M J Hÿtch

C Gatel

Phase detection limits in off-axis electron holography from pixelated detectors: gain variations, geometric distortion and failure of reference-hologram correction

Phase detection limits in off-axis electron holography from pixelated detectors: gain variations, geometric distortion and failure of reference-hologram correction

Martin Hÿtch, Christophe

Introduction

In electron holography, there is a continual quest to improve the precision and accuracy of phase measurements [START_REF] Ruijter | Detection limits in quantitative off-axis electron holography[END_REF][START_REF] Ru | Principle and application of phase-shifting electron holography[END_REF][START_REF] Harscher | Experimental study of amplitude and phase detection limits in electron holography[END_REF][START_REF] Cooper | Medium resolution off-axis electron holography with millivolt sensitivity[END_REF][START_REF] Lichte | Performance limits of electron holography[END_REF][START_REF] Voelkl | Approaching routine 2π/1000 phase resolution for off-axis type holography[END_REF][START_REF] Mcleod | Phase measurement error in summation of electron holography series[END_REF]. This is particularly the case at medium resolution where the phase is directly related to the local field strengths measured in quantitative experiments, whether they be electric [START_REF] Rau | Twodimensional mapping of the electrostatic potential in transistors by electron holography[END_REF][START_REF] Twitchett | Quantitative electron holography of biased semiconductor devices[END_REF][START_REF] Mccartney M R, Dunin-Borkowski | Quantitative measurement of nanoscale electrostatic potentials and charges using off-axis electron holography: developments and opportunities[END_REF], magnetic [START_REF] Dunin-Borkowski R E | Towards quantitative electron holography of magnetic thin films using in situ magnetization reversal[END_REF][START_REF] Gatel | Size-specific spin configurations in single iron nanomanet: from flower to exotic vortices[END_REF] or crystalline strain [START_REF] Hÿtch | Nanoscale holographic interferometry for strain measurements in electronic devices[END_REF][START_REF] Hÿtch | Dark-field electron holography for the measurement of geometric phase[END_REF]. Most attention has been focussed on the detection limits imposed by the random errors caused by counting statistics [START_REF] Lichte | Electron noise in off-axis image plane holography[END_REF]. However, there are also many other sources of error, more often systematic in nature, such as specimen interactions and preparation artefacts, optical parameters, hologram processing and the main object of this paper, the detector characteristics. With the advent of almost unlimited exposure times [START_REF] Gatel | Unlimited acquisition time in electron holography by automated feedback control of transmission electron microscope[END_REF], systematic errors will determine the ultimate accuracy, and even precision in some sense, of electron holography experiments.

Random errors indeed arise essentially from the counting statistics. Electron holograms are acquired with a finite number of electrons which leads inevitably to random fluctuations in the measured intensities, which in turn leads to noise in the reconstructed phase images [START_REF] Lichte | Electron noise in off-axis image plane holography[END_REF]. Numerous papers, both theoretical and experimental, have been dedicated to the phase detection limits imposed by counting statistics. They show that the noise is random and can be reduced by increasing the electron dose [START_REF] Harscher | Experimental study of amplitude and phase detection limits in electron holography[END_REF][START_REF] Cooper | Medium resolution off-axis electron holography with millivolt sensitivity[END_REF][START_REF] Mcleod | Phase measurement error in summation of electron holography series[END_REF]. Systematic errors come from various sources, first and foremost being the sample under investigation. Whilst the sample is providing the fields we are trying to measure, real specimens also introduce a number of additional phenomena which can produce systematic shifts in the phase. An uneven sample thickness can introduce unwanted phase variations through the action of the mean-inner potential of the material [START_REF] Rau | Twodimensional mapping of the electrostatic potential in transistors by electron holography[END_REF] or dynamic diffraction effects [START_REF] Gajdardziska-Josifovska | Accurate measurements of mean inner potential of crystal wedges using digital electron holograms[END_REF]. Sample charging under the electron beam also produces spurious phase variations [START_REF] Mccartney | Characterization of charging in semiconductor device materials by electron microscopy[END_REF][START_REF] Park | Impact of electron irradiation on electron holographic potentiometry[END_REF][START_REF] Suzuki | Secondary electron effect on electron beam induced charging of SiO2 particle analysed by electron holography[END_REF] and damage from specimen preparation may diminish the phase change from electric fields in a systematic way [START_REF] Twitchett | Quantitative electron holography of biased semiconductor devices[END_REF][START_REF] Cooper | Dopant profiling of focused ion beam milled semiconductors using off-axis electron holography: reducing artifacts, extending detection limits and reducing the effects of gallium implantation[END_REF][START_REF] Gribelyuk | Sample preparation by focused ion beam without argon ion milling for quantitative electron holography of p-n junctions[END_REF].

A number of artefacts arise from the imaging conditions, such as the curvature of the elliptic incident wave [START_REF] Lehmann | Influence of the elliptical illumination on acquisition and correction of coherent aberrations in high-resolution electron holography[END_REF] or from charging of biprisms or their associated Fresnel fringes when used singly [START_REF] Lichte | Artefacts in electron holography[END_REF]. The projector lenses of the microscope will introduce long-range phase changes, just like in high-resolution electron microscopy [START_REF] Hüe | Calibration of projector lens distortions[END_REF]. Finally, the detector introduces systematic phase variations from the packing of the fibre-optic coupling of charge-coupled devices (CCD) [START_REF] Ruijter | Detection limits in quantitative off-axis electron holography[END_REF][START_REF] Voelkl | Approaching routine 2π/1000 phase resolution for off-axis type holography[END_REF].

The best way to deal with systematic errors is of course to eliminate them at the source, such as employing better sample preparation techniques [START_REF] Gribelyuk | Sample preparation by focused ion beam without argon ion milling for quantitative electron holography of p-n junctions[END_REF] or using two biprisms to avoid almost entirely Fresnel fringes in holograms [START_REF] Harada | Double-biprism electron interferometry[END_REF]. To correct for the remaining systematic errors, the standard procedure is to acquire a reference hologram taken in a field-free region of the vacuum [START_REF] Ruijter | Detection limits in quantitative off-axis electron holography[END_REF]. Apart from sample artefacts (and Fresnel fringes) the method is considered robust for all other systematic errors. In this paper, we will show that for pixelated detectors, such as CCDs or direct-detection devices (DDD), a single reference hologram is not sufficient and will furthermore introduce additional systematic errors. This paper was written in the context of dynamic automation and the stabilisation of holographic fringes [START_REF] Gatel | Unlimited acquisition time in electron holography by automated feedback control of transmission electron microscope[END_REF]. We have shown that controlling the microscope with feedback loops allows us to stabilise the phase of the hologram during very long exposure times, without the need of image stacks [START_REF] Mcleod | Phase measurement error in summation of electron holography series[END_REF][START_REF] Niermann | Averaging scheme for atomic resolution off-axis electron holograms[END_REF]. The aim is to reduce significantly the random errors from counting statistics and hence reach the ultimate phase detection limits in electron holography experiments. Of course the dose that samples can submit has a limit. But in some cases, the important factor is the dose rate, meaning that a high number of electrons can be accumulated over very long exposure times at low beam intensity. In addition, some fields are measured in the vacuum where there are no such limitations.

As random errors are gradually squeezed out of experiments, all sources of systematic error must be ascertained and removed. Control of the average phase of the hologram also allows more sophisticated phase reconstruction methods such as the fringe-shifting method [START_REF] Volkov | Double resolution electron holography with simple Fourier transform of fringe-shifted holograms[END_REF] used in the pi-shift variant in image stacks [START_REF] Boureau | Off-axis electron holography combining summation of hologram series with double-exposure phase-shifting: theory and application[END_REF] or through automation [START_REF] Gatel | Unlimited acquisition time in electron holography by automated feedback control of transmission electron microscope[END_REF]. The method is particularly useful for separating the centre-band from the side-band in the reconstruction process. We will be making use of these possibilities in the theoretical development. We will not consider sample artefacts and implicitly treat measurements carried out in the vacuum either containing stray fields or in field-free reference hologram regions. We will also consider a double biprism configuration was successful in removing the Frensel fringes [START_REF] Harada | Double-biprism electron interferometry[END_REF]. The only remaining source of error is the detector. Pixelated digital detectors, originally in the form of CCD and now DDD, have revolutionized electron microscopy [START_REF] Krivanek | Applications of slow-scan CCD cameras in transmission electron microscopy[END_REF] and were key to the development of quantitative techniques, in particular electron holography [START_REF] Ruijter | Imaging properties and applications of slow-scan charge-coupled device cameras suitable for electron microscopy[END_REF]. Nevertheless, they are not perfect and suffer, for example, from non-uniform modulation transfer functions (MTF) or lower than ideal Detective Quantum Efficiency (DQE). Here, we will be examining the consequences for electron holography of two less studied imperfections: gain variations and geometric displacement of pixels.

Theory

The case of a perfectly regular set of interference fringes illuminating a pixelated detector will be studied. In addition, it will be assumed that the signal is sufficiently intense to ignore counting statistics. The input signal, S, can then be described as:

𝑆𝑆(𝒓𝒓, 𝜙𝜙) = 1 + 2𝑎𝑎 cos(2𝜋𝜋𝒒𝒒. 𝒓𝒓 + 𝜙𝜙) ( 1
)
where q is the carrier frequency vector of the interference fringes, r the position coordinate on the detector in the x-y plane, 'a' the amplitude (with this definition equal to half the visibility) of the fringes and φ their phase. The amplitude is a real number less than one half and the phase of the fringes will depend on the choice of origin of the coordinate system with respect to the detector. In what follows, the assumed units for r and q will be pixels and pixels -1 respectively.

