (in pixels per second) as follows [START_REF] Hao Pan | LCD motion blur modeling and analysis[END_REF]:

W = aV, (1)
with a a parameter which depends on temporal aperture of the display. The computation of an average edges velocity along the sequence enables to estimate a global magnitude of perceived motion blur. It appears that for sequences with significant movements the loss of quality on LCD devices is linearly related to this magnitude [START_REF] Tourancheau | Prediction of perceived quality differences between CRT and LCD displays based on motion blur[END_REF].

Influence of resolution

The results of subjective quality assessment tests presented in this paper show that the difference of quality between CRT and LCD is larger with HDTV sequences than with SDTV ones. In other terms, the increase of display resolution seems to amplify LCD visual defects. In the particular case of LCD motion blur, this could be easily explained. In SDTV, the velocity of moving objects is reduced by two in terms of resolution (pixels per second) with respect to HDTV. According to previous statements, the magnitude of perceived motion blur is reduced in the same proportions. The loss of quality on LCD should be roughly two times less important in SDTV than in HDTV.

Previous results [START_REF] Pinson | The impact of monitor resolution and type on subjective video quality testing[END_REF] show that some sequences at NTSC format (525i) obtain best subjective quality scores on a 5.5 inch LCD monitor with a CIF resolution (352× 288) than on a 20 inch CRT monitor. The perceptual quality on LCD is better than those on CRT at low resolution (CIF format). These results are in continuity with ours: LCD and CRT visual quality score are almost the same in intermediate resolution (SDTV format) and CRT perceptual quality is higher at high resolution (HDTV format). The display resolution has an important influence on the visual defects affecting liquid crystal displays. These visual defects seems to be more visible at high resolution than at low resolution.

CONCLUSION

Subjective video quality assessment tests have highlighted a difference of perceptual quality between CRT and LCD devices. Moreover, this difference is very more important on high resolution sequences than on standard resolution ones. This loss of quality on LCD can be explained by the visual defects affecting this new display technology. The influence of the display resolution on the perceptual video quality on LCD has been explained for the motion blur issue.

Of course, the increase of pictures resolution in HDTV leads to broadcasting issues such as the increase of minimal acceptable bit-rates. But less blatantly, it appears that visual shortcomings relative to new display technologies are worsened by this display resolution increase. Liquid crystal display technology has to be improved in order to reach the visual quality of CRT, particularly in HDTV. However, more reduced video applications supports such as laptop, cell phone, personal digital assistants, etc. seem to not be as much affected by this visual defects due to their smaller display resolutions.

Fig. 1 .

 1 Fig. 1. Quality mean opinion scores obtained with HD, for the two types of displays, as a function of bit-rate, for the sequence Shields. The horizontal shift for each data point is for clarity.

Fig. 2 .

 2 Fig. 2.Quality mean opinion scores obtained with SD, for the two types of displays, as a function of bit-rate, for the sequence Shields. The horizontal shift for each data point is for clarity.

Fig. 3 .

 3 Fig. 3. LCD quality mean opinion scores as a function of CRT quality mean opinion scores.