For most of the following development, the signal will be expressed in the complex form:

𝑆𝑆(𝒓𝒓, 𝜙𝜙) = 1 + 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑒𝑒 2𝜋𝜋𝑖𝑖𝒒𝒒.𝒓𝒓 + 𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑒𝑒 -2𝜋𝜋𝑖𝑖𝒒𝒒.𝒓𝒓 (2)
which allows easy passage to and from the Fourier-space representation:

𝑆𝑆 ̃(𝒌𝒌, 𝜙𝜙) = 𝛿𝛿(𝒌𝒌) + 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝛿𝛿(𝒌𝒌 -𝒒𝒒) + 𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝛿𝛿(𝒌𝒌 + 𝒒𝒒)

where k is the frequency coordinate in Fourier space. The signal is separated into three parts: a centre-band centred at k = 0 and two side bands at q and -q.

Pi-shifted holograms

We will also be making use of pi-shifted holograms during the development [START_REF] Volkov | Double resolution electron holography with simple Fourier transform of fringe-shifted holograms[END_REF]. This consists in recording a second hologram, shifted in phase by exactly pi:

𝑆𝑆(𝒓𝒓, 𝜙𝜙 + 𝜋𝜋) = 1 + 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖+𝑖𝑖𝜋𝜋 𝑒𝑒 2𝜋𝜋𝑖𝑖𝒒𝒒.𝒓𝒓 + 𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖-𝑖𝑖𝜋𝜋 𝑒𝑒 -2𝜋𝜋𝑖𝑖𝒒𝒒.𝒓𝒓 = 1 -𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑒𝑒 2𝜋𝜋𝑖𝑖𝒒𝒒.𝒓𝒓 -𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑒𝑒 -2𝜋𝜋𝑖𝑖𝒒𝒒.𝒓𝒓 (4)
If the two hologram signals are summed, we obtain the centre-band only:

𝑆𝑆 (+) (𝒓𝒓, 𝜙𝜙) ≝ 𝑆𝑆(𝒓𝒓, 𝜙𝜙) + 𝑆𝑆(𝒓𝒓, 𝜙𝜙 + 𝜋𝜋) = 2 (5)
and subtraction allows the centre-band to be eliminated:

𝑆𝑆 (-) (𝒓𝒓, 𝜙𝜙) ≝ 𝑆𝑆(𝒓𝒓, 𝜙𝜙) -𝑆𝑆(𝒓𝒓, 𝜙𝜙 + 𝜋𝜋) = 2𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑒𝑒 2𝜋𝜋𝑖𝑖𝒒𝒒.𝒓𝒓 + 2𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑒𝑒 -2𝜋𝜋𝑖𝑖𝒒𝒒.𝒓𝒓 (6)
The factor of two indicates that twice as many electrons were used to create the pi-shifted hologram than the single hologram.

Phase reconstruction

The extraction of a side-band from the hologram can be described as follows [START_REF] Hÿtch | Analysis of variations in structure from high resolution electron microscope images by combining real space and Fourier space information[END_REF]:

𝑆𝑆 ̃𝑞𝑞(𝒌𝒌, 𝜙𝜙) ≝ 𝑆𝑆 ̃(𝒌𝒌 + 𝒒𝒒, 𝜙𝜙)𝑀𝑀 � (𝒌𝒌) (7)
where 𝑀𝑀 � (𝒌𝒌) is the masking function. The mask can take different shapes but will be considered to be real, centrosymmetric and unity at the origin:

𝑀𝑀 � (𝒌𝒌) = 𝑀𝑀 � * (𝒌𝒌), 𝑀𝑀 � (𝒌𝒌) = 𝑀𝑀 � (-𝒌𝒌) and 𝑀𝑀 � (𝟎𝟎) = 1 (8)
The simplest mask shape is a hard circular mask, with a radius of kM that we will assume to be less than q/2 in order to separate side-band from centre-band:

∀ 𝑘𝑘 < 𝑘𝑘 𝑀𝑀 , 𝑀𝑀 � (𝒌𝒌) = 1, otherwise 0 (9)
Another form is the cosine mask, which we will be using later to avoid ringing artefacts in the processing of the simulated holograms:

∀ 𝑘𝑘 < 𝑘𝑘 𝑀𝑀 , 𝑀𝑀 � (𝒌𝒌) = cos � 𝜋𝜋 2 𝑘𝑘 𝑘𝑘 𝑀𝑀 � , otherwise 0 (10)
It is easy to show that the two extracted side-bands are conjugate symmetric [START_REF] Hÿtch | Analysis of variations in structure from high resolution electron microscope images by combining real space and Fourier space information[END_REF], as for Fourier components of real images:

𝑆𝑆 ̃-𝑞𝑞 (𝒌𝒌, 𝜙𝜙) = 𝑆𝑆 ̃𝑞𝑞(𝒌𝒌, 𝜙𝜙) * (11)
It is therefore only necessary to consider one of the side-bands, the positive one in our case.

Taking the perfect hologram described in Fourier space, Eq. (3), and applying Eq. ( 7) to extract the side-band and centre-band produces the following trivial result:

𝑆𝑆 ̃𝑞𝑞(𝒌𝒌, 𝜙𝜙) = a𝑒𝑒 𝑖𝑖𝑖𝑖 𝛿𝛿(𝒌𝒌) and 𝑆𝑆 ̃0(𝒌𝒌, 𝜙𝜙) = 𝛿𝛿(𝒌𝒌) [START_REF] Gatel | Size-specific spin configurations in single iron nanomanet: from flower to exotic vortices[END_REF] and in real-space:

𝑆𝑆 𝑞𝑞 (𝒓𝒓, 𝜙𝜙) = a𝑒𝑒 𝑖𝑖𝑖𝑖 and 𝑆𝑆 0 (𝒌𝒌, 𝜙𝜙) = 1

Naturally, the exact shape of the mask does not matter for perfect fringes with delta-function side and centre-bands.

For the pi-shifted holograms the results for the side-band reconstruction are:

𝑆𝑆 𝑞𝑞 (-) (𝒓𝒓, 𝜙𝜙) = a𝑒𝑒 𝑖𝑖𝑖𝑖 and 𝑆𝑆 𝑞𝑞 (+) (𝒓𝒓, 𝜙𝜙) = 0 (14)
and for the centre-band reconstruction:

𝑆𝑆 0 (-) (𝒓𝒓, 𝜙𝜙) = 0 and 𝑆𝑆 0 (+) (𝒓𝒓, 𝜙𝜙) = 1 (15)
This is the expected result from the pi-shifting algorithm: the side-band and centre-band are obtained from the subtracted and summed holograms, respectively. For a uniform and perfect hologram, the process has no interest. However, the technique will be of great value in the presence of spatial variations, allowing the centre and side-bands to be separated.

Gain variations

We will now consider the effect of an imperfect pixelated detector, and firstly with a variable gain.

Let the gain, G, be a function of (pixel) position:

𝐺𝐺(𝒓𝒓) = 1 + 𝑔𝑔(𝒓𝒓) (16)
where g(r) is the deviation from the perfect value of 1 and assumed to have an average of zero.

The gain can have both low and high-frequency variations. In this paper, we will be particularly interested in the latter. Later on, the gain will be modelled as random variations, uncorrelated from pixel to pixel, but fixed for a given experiment. The Fourier transform of the gain will therefore be uniformly filled across all frequencies: equivalent to a white noise distribution. The following development will, however, remain general.

For a general signal, S, the image intensity recorded by the detector will be:

𝐼𝐼(𝒓𝒓) = 𝑆𝑆(𝒓𝒓)𝐺𝐺(𝒓𝒓) (17)
In Fourier space the gain will be introduced as a convolution:

𝐼𝐼 ̃(𝒌𝒌) = 𝑆𝑆 ̃(𝒌𝒌)⨂𝐺𝐺 � (𝒌𝒌) (18)
which is the source of the complications for the phase reconstruction. For the perfect hologram fringes, the result in Fourier space will be, explicitly:

𝐼𝐼 ̃(𝒌𝒌, 𝜙𝜙) = 𝑆𝑆 ̃(𝒌𝒌, 𝜙𝜙) ⊗ 𝐺𝐺 � (𝒌𝒌) = �𝛿𝛿(𝒌𝒌) + 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝛿𝛿(𝒌𝒌 -𝒒𝒒) + 𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝛿𝛿(𝒌𝒌 + 𝒒𝒒)� ⊗ (𝛿𝛿(𝒌𝒌) + 𝑔𝑔 �(𝒌𝒌)) = 𝑆𝑆 ̃(𝒌𝒌, 𝜙𝜙) + 𝑔𝑔 �(𝒌𝒌) + 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑔𝑔 �(𝒌𝒌 -𝒒𝒒) + 𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑔𝑔 �(𝒌𝒌 + 𝒒𝒒) (19)
The convolution is illustrated schematically in Figure 1a where the Fourier transform of the gain is convoluted with the centre-band and side-bands to create overlap. After extraction of the positive side-band:

𝐼𝐼 ̃𝑞𝑞(𝒌𝒌, 𝜙𝜙) = �𝛿𝛿(𝒌𝒌 + 𝒒𝒒) + 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝛿𝛿(𝒌𝒌) + 𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝛿𝛿(𝒌𝒌 + 2𝒒𝒒) + 𝑔𝑔 �(𝒌𝒌 + 𝒒𝒒) + 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑔𝑔 �(𝒌𝒌) + 𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑔𝑔 �(𝒌𝒌 + 2𝒒𝒒)� 𝑀𝑀 � (𝒌𝒌) = 𝑀𝑀 � (-𝒒𝒒) + 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑀𝑀 � (𝟎𝟎) + 𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑀𝑀 � (-2𝒒𝒒) + 𝑔𝑔 �(𝒌𝒌 + 𝒒𝒒)𝑀𝑀 � (𝒌𝒌) + 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑔𝑔 �(𝒌𝒌)𝑀𝑀 � (𝒌𝒌) + 𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑔𝑔 �(𝒌𝒌 + 2𝒒𝒒)𝑀𝑀 � (𝒌𝒌) (20)
Since the mask is zero by design for values of k > q/2, and unity at the origin, some of the terms will disappear:

𝐼𝐼 ̃𝑞𝑞(𝒌𝒌, 𝜙𝜙) = 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 + 𝑔𝑔 �(𝒌𝒌 + 𝒒𝒒)𝑀𝑀 � (𝒌𝒌) + 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑔𝑔 �(𝒌𝒌)𝑀𝑀 � (𝒌𝒌) + 𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑔𝑔 �(𝒌𝒌 + 2𝒒𝒒)𝑀𝑀 � (𝒌𝒌) (21)
To simplify the expression, the same definition as for the side-band extraction, Eq. ( 7), can be exploited to describe the different parts of the gain function:

𝐼𝐼 ̃𝑞𝑞(𝒌𝒌, 𝜙𝜙) = 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 + 𝑔𝑔 � 𝑞𝑞 (𝒌𝒌) + 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑔𝑔 � 0 (𝒌𝒌) + 𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑔𝑔 � 2𝑞𝑞 (𝒌𝒌) (22)
The three contributions are illustrated in Figure 1b which we will be calling the low, medium and high-frequency components of the gain variation, bearing in mind of course that their actual frequency range will depend on the carrier frequency q. In real-space the side-band becomes:

𝐼𝐼 𝑞𝑞 (𝒓𝒓, 𝜙𝜙) = 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 + 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑔𝑔 0 (𝒓𝒓) + 𝑔𝑔 𝑞𝑞 (𝒓𝒓) + 𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑔𝑔 2𝑞𝑞 (𝒓𝒓) (23)
The low-frequency term, 𝑔𝑔 0 (𝒓𝒓), is real with no imaginary part, but the medium, 𝑔𝑔 𝑞𝑞 (𝒓𝒓), and highfrequency, 𝑔𝑔 2𝑞𝑞 (𝒓𝒓), terms are in general complex.

q 2q 0 q 0 -q a b
The side-band (and centre-band) can in turn be described in terms of the reconstructed amplitude, 𝐴𝐴(𝒓𝒓, 𝜙𝜙), and phase, Φ(𝒓𝒓, 𝜙𝜙), defined by: 𝐼𝐼 𝑞𝑞 (𝒓𝒓, 𝜙𝜙) ≝ 𝐴𝐴(𝒓𝒓, 𝜙𝜙)𝑒𝑒 𝑖𝑖Φ(𝒓𝒓,𝑖𝑖) (

where the capitals emphasise their difference from the input, or original, amplitude and phase. To evaluate the variation from the perfect detector case where the recorded side-band would equal the signal side-band, we introduce the normalised functions such that:

𝐼𝐼 ̂𝑞𝑞(𝒓𝒓, 𝜙𝜙) = 𝐼𝐼 𝑞𝑞 (𝒓𝒓, 𝜙𝜙) 𝑆𝑆 𝑞𝑞 (𝒓𝒓, 𝜙𝜙) ⁄ ( 25
)
and for the amplitudes and phases, respectively:

𝐴𝐴(𝒓𝒓, 𝜙𝜙) = Amp�𝑆𝑆 𝑞𝑞 (𝒓𝒓, 𝜙𝜙)� × Amp�𝐼𝐼 ̂𝑞𝑞(𝒓𝒓, 𝜙𝜙)� ≡ 𝑎𝑎𝐴𝐴 ̂(𝒓𝒓, 𝜙𝜙) (26)
and

Φ(𝒓𝒓, 𝜙𝜙) = Phase�𝑆𝑆 𝑞𝑞 (𝒓𝒓, 𝜙𝜙)� + Phase�𝐼𝐼 ̂𝑞𝑞(𝒓𝒓, 𝜙𝜙)� ≡ 𝜙𝜙 + Φ � (𝒓𝒓, 𝜙𝜙) (27)
For both the side-bands and the centre-band, the amplitude of the normalised function gives the relative variation, or percentage error, in the amplitude compared to the input signal and the phase of the normalised side-band gives directly the phase error. We will be assuming that the gain variations are small and hence the amplitude of the normalised function be close to one and the imaginary part close to zero. In that case, the corresponding real and imaginary parts will equal the amplitude and phase, respectively, to a very good approximation:

𝐴𝐴 ̂(𝒓𝒓, 𝜙𝜙) ≅ 𝑅𝑅𝑒𝑒�𝐼𝐼 ̂𝑞𝑞(𝒓𝒓, 𝜙𝜙)� (28)
and Φ � (𝒓𝒓, 𝜙𝜙) ≅ 𝐼𝐼𝐼𝐼�𝐼𝐼 ̂𝑞𝑞(𝒓𝒓, 𝜙𝜙)� (29)

Side-band analysis

For the side-band, the result is:

𝐼𝐼 ̂𝑞𝑞(𝒓𝒓, 𝜙𝜙) = 1 + 𝑔𝑔 0 (𝒓𝒓) + 𝑎𝑎 -1 𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑔𝑔 𝑞𝑞 (𝒓𝒓) + 𝑒𝑒 -2𝑖𝑖𝑖𝑖 𝑔𝑔 2𝑞𝑞 (𝒓𝒓) (30)
Firstly, the function is clearly complex and will have an effect on both the reconstructed amplitude and phase. Secondly, and perhaps even more crucially, this effect will depend on the phase of the input hologram, 𝜙𝜙. To understand better, we need to look at the real and imaginary parts:

𝑅𝑅𝑒𝑒�𝐼𝐼 ̂𝑞𝑞(𝒓𝒓, 𝜙𝜙)� = 1 + 𝑔𝑔 0 (𝒓𝒓) + 𝑎𝑎 -1 𝑅𝑅𝑒𝑒�𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑔𝑔 𝑞𝑞 (𝒓𝒓)� + 𝑅𝑅𝑒𝑒�𝑒𝑒 -2𝑖𝑖𝑖𝑖 𝑔𝑔 2𝑞𝑞 (𝒓𝒓)� (31)
𝐼𝐼𝐼𝐼�𝐼𝐼 ̂𝑞𝑞(𝒓𝒓, 𝜙𝜙)� = 𝑎𝑎 -1 𝐼𝐼𝐼𝐼�𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑔𝑔 𝑞𝑞 (𝒓𝒓)� + 𝐼𝐼𝐼𝐼�𝑒𝑒 -2𝑖𝑖𝑖𝑖 𝑔𝑔 2𝑞𝑞 (𝒓𝒓)� (32)
where we have used the fact that 𝑔𝑔 0 (𝒓𝒓) is real.

Both the amplitude and the phase depend on the original phase of the hologram, 𝜙𝜙, except for the low-frequency term present in the amplitude. A subtler feature is that the relative weighting of the terms depends on the input amplitude of the fringes, a. As the visibility improves, the mediumfrequency 𝑔𝑔 𝑞𝑞 (𝒓𝒓) term diminishes in influence compared with the low 𝑔𝑔 0 (𝒓𝒓) and high-frequency

𝑔𝑔 2𝑞𝑞 (𝒓𝒓) terms.
The interim conclusion is that the usual procedure of subtracting the phase of a reference hologram to correct the systematic errors will run into problems, as we will show later in detail.

Centre-band analysis

The result for the centre-band, whether normalised or not, is:

𝐼𝐼 0 (𝒓𝒓, 𝜙𝜙) = 𝐼𝐼 ̂0(𝒓𝒓, 𝜙𝜙) = 1 + 𝑔𝑔 0 (𝒓𝒓) + 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑔𝑔 -𝑞𝑞 (𝒓𝒓) + 𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑔𝑔 𝑞𝑞 (𝒓𝒓) = 1 + 𝑔𝑔 0 (𝒓𝒓) + 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑔𝑔 𝑞𝑞 (𝒓𝒓) * + 𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑔𝑔 𝑞𝑞 (𝒓𝒓) = 1 + 𝑔𝑔 0 (𝒓𝒓) + 2𝑎𝑎𝑅𝑅𝑒𝑒�𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑔𝑔 𝑞𝑞 (𝒓𝒓)� (33)
Again, we see that the phase of the hologram will change the contribution from the gain but is not influenced by the high-frequency gain variations.

Pi-shifted holograms

Side-band analysis

The results for the pi-shifted holograms are the following for the side-band:

𝐼𝐼 ̂𝑞𝑞 (-) (𝒓𝒓, 𝜙𝜙) = 1 + 𝑔𝑔 0 (𝒓𝒓) + 𝑒𝑒 -2𝑖𝑖𝑖𝑖 𝑔𝑔 2𝑞𝑞 (𝒓𝒓) (34)
where the normalisation is carried out with respect to the signal, Eq. ( 25), which is twice as strong as for a single hologram. If we compare with the result from a single hologram, Eq. ( 30), we can see that the medium-frequency noise term has been eliminated. This term came from the convolution of the signal centre-band with the gain variation, the influence of the centre-band being eliminated from the pi-shifted hologram.

The real and imaginary parts are the following:

𝑅𝑅𝑒𝑒 �𝐼𝐼 ̂𝑞𝑞 (-) (𝒓𝒓, 𝜙𝜙)� = 1 + 𝑔𝑔 0 (𝒓𝒓) + 𝑅𝑅𝑒𝑒�𝑒𝑒 -2𝑖𝑖𝑖𝑖 𝑔𝑔 2𝑞𝑞 (𝒓𝒓)� (35)
𝐼𝐼𝐼𝐼 �𝐼𝐼 ̂𝑞𝑞 (-) (𝒓𝒓, 𝜙𝜙)� = 𝐼𝐼𝐼𝐼�𝑒𝑒 -2𝑖𝑖𝑖𝑖 𝑔𝑔 2𝑞𝑞 (𝒓𝒓)� (36)
For completeness, we will include the result for the addition of the pi-shifted side-bands, which in the ideal case would produce zero (see Eq. ( 14)). However, in the presence of gain variations a non-zero result is obtained:

𝐼𝐼 ̂𝑞𝑞 (+) (𝒓𝒓, 𝜙𝜙) = 𝑎𝑎 -1 𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑔𝑔 𝑞𝑞 (𝒓𝒓) (37)
In this case, the result has been normalised with respect to the side-band signal. The mediumfrequency noise has not been eliminated by the pi-shifting algorithm but displaced.

Centre-band analysis

The centre-band of the pi-shifted holograms should give a uniform distribution in the absence of noise, Eq. ( 15), but now will give:

𝐼𝐼 ̂0 (-) (𝒓𝒓, 𝜙𝜙) = 2𝑎𝑎𝑅𝑅𝑒𝑒�𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑔𝑔 𝑞𝑞 (𝒓𝒓)� (38)
and

𝐼𝐼 ̂0 (+) (𝒓𝒓, 𝜙𝜙) = 1 + 𝑔𝑔 0 (𝒓𝒓) (39)
The centre-band result has no imaginary part, as must be the case given the masks used (Eq. ( 8)).

Even for the pi-shifted holograms will be affected by the original phase of the signal but to a lesser degree.

Pixel displacements

After having analysed gain variations, we will now consider the fact that the pixels of a detector never lie on a perfectly regular square array. Their effective positions are slightly displaced from their ideal positions, either because of the fibre optic coupling or the lay-out of the detection electronics [START_REF] Coleman | Imaging characteristics of rigid coherent fiber optic tapers[END_REF]. These are known to produce phase variations [START_REF] Ruijter | Detection limits in quantitative off-axis electron holography[END_REF][START_REF] Voelkl | Approaching routine 2π/1000 phase resolution for off-axis type holography[END_REF] but the consequences of highfrequency components from abrupt pixel-to-pixel displacements have not been fully appreciated.

We can model these distortions as a vector displacement field, u(r). Following the same argument as for geometric phase analysis [START_REF] Hÿtch | Quantitative measurement of displacement and strain fields from HREM micrographs[END_REF], the recorded intensity is related to the input signal in the following way:

𝐼𝐼(𝒓𝒓) = 𝑆𝑆(𝒓𝒓 -𝒖𝒖(𝒓𝒓)) (40)
For the perfect hologram fringes, this becomes:

𝐼𝐼(𝒓𝒓, 𝜙𝜙) = 1 + 𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑒𝑒 -2𝜋𝜋𝑖𝑖𝒒𝒒.𝒖𝒖(𝒓𝒓) 𝑒𝑒 2𝜋𝜋𝑖𝑖𝒒𝒒.𝒓𝒓 + 𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑒𝑒 2𝜋𝜋𝑖𝑖𝒒𝒒.𝒖𝒖(𝒓𝒓) 𝑒𝑒 -2𝜋𝜋𝑖𝑖𝒒𝒒.𝒓𝒓 (41)

𝑤𝑤here the familiar geometric phase terms appear. At this point, it is interesting to note that the centre-band is not directly affected by the displacement field, contrary to gain variations. In Fourier space, only the two side-bands will be convoluted with the displacement field distribution.

This has consequences for the pi-shifting algorithm later on.

The displacements will be considered small, small compared to the pixel spacing, so to a good approximation:

𝐼𝐼(𝒓𝒓, 𝜙𝜙) = 𝑆𝑆(𝒓𝒓, 𝜙𝜙) -2𝜋𝜋𝜋𝜋𝒒𝒒. 𝒖𝒖(𝒓𝒓)𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑒𝑒 2𝜋𝜋𝑖𝑖𝒒𝒒.𝒓𝒓 + 2𝜋𝜋𝜋𝜋𝒒𝒒. 𝒖𝒖(𝒓𝒓)𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑒𝑒 -2𝜋𝜋𝑖𝑖𝒒𝒒.𝒓𝒓 (42)
To simplify the appearance of the equations, we will introduce the scalar function, u, which is the component of the displacement in the direction of the carrier frequency vector, q:

𝑢𝑢(𝒓𝒓) ≝ 𝒒𝒒. 𝒖𝒖(𝒓𝒓) 𝑞𝑞 ⁄ ( 43
)
where q is the modulus of the vector q. In Fourier space, the recorded intensity then becomes:

𝐼𝐼 ̃(𝒌𝒌, 𝜙𝜙) = 𝑆𝑆 ̃(𝒌𝒌, 𝜙𝜙) -2𝜋𝜋𝜋𝜋𝑞𝑞𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑢𝑢 �(𝒌𝒌 -𝒒𝒒) + 2𝜋𝜋𝜋𝜋𝑞𝑞𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑢𝑢 �(𝒌𝒌 + 𝒒𝒒) (44)
Following the same procedure as for the gain that resulted in Eq. ( 30), we can arrive at the following relations for the normalised side-bands:

𝐼𝐼 ̂𝑞𝑞(𝒓𝒓, 𝜙𝜙) = 1 -2𝜋𝜋𝜋𝜋𝑞𝑞𝑢𝑢 0 (𝒓𝒓) + 2𝜋𝜋𝜋𝜋𝑞𝑞𝑒𝑒 -2𝑖𝑖𝑖𝑖 𝑢𝑢 2𝑞𝑞 (𝒓𝒓) (45)
Given that the low-frequency displacement function is real, the result in terms of the real and imaginary parts, and hence the measured amplitude and phase, is the following:

𝑅𝑅𝑒𝑒�𝐼𝐼 ̂𝑞𝑞(𝒓𝒓, 𝜙𝜙)� = 1 -2𝜋𝜋𝑞𝑞𝐼𝐼𝐼𝐼�𝑒𝑒 -2𝑖𝑖𝑖𝑖 𝑢𝑢 2𝑞𝑞 (𝒓𝒓)� (46) 𝐼𝐼𝐼𝐼�𝐼𝐼 ̂𝑞𝑞(𝒓𝒓, 𝜙𝜙)� = -2𝜋𝜋𝑞𝑞𝑢𝑢 0 (𝒓𝒓) + 2𝜋𝜋𝑞𝑞𝑅𝑅𝑒𝑒�𝑒𝑒 -2𝑖𝑖𝑖𝑖 𝑢𝑢 2𝑞𝑞 (𝒓𝒓)� (47)
The amplitude of the reconstructed side-band will therefore have systematic noise from the highfrequency displacement field, 𝑢𝑢 2𝑞𝑞 (𝒓𝒓), dependent on the initial phase. The phase of the reconstruction will be modified by the low-frequency component of the displacement field, 𝑢𝑢 0 (𝒓𝒓).

It is this component which is usually corrected for using a reference hologram, and accurately so,

given that this noise is independent of the initial phase of the hologram.

However, there is also noise from the high-frequency component of the displacement field, 𝑢𝑢 2𝑞𝑞 (𝒓𝒓), which will not be eliminated in general by subtracting the phase of a single reference hologram.

Contrary to gain variations, Eq. ( 30), however, the term in frequency q of the noise, 𝑢𝑢 𝑞𝑞 (𝒓𝒓), is absent. This is due to the fact that displacements do not modify the centre-band of the signal directly. However, this term does appear in the centre-band of the reconstruction:

𝐼𝐼 ̂0(𝒓𝒓, 𝜙𝜙) = 𝐼𝐼 0 (𝒓𝒓, 𝜙𝜙) = 1 -2𝜋𝜋𝜋𝜋𝑞𝑞𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑢𝑢 -𝑞𝑞 (𝒓𝒓) + 2𝜋𝜋𝜋𝜋𝑞𝑞𝑎𝑎𝑒𝑒 𝑖𝑖𝑖𝑖 𝑢𝑢 𝑞𝑞 (𝒓𝒓) (48)
Using the symmetry between conjugate components, Eq. ( 11), this simplifies to:

𝐼𝐼 ̂0(𝒓𝒓, 𝜙𝜙) = 𝐼𝐼 0 (𝒓𝒓, 𝜙𝜙) = 1 -4𝜋𝜋𝑞𝑞𝐼𝐼𝐼𝐼�𝑎𝑎𝑒𝑒 -𝑖𝑖𝑖𝑖 𝑢𝑢 𝑞𝑞 (𝒓𝒓)� (49)
As expected, the centre-band reconstruction is real and has no phase component. However, the amplitude is indeed modulated by noise from the pixel-to-pixel displacements, this time the medium-frequency component, 𝑢𝑢 𝑞𝑞 (𝒓𝒓). This noise depends on the initial phase of the fringes, just like the side-band.

There is no point in considering the pi-shifted holograms for the pixel-to-pixel distortions as we can see that the reconstructed side band, Eq. ( 45), is unaffected by a change of pi of the signal hologram. No components of the noise will therefore be eliminated using the algorithm and is a direct consequence of the centre-band of the signal not influencing the results, as mentioned previous with respect to Eq. ( 41).

Results

To test the theory, we have carried out some simulations using scripting in DigitalMicrograph (Gatan Inc.) and hologram processing with HoloDark (HREM Research Inc.), a plugin for DigitalMicrograph. The detector was modelled as having 512 by 512 pixels and illuminated by a perfect hologram with 6-pixel fringe spacing at an angle of 20° to the horizontal. The visibility of the fringes was 40% giving a fringe amplitude, a, of 0.2 and had different input phases, φ, of zero, pi/2 and pi. The detector was considered to have variable gain and pixel-to-pixel distortions but otherwise perfect. The hologram signal was simulated using Eq. ( 1) and the recorded hologram using Eq. ( 17) for the gain variations and Eq. ( 41) for the displacements.

Gain variation simulations

Random numbers with a Gaussian distribution were generated for each pixel and then fixed for all the simulations (see Figure 2a). The standard deviation of the gain variations was 5%. The low, medium and high-frequency gain components appearing in Eq. ( 23) were extracted from the Fourier transform (see Figure 2b) using Eq. ( 7) with circular cosine masks, Eq. ( 10) of radius q/2. As expected, the power spectrum of the gain (Fig. 2b) shows the characteristic white noise speckle, uniformly spread across all frequencies. We can see that the low-frequency term, 𝑔𝑔 0 (𝒓𝒓), is indeed real with no imaginary part (Fig. 2d) but that the medium, 𝑔𝑔 𝑞𝑞 (𝒓𝒓), and high-frequency, 𝑔𝑔 2𝑞𝑞 (𝒓𝒓), terms are complex (Eq. ( 23)). It is important to point out that there is no correlation between the different components, as they come from different parts of the Fourier transform, and have completely uncorrelated spatial distributions even between real and imaginary parts.

For gain variations of 5% shown here, the standard deviations of the real part of low-frequency term 𝑔𝑔 0 (𝒓𝒓) is 0.40% whereas the real and imaginary parts of 𝑔𝑔 𝑞𝑞 (𝒓𝒓), and high-frequency, 𝑔𝑔 2𝑞𝑞 (𝒓𝒓), have standard deviations of 0.28%. In fact, the standard deviation of the modulus of all the components is 0.40%, the reason for the apparent difference being that the low-frequency term has no imaginary part. The much lower variability in the components compared with the original gain comes from the mask used in the Fourier processing detailed later.

A hologram, simulated and adjusted for the gain directly using Eqs. ( 1) and ( 17), is shown in Figure 3a. The side-band was extracted using Eq. ( 7) with the same cosine mask as previously (Figs. 3c andd). The standard deviation of the amplitude is 1.5% of the average value and the phase 14.1 mrad. To test the theoretical development, the corresponding amplitude and phase (Figs. 3e and3f) were then calculated instead using Eqs. ( 31) and ( 32), respectively, and injecting the gain components calculated in Figure 2. The results of simulations and theory match to within numerical errors. to that from the hologram with an initial phase of zero (Fig. 3c and3d). Subtracting the two hologram phases would therefore increase, not eliminate, the noise from gain variations. We can easily calculate the relative phase for any other value of initial phase, φ, using Eq. ( 32).

These are shown on Figure 4 for initial phases from 0 to 2π in steps of 𝜋𝜋 4 ⁄ . The phase noise changes gradually over the cycle. Because the two terms in Eq. ( 32) cycle differently, every phase image is different. The dominant term depends on the medium-frequency component 𝑔𝑔 𝑞𝑞 (𝒓𝒓) which inverts on changing the initial phase by π. However, the other term depending on the highfrequency term 𝑔𝑔 2𝑞𝑞 (𝒓𝒓) does not.

The analysis of a simulated pi-shifted hologram, calculated from two simulated holograms using Eq. ( 6), is shown in Figure 5. The centre-band is indeed absent from the Fourier transform (Fig. 5b). The amplitude and phase variations are much smaller (Figs. 5c and5d) than in Figure 3, given the absence of terms in the theoretical expressions Eqs. ( 35) and ( 36) respectively (Figs. 5e and5f). Indeed, the standard deviations for the relative amplitude and phase are now only 0.51% and 2.9 mrad, respectively. (c,e,g) 0.51% of the amplitude; (d,f,h) 2.9 mrad in the phase.

Figures 5g and5h show the amplitude and phase determined from a pi-shifted hologram with calculated from the holograms of initial phases of π/2 and 3π/2. The phase (Fig. 5h) is exactly inverted with respect to pi-shifted hologram at zero initial phase (Fig. 5d), as expected from Eq.

(36). In this case, subtracting the two reconstructed phases would exactly double the noise.

Pixel displacement simulations

We can carry out a similar analysis for the pixel-to-pixel displacements. This time, a random displacement field was simulated having a standard deviation of 0.05 pixels in both x and y directions. The corresponding holograms were calculated using Eq. ( 41) and the same fringe visibility and carrier frequency as before. A hologram having and initial phase of zero is shown in Figure 6a. and (47), agree to within numerical errors. The standard deviations for the relative amplitude and phase are 0.29% and 5.1 mrad respectively. For an initial phase of π/2, the spatial distribution of the noise is different (Figs. 6g and6h). In fact, the amplitude noise is exactly inverted, whereas the phase appears uncorrelated. This is because only the high-frequency distortion terms inverts (Eq. ( 46)) whilst the low-frequency distortion term stays the same.

Discussion

Gain variations and pixel displacements produce phase variations which look like random noise (see Figs. 3456). However, the variations can be determined directly from the static gain and displacement distributions, through Eqs. ( 32) and (47) respectively. Unfortunately, these distributions are difficult to determine experimentally with sufficient accuracy. Gain varies from day to day and geometric distortions, though stable, are difficult to measure accurately at the scale of individual pixels. The only solution is to consider them as a fatality, similar to random noise, or try to correct them. We will begin by estimating the resulting standard deviation in the phase.

The variations of the relevant gain and displacement components will be lower than the initial standard deviation (cf Fig. 2) depending on the mask used. The reduction factor has been calculated for hard binary masks [START_REF] Chang | Performance of a direct detection camera for off-axis electron holography[END_REF]. An extension to other masks, like the cosine mask used here, shows that the variance is multiplied by a factor equal to the integral of the square of the mask in Fourier space. Some tedious integration of the cosine mask, Eq. ( 10), gives the following result for the standard deviation factor, fM:

𝑓𝑓 𝑀𝑀 = 𝑘𝑘 𝑀𝑀 � 𝜋𝜋 2 - 2 𝜋𝜋 = 0.967𝑘𝑘 𝑀𝑀 (50)
The mask radius, kM, expressed in pixels -1 , is less than 0.5 (Nyquist) and therefore the factor is always less than one for any mask. In our case, the mask radius is q/2 giving:

𝑓𝑓 𝑀𝑀 = q 2 � 𝜋𝜋 2 - 2 𝜋𝜋 = 0.484𝑞𝑞 (51)
By comparison, a hard square mask of side q would give [START_REF] Chang | Performance of a direct detection camera for off-axis electron holography[END_REF]:

𝑓𝑓 𝑀𝑀 = 𝑞𝑞 (52)
The standard deviation of the real and imaginary parts of the medium and high-frequency components are √2 smaller when taken individually. For a given standard deviation of the gain variations, σg, they follow the relation:

𝜎𝜎�𝑔𝑔 0 (𝒓𝒓)� = √2𝜎𝜎�𝑅𝑅𝑒𝑒�𝑔𝑔 𝑞𝑞 (𝒓𝒓)�� = √2𝜎𝜎�𝐼𝐼𝐼𝐼�𝑔𝑔 𝑞𝑞 (𝒓𝒓)�� = 𝑓𝑓 𝑀𝑀 𝜎𝜎 𝑔𝑔 (53)
For our fringe spacing of 6 pixels, the mask factor is 0.081 using Eq. ( 51), giving for a gain variation of 5% a standard deviation of 0.40% for the low-frequency component and 0.28% for the real and imaginary parts of the others. These values reproduce exactly the standard deviations measured from the simulations (Fig. 3).

We have already seen that the noise appears lower in the pi-shifted holograms in the presence of gain variations (cf Figs. 3 and5). Assuming that the different components are indeed uncorrelated, the standard deviation of the phase due to non-uniform gain can be estimated from Eq. ( 32) by summing the variances:

𝜎𝜎 𝑔𝑔 �Φ � � = 1 2
𝑓𝑓 𝑀𝑀 𝜎𝜎 𝑔𝑔 √𝑎𝑎 -2 + 1 (54)

The corresponding result for the pi-shifted holograms, Eq. (47), is:

𝜎𝜎 𝑔𝑔 �Φ � (-) � = 1 2
𝑓𝑓 𝑀𝑀 𝜎𝜎 𝑔𝑔 (55)

We would therefore expect the pi-shifted hologram to have noise lowered by a factor of √𝑎𝑎 -2 + 1, or in our case about 5. The actual values from the simulations are 14.1 mrad (Fig. 3) and 2.9 mrad (Fig. 4) respectively, giving a reduction of 4.9. Indeed, any improvement seen in experiments on pi-shifting could be indicative of poor gain normalisation.

For the distortion, similar analysis applied to Eq. (47) results in: Gain variations were 5%, pixel displacements 0.05, hologram fringe spacing 6 pixels, visibility 40% and cosine mask radius q/2.

Conclusions

Residual gain variations and non-uniformity in the spatial distribution of pixels will introduce amplitude and phase variations which look like random noise. They are, however, completely deterministic given complete knowledge of the detector characteristics. Analytical expressions have been given (see for example, Eqs. ( 32) and ( 47)) and estimates of the standard deviations proposed (see Eqs. ( 54) and ( 56)). Gain variations will have a much lesser effect on pi-shifted holograms, confirming if need be the interest of this technique.

With perfect knowledge of the detector in terms of gain variation and pixel distortions, these errors could be calculated and possibly removed from the reconstructed phase. The case of a varying object phase has not been analysed neither has the combination of gain and pixel distortions. It also remains to be seen if experimental methods can be developed to measure these characteristics independently with sufficient ease and accuracy. The standard procedure of using a reference hologram cannot eliminate the noise given that the phase variations depend on the precise position of the hologram fringes with respect to the detector. Only when the phase of the object and reference holograms are identical over the whole field of view will the noise be eliminated, a rather limited case.

Nevertheless, it should not be concluded that reference holograms are not useful. In fact, they are essential to correct all the other sources of systematic error in the phase. For example, projector lens distortions and biprism charging produce phase variations orders of magnitude larger than those presented here. We are saying, however, that they will not eliminate all sources of error.

Indeed, it is likely that the only sources of error remaining in holography experiments, in the vacuum at least, will be the errors from the detector whether they be CCD or DDD. In the light of these results, new procedures for camera calibration may require investigation.

Figure 1 .

 1 Figure 1. Fourier transform of image intensity described as a convolution: (a) Fourier transform of the gain variation is convoluted with the centre-band (green), positive side-band (yellow) and negative side-band (orange); (b) Contributions to positive side-band from different parts of the Fourier transform of the gain.

  The real and imaginary parts of the functions are shown in Fig.2b-h. To avoid any possible edge effects, only the central 256 square area is shown for all the Figures.

Figure 2 .

 2 Figure 2. Simulated gain variations: (a) random gain variations; (b) power spectrum of gain variations with masks indicated for low (c,d), medium (e,f) and high frequency components (g,h). Real and Imaginary parts shown for central 256 square area of detector. Grey levels (a) ±15% minimum to maximum (c-h) ±1% minimum to maximum. Standard deviations are: (a) 5% (c) 0.40% (e-h) 0.28%.

Figure 3 .

 3 Figure 3. Simulated hologram with 5% gain variations: (a) hologram intensity simulated using Eq. (17); (b) power spectrum with mask indicating side-band extraction; (c) reconstructed relative amplitude; (d) corresponding reconstructed phase (in colour); (e) theoretical relative amplitude using Eq. (31); (f) theoretical phase using Eq. (32); (g) reconstructed relative amplitude for simulation with initial phase of π/2; (h) corresponding relative phase. Grey levels (c,e,g) ±1% minimum to maxium; Colour range (d,f,h) ±50 mrad black to white. Standard deviations are: (c,e,g) 1.5% of the amplitude; (d,f,h)14.1 mrad in the phase. Also shown on Figure 3 are the amplitude and phase for a simulated hologram with initial phase of 𝜋𝜋 2 ⁄ . The amplitude and phase systematic noise (Figs. 3g and 3h respectively) are very different

Figure 4 .

 4 Figure 4. Phase noise from 5% gain variation as a function of initial phase, φ: theoretical relative phase Φ � (𝒓𝒓, 𝜙𝜙) calculated using Eq. (32). Colour range: ±50 mrad black to white.

Figure 5 .

 5 Figure 5. Simulated pi-shifted hologram with 5% gain variation: (a) pi-shifted hologram intensity; (b) power spectrum with mask indicating side-band extraction; (c) reconstructed relative amplitude; (d) corresponding reconstructed phase (in colour); (e) theoretical relative amplitude using Eq. (35); (f) theoretical phase using Eq. (36); (g) reconstructed relative amplitude for initial phase of π/2; (h) corresponding relative phase. Grey levels (c,e,g) ±1% minimum to maxium; Colour range (d,f,h) ±50 mrad black to white. Standard deviations are:

Figure 6 .

 6 Figure 6. Simulated hologram with pixel-to-pixel displacements: (a) hologram simulated using Eq. (41); (b) power spectrum with mask indicating side-band extraction; (c) reconstructed relative amplitude; (d) corresponding reconstructed phase (in colour); (e) theoretical relative amplitude using Eq. (46); (f) theoretical phase using Eq. (47); (g) reconstructed relative amplitude for initial phase of π/2; (h) corresponding relative phase. Grey levels (c,e,g) ±1% minimum to maxium; Colour range (d,f,h) ±50 mrad black to white. Standard deviations are: (c,e,g) 0.29% of the amplitude; (d,f,h) 5.1 mrad in the phase. Again, the simulation (Figs. 6c and 6d) using Eq. (41) and theory (Figs. 6e and 6f) using Eqs. (46)

Figure 7 .

 7 Figure 7. Phase errors before (raw) and after reference hologram correction (corrected), as a function of the phase-shift between object and reference holograms: (a) gain; (b) distortion.

Acknowledgements

The research leading to these results has received funding from the European Union Horizon 2020 research and innovation programme under grant agreement No. 823717 -ESTEEM3. This work has been supported by the French National Research Agency under the "Investissement d'Avenir" program reference No. ANR-10-EQPX-38-01.

The standard deviation calculated for the simulation is 5.1 mrad, which agrees with the measured value (Figure 6). A systematic increase in the phase noise with carrier frequency (but keeping the same mask size, see Eqs. ( 50) and (51)) might be indicative of high-frequency pixel displacements.

Unfortunately, other sources of noise typically increase with carrier frequency. No improvement should be visible in the pi-shifted holograms.

The question is whether these values can be improved by correction using the standard procedure of subtracting the phase of a reference hologram, given that we have shown that the variations depend on the original phase of the signal (cf Fig. 4). If all the conditions are identical, including the initial phase of the hologram, the systematic terms will all be eliminated. However, there is no certainty that the reference hologram would be acquired with the same phase, barring automation [START_REF] Gatel | Unlimited acquisition time in electron holography by automated feedback control of transmission electron microscope[END_REF]. Indeed, the hologram of a real object will have a varying phase over the field of view. Locally, the object hologram and reference hologram will not have the same phase in general.

Imagine correcting the phase using a reference hologram of phase, 𝜙𝜙 ref :

From Eq. ( 29), it follows that:

For gain variations, Eq. ( 32), this means:

Now expressing the phase difference as:

we come to the following result:

There are two terms with different behaviour. If the phase of the reference hologram is the same as the target hologram, the corrected phase is zero. This is the signature of systematic errors:

repeating an identical experiment will produce exactly the same result. However, the reference hologram is more likely to have a different phase as explained previously. If the phase difference is π, for example, then:

and the noise is far from being eliminated on correction.

The pi-shifted holograms, however, have only one term, giving the following result:

This shows that the maximum error is obtained when the two holograms are 𝜋𝜋 2 ⁄ out of phase, doubling the initial noise level from the gain (cf Figs. 5d and5h). Using the previous expressions for the standard deviations, Eq. ( 53):

and for the pi-shifted hologram:

For the distortions, Eq. ( 47), the result is similar to the pi-shifted gain variations:

The low-frequency term has been completely eliminated, leaving only the high-frequency term.

The corresponding standard deviation of the corrected phase will be:

Using the values for our simulations, we can present Eqs. ( 64), ( 65) and (67) as a function of the phase shift between object and reference hologram to demonstrate the salient features (Figure 7).

We can see that for the gain variations (Fig. 7a) the residual phases are in general much smaller for the pi-shifted holograms than for the individual holograms, corrected or otherwise, and that only very particular values of the relative phases completely eliminate the noise. For intermediate values, the noise in the corrected phase can even be bigger than for the raw holograms. The results for the pixel displacements (Fig. 7b) follow a similar trend. However, correction usually improves the phase noise because the low-frequency component is eliminated.

Of course, the actual amount of detector noise, as illustrated in Fig. 7, will depend on the mask size and the standard deviations of the gain and pixel displacements. We chose the values of 5% gain variation and 5% of the pixel spacing as being representative. These values could be the residuals after (imperfect) calibration and correction of gain and geometric distortion. Indeed, gain normalisation is not necessarily easy to perform to high accuracy for every experiment. Even DDD cameras operating in so-called counting mode, where electrons are detected individually, have a gain variation representing the fact that some pixels have a higher probability of detecting an electron than others. On the other hand, pixel displacements are likely to be smaller in DDD cameras than CCD cameras, due to the lack of fibre-optic coupling, but without experimental verification it is hard to say.


Louwanda Lakiss
email: louwanda.lakiss@ensicaen.fr

Cassandre Kouvatas

Jean-Pierre Gilson

Hristiyan A Aleksandrov

Georgi N Vayssilov

Nikolai Nesterenko

Svetlana Mintova

Valentin Valtchev
email: valentin.valtchev@ensicaen.fr

Unlocking the potential of hidden sites in FAUJASITE: new insights in a proton transfer mechanism

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The rare aluminosilicate mineral Faujasite (FAU-type structure) was discovered in 1842. [1] Its synthetic counterpart comprises two different framework compositions, the Y (Si/Al  2.5) and X (Si/Al  1.0) zeolites. A so-called ultra-stable hierarchical derivative of Y, USY (Si/Al  3.0-25), is widely used in oil refining and petrochemistry and holds much promises in biomass upgrading. [2][3][4][START_REF] Rigutto | Zeolites in Hydrocarbon Processing[END_REF][START_REF] Kulprathipanja | Zeolites in Industrial Separation and Catalysis[END_REF][START_REF] Ennaert | [END_REF][START_REF] Flanigen | Zeolites Ind. Sep. Catal[END_REF][START_REF] Guisnet | Zeolites for Cleaner Technologies[END_REF][START_REF] Martínez | [END_REF][11] Its introduction in hydrocracking and fluid catalytic cracking (FCC) processes in the early 1960's, is a defining moment in heterogeneous catalysis applied to the production of quality transportation fuels and luboils. [START_REF] Marcilly | Acido-Basic Catalysis : Application to Refining and Petrochemistry[END_REF] The importance of FCC and hydrocracking catalysts in the ongoing energy transition will remain as they are also able to process biomass-derived liquids and produce light olefins. [3] Zeolite X, Y, and their derivatives are not only used in industrially relevant catalytic and separation processes but also in emerging applications as nanotechnology, environmental protection, biochemistry, medicine… [START_REF] Pina | [END_REF][14][15][16][17][18] The three-dimensional FAU structure is built from sodalite cages (cubo-octahedral units of 24 T atoms, t-toc) linked by hexagonal prisms (double-six rings of 12 T atoms, t-hpr), generating a supercage with a diameter of 1.3 nm. [START_REF]FAU: Framework Type[END_REF] This supercage is accessible through 12 membered ring windows with a diameter of 0.73 nm (Figure S1). [START_REF]FAU: Framework Type[END_REF][START_REF] Kaduk | [END_REF] Zeolite Y develops a high specific surface area (ca. 800 m 2 g -1 ) and micropore volume (0.33 cm 3 g -1 ). Its chemical composition can be easily altered by ion-exchange and other post-synthesis treatments to increase thermal and chemical stability and fine-tune key properties. Such features explain the remarkable resilience of the "first lady" of zeolite catalysis. [3] Although the FAU-type structure is one of the most open amongst the known zeolites, only a third of its active sites, those located in the supercage, are accessible to reactants. [START_REF] Van Den Berg | Zeolites Catal[END_REF][START_REF] Chen | Molecular Transport and Reaction in Zeolites: Design and Application of Shape Selective Catalysis[END_REF][START_REF] Lakiss | [END_REF][24] The active sites analysis by probe molecules of different size and basicity shows that Brønsted acid located in sodalite cage and hexagonal prism remain inaccessible due to the restricted access through the 6 and 4 member rings windows. [25][26][27] Noteworthy, the sodalite cage and hexagonal prism are not accessible even to a molecule as small as CO. [26] Therefore, different active sites accessibility must be considered when the FAU zeolite is used as a catalyst. The post-synthesis transformation of zeolite Y into USY involves a high temperature steaming during which some framework Al is expelled to extra-framework (EFAl) positions and replaced by Si via a T-jump mechanism, decreasing the number of accessible Brønsted active sites. [27] This atomic reshuffling also generates a hierarchical material by adding mesopores to the native zeolite microporosity. [11,27] Although USY contains far fewer active sites than its Y parent, it exhibits a much higher catalytic activity. [28,29] Numerous studies attribute this superior performance to a synergy between the extraframework aluminum species (EFAl) generated during steaming [29][30][31] and vicinal Brønsted acid sites. This intimate interaction creates so-called enhanced or highly reactive acidic sites. Even though several papers reported that EFAl species increase some hydrocarbons (i-C4H10, [30] n-C5H12, [32] and n-C6H14 [33] ) cracking rates, they all failed to find a direct correlation between these species and the catalytic activity. Gounder et al. proposed that the presence of EFAl increased the confinement of reactants by decreasing the effective size of the supercage. [34] However, the EFAl, comprising mostly highly polymerized aluminum oxides, are either preferentially located on the USY external surface or often removed by acid washing. For instance, the widely studied commercial USY from Zeolyst, CBV760, contains few EFAl (Lewis acid sites), typically less than 0.1 mmol/g, although they display a higher catalytic activity than its Y parent. [START_REF] Lakiss | [END_REF] In a nutshell, the fundamentals of such an unexpectedly high catalytic activity in USY are still debated after about 60 years of academic research and impressive commercial successes. Answering this question is essential to design superior Y zeolites, able to meet the current and emerging catalytic challenges in the ongoing energy, environmental and chemical transition. We recently reported that a novel chemical treatment, the unbiased leaching of framework Si and Al by a NH4F solution, could open some sodalite cages in Y zeolite. [35] Our starting hypothesis is that partial destruction of some sodalite cages in zeolite Y might also occur during the steaming process leading to its USY derivative. Herein, a combination of advanced spectroscopic techniques and molecular modeling is employed to investigate the structural and chemical changes occurring during the USY zeolite preparation and thus to highlight the origin of its unique properties. The accessibility of the active sites in the supercage and sodalite cage, is investigated by infrared spectroscopy using probe molecules with different kinetic diameters () and proton affinities (PA) -CO (=0.38 nm, PA=598 kJ/mol), pyridine (=0.57 nm, PA=912 kJ/mol) and 2,6-Di-tertbutylpyridine (=0.83 nm, PA=983 kJ/mol). [27] The interactions between these probes and the zeolites are further investigated by solid-state NMR spectroscopy and density functional theory (DFT) modeling.

Results and discussion

Highly crystalline samples of zeolite Y and USY were used in this study (Figure S2). The physicochemical properties of two zeolites are presented in Table 1. The major differences between Y and USY zeolites are the presence of mesopores and a much higher Si/Al ratio of the latter. The mesoporosity of USY is a consequence of partial destruction of the zeolite framework, including the sodalite cages, during the steaming procedure, followed by acid leaching of extra framework aluminum.

Spectroscopic study

Infrared spectroscopy

Figure 1A shows the hydroxyls stretching vibration (4000-3000 cm -1 window) after CO adsorption to saturation (T = 100 K) on zeolite Y. The corresponding CO stretching (2250-2000 cm -1 ) is reported in Figure S3A. Upon CO adsorption, the band associated with the bridged hydroxyls located in the supercage (OHSuper) at 3650 cm -1 is shifted to 3350 cm -1 due to its interaction with CO, while those in sodalite cages (OHSod: 3540 cm -1 ) are unaffected. The acid site concentrations of the OHSuper and OHSod, as well as the OHSuper interacting with CO, are summarized in Table 2. The amount of tetrahedral aluminum (Al IV ), calculated from the Si/Al ratio determined by 29 Si MAS NMR, provided the total number of acid sites (OHSuper and OHSod). In Y, the numbers of acid sites determined by quantifying the bridged hydroxyls (OHSuper + OHSod = 3.40 mmol/g) and the tetrahedral aluminum (Al IV = 3.56 mmol/g) were very close. In USY, the number of bridged hydroxyls determined by IR is only 60 % of the value derived from 29 Si NMR due to partial amorphization (30%) and steaming. [36] The spectral evolution of USY interacting with CO (Figure 1B and Figure S3B) is similar to that of Y zeolite, i.e., only OHSuper are accessible, and the associated band shifts to 3275 cm -1 . The low-intensity band at 3670 cm -1 corresponds to extraframework -(AlOH)-species. [37,38] The total acidity probed by CO is very close to the number of OHSuper. This confirms again that, despite its small size, CO does not interact with OHSod in both Y and USY zeolites. The stretching band of OHsuper shifts upon CO adsorption (OH) on Y (300 cm -1 ) and USY (360) (Figure 1), while the associated stretching bands of adsorbed CO (CO) are 35 cm -1 and 40 cm -1 (Figure S3). As CO is indicative of the acid strength of the corresponding O-H [16] , the acid sites (OHSuper) in USY appear stronger than those of zeolite Y. It is not surprising that CO (kinetic diameter 0.38 nm) cannot access the sodalite cage of zeolite Y through the 0.28 nm 6MR window. However, for USY, it is somewhat surprising that even after partial destruction of the framework, CO cannot access a portion of sodalite cages. DFT modeling is used to further elucidate this inaccessibility; more methodological details are available in the supporting information. For USY, we consider models with two and four T-atom vacancies (Figures S4A,B) occupied by hydrogenbonded silanols (nests), preventing adsorbates from entering the sodalite cage. Each model contains a Brønsted acid site with the acidic proton oriented towards the sodalite cage and the hexagonal prism; thus it is not accessible from the adjacent supercage. One may assume that after removal of silicon centers from the hexagonal prism and formation of silicon vacancies, the acidic proton will be accessible for the guest molecules(Figures S4C,D). However, even in the four T-atoms vacancies model (Figure S5), probe molecules cannot interact directly with the OHSod and therefore be polarized or protonated. The reason for this is the blocking of the access to the bridging OHSod group by the silanols -the interatomic distance between the oxygen atoms of the first group of silanols, close to the supercage, is in the range 0.48 -0.50 nm, which is sufficiently large the Py molecule to enter (the sum of VDW radia of two O and N is 0.46 nm), but the distance between oxygens of the second group of silanols is too short, 0.42 -0.44 nm, for Py to enter. The same is valid for the CO molecules, for which the sum of the VDW radia is 0.47 nm. This modeling fits well with the experimental data of CO adsorbed on USY zeolite; it calculates that the presence of silanol nests reduces the Edeprot of bridge hydroxyls by 23 kJ/mol (more in SI). Thus, when a silanol nest is close to a bridging OH, the latter's acidity increases, in good agreement with the shifts of 360 cm -1 for the OHSuper (OH) and 40 cm -1 for CO (CO) vibrations upon CO adsorption on a Y and USY, vide supra. Si MAS NMR [23] [b] OH in supercage and sodalite cage determined using the molar absorption coefficients of the respective OH bands [39] [c]OH probed by CO using the molar absorption coefficient of the CO-OH band at 2172 cm -1 [16] [d] OH probed by pyridine and 2,6-di-tert-butylpyridine using the molar absorption coefficient of their protonated band at 1545 cm -1 and 3360 cm - 1 respectively [START_REF] Lakiss | [END_REF] Upon adsorption of Py and DTBPy, the OHsuper bands disappear gradually in both zeolites (Figures 1C,D). However, the OHSod in Y and USY behave differently. Py and DTBPy do not interact with the OHSod in Y (Figure 1C) as the 3540 cm -1 band remains intact after either Py or DTBPy adsorption after evacuation at 423 K. A weak partial interaction of OHSod with pyridine (a small negative peak at 3562 cm-1 and a broadening of the peak appear in the difference spectrum (Figure S12) is observed. The amount of these probes protonated on Y is in line with the concentration of OHSuper as well as all OH titrated by CO (Table 2). In agreement with this result, DFT modeling indicates that on Y zeolite, pyridine and 2,6-di-tertbutylpyridine are protonated only by the acidic OHSuper with binding energies of -168 kJ and -222 kJ/mol, respectively (Table S1). On the other hand, upon Py and DTBPy adsorption on USY, the OHSod band disappears, indicating a strong interaction between these probes and OHSod, Figure 1D. Moreover, the number of acidic sites probed by Py and DTBPy is close to the sum of OHSuper and OHSod, Table 2. The fraction of Py (0.14 mmol/g) and DTBPy (0.16 mmol/g) interacting with OHSod, is calculated and in good agreement with the amount of OHSod calculated by integrating its corresponding band (3540 cm -1 ) intensity. Py and DTBPy would therefore be protonated by Brønsted acid sites located in the USY sodalite cages. To evaluate the validity of the results obtained on CBV760 we have studied two more USY samples, CBV712 (Si/Al=7) and CBV720 (Si/Al=16). Similar behavior has been observed for all USY samples, no matter of different Si/Al ratios. Both the OHSuper and OHSod disappeared after pyridine adsorption (Figure S14).This is, at first glance, surprising since both molecules cannot penetrate sodalite cages. [START_REF]SOD: Framework Type[END_REF] Moreover, flipping all OHSod in the supercage to reach Py and DTBPy is impossible due to the crystalline framework's rigidity. [16] Indeed, proton flipping or jumping between two oxygen atoms surrounding the aluminum (first coordination sphere) might occur, but it is an isolated phenomenon that involves only a few of the OH situated in the sodalite cage. Proton mobility or proton hopping in zeolites was already investigated by several groups using temperature-dependent 1H NMR spectroscopy, temperature-dependent infrared spectroscopy, DFT and ab Initio calculations. [START_REF] Sarv | [END_REF][42][43][44][45] For instance, Sierka and Sauer [44] studied the dynamics of Brønsted acidic sites in MFI (ZSM-5), CHA (SSZ-13) and FAU (Y) zeolites by ab initio calculations and demonstrated that proton jumps occur in FAU and CHA structures, but to a lower extent than in ZSM-5. Such a mechanism is highly dependent on the flexibility of the structure and the temperature. They also concluded that this tunneling effect is negligible at temperatures above room temperature. Hence, a proton transfer between an acidic OHSod and a bulky probe molecule in USY should occur by an as yet unreported mechanism since a direct proton jump is impossible. DFT modeling of the interaction between Py and DTBPy with USY could provide insights to the protonation mechanism. A structural model with one Al center (generating an inward-looking bridging hydroxyl in the sodalite cage, OHSod) and two nearby T-atom vacancies is considered together with three states of the basic probes: (1) unprotonated (initial state), (2) protonated by one of the silanols near the bridging hydroxyl (intermediate state), and (3) a proton from the bridging hydroxyl is transferred to restore the neighboring silanol group (Figure 2). In such a scenario, the Brønsted acid site protonates the basic probe via a two-step proton transfer (aka proton shuttle). The binding energies of DTBPy of complexes 1, 2 and 3 are -65, -112 and -246 kJ/mol (Table S1 in SI), thus, the two steps of the proton transfer are exothermic by 47 kJ/mol and 134 kJ/mol, respectively. The same trends are found for Py protonation. The calculated Gibbs free energies of different complexes (Table S1) also suggest that the protonation of DTBPy is favorable with ΔG values of -39 and -126 kJ/mol for the first and for the second step, respectively. In addition, we searched for transitions states of the proton transfer and the calculated activation barriers for the two steps are 2 and 65 kJ/mol. One may note that DTBPy protonation by the silanol is essentially barrier less since it is easily accessible for nitrogen atom of the base molecule, while the second step requires more distant proton transfer from the bridging hydroxyl to the deprotonated silanol. In any case, the low values suggest that the proton transfer via the proton shuttle mechanism mediated by silanol group occurs easy. A model with four T-atom vacancies will accommodate a larger nest with 8 silanols (Figure S5). However, neither Py nor DTBPy can enter the sodalite cage since the silanol nest blocks its entrance, as described above. The Py and DTBPy basicities are sufficiently high to allow their protonation by a silanol from the nest. The binding energies of the protonated Py and DTBPy are -107 kJ/mol and -113 kJ/mol, respectively; in turn, the acidity of the corresponding silanol from the nest is enhanced since the remaining silanols stabilize the deprotonated oxygen by H-bonding.The calculated Gibbs free energies for the Py and DTBPy models protonated by the silanol nests are -60 and -60 kJ/mol, i.e. the protonation process is exergonic with respect to the sorption energy of molecules before protonation, 12 and -23 kJ/mol, respectively (Table S1 The model that best fits the IR results is one with two T-atom vacancies near a Brønsted acid site: Py and 2,6-DTBPy are protonated by a proton transfer via a silanol. The initial proton is taken from a silanol, and the deprotonated oxygen then accommodates a proton transferred from the most acidic bridging hydroxyl group inside the sodalite cage (Figure 3). The increased flexibility of the zeolite framework near the T vacancy also facilitates a partial reorientation of this bridging hydroxyl. Both, DFT modeling and IR spectroscopy,points out that a proton transfer occurs when a few T atoms are replaced with bridging hydroxyls. Figure 3 illustrates the proton transfer mechanism from the sodalite cage to the supercage when a probe molecule is in close proximity. The protonation through a "gate-keeper" silanol is followed by a proton transfer from a Brønsted site located in the sodalite cage to restore the silanol. In that way, the Brønsted acid site in the sodalite cage is indirectly involved during a chemical reaction. As a result, a substantial fraction of the Brønsted acid sites located in the USY sodalite cage, spectators in a conventional zeolite Y, could participate and contribute to the chemical reaction.

NMR spectroscopy

Solid-state NMR is frequently used in conjunction with IR to probe the local structure of zeolites and their hydroxyls in particular. In this study, we combined several NMR techniques to shed more light on the proton transfer in USY revealed by the IR and modeling studies.

H MAS NMR spectroscopy

In the 1 H MAS NMR spectrum of USY (Figure S6), four distinct signals appear at 1.9 ppm (H1), 3.2 ppm (H2), 4.5 ppm (H3), and 6.5 ppm (H4). H1 is commonly assigned to external or internal silanols while H2 and H3 correspond to Brønsted acid sites located in supercage and sodalite cage, respectively. [46][47][48][49] H4 corresponds to acid sites with enhanced acidity tentatively associated with isolated Brønsted sites (AlOHSi). [50] In our case, H4 is not related to any ammonium residue as the { 1 H}- 15 N CP NMR spectrum does not show any ammonium cations, in agreement with the zeolite IR spectrum where the characteristic deformation vibration band of ammonium at 1455 cm -1 is absent, Figure S7. [16] The remaining signal at 2.6 ppm corresponds to extraframework aluminum species HEFAL. 1 H-{ 27 Al } TRAPDOR MAS NMR (TRAnsfer of Population in DOubleResonance) 1 H-{ 27 Al } TRAPDOR MAS NMR [51] distinguishes between internal and external silanols by resolving the H1 (1.9 ppm) signal into two distinct ones H1' (internal silanols, 2 ppm) and H1'' (external silanols, 1.8 ppm).(Figure S8) Since silanol peaks are impacted by 27 Al irradiation during the TRAPDOR experiment, they should be close to Al species (see SI). This closeness is further highlighted by a Back-to-Back (BABA) SQ-DQ 1 H-1 H spectrum (Figure 4A), probing spatial proximities between hydrogen species through correlations reported in Table S2. Internal silanols are spatially close to acidic OH in both environments (OHSod and OHSuper), in turn, close to each other. In addition, Brønsted acid sites in the supercage are near EFAL species but also close to acidic OH hydrogen-bonded to neighboring framework oxygen. Since the BABA sequence probes distance in the 2-3 Å range, H4 can be attributed to the H bonding of an acidic OHSuper.

Proton mobility probed by 1 H-1 H magnetization exchange

To investigate the proton transfer mechanism revealed by the IR and modeling studies, the 1 H-1 H magnetization exchange in USY is monitored at mixing times of 5, 50, and 500 ms. (Figures 3B-D) The off-diagonal peaks (crosspeaks) usually originate from spin diffusion or chemical exchange between close 1 H spins. However, such magnetization transfer between 1 H species can also occur at longer distances when mobile species, like water, are involved. As our experiments are performed on dried samples (see Experimental section), the cross-peaks are not due to mobile water. Cross correlations of the magnetization exchange are reported in Table 3. At a short mixing time of 5 ms (Figure 4B), no cross-peaks appear since spin diffusion does not occur with such a short mixing time. At 50 ms (Figure 4C), a strong cross-peak appears between silanols and acidic OHSod indicating exchange between these two sites, i.e., their spatial proximity already highlighted in the BABA experiment, vide supra. With a 500 ms mixing time, a second cross-peak appears between silanols and isolated Brønsted acid sites. (Figure 4D). The long time scale (500 ms) required to observe this cross-peak implies a rather slow exchange between these species, relatively distant from each other as no corresponding cross-peak appears in the BABA spectrum. 1 H-1 H magnetization exchange in combination with BABA experiments prove that internal silanols are spatially close to acidic OHSod, favoring the proton exchange observed at the 50 ms mixing time. Moreover, CP { 1 H}- 29 Si MAS (Figure S9) and 2D CP HETCOR { 1 H}- 29 Si (Figure S10) experiments performed with contact times of 7.5 and 1 ms concur with this hypothesis. Indeed, combining 29 Si MAS, { 1 H} 29 Si CPMAS and 2D { 1 H} 29 Si HETCOR MAS NMR confirms i) the existence of Q 4 species close to Brønsted acid sites in sodalite cages, and ii) a spatial proximity between some of them and SiOH species. These conclusions are also in agreement with BABA experiments indicating that silanols are spatially close to acidic OHSod.

Discussion

The experimental data show that the unique properties of USY zeolite vs its parent sample (FAU-Y) result from several inter-related structural modifications that lead to a new property altering the catalytic performance. First, the common supercage acid sites OHSuper present in both samples, Y and USY, are stronger in USY. This was confirmed by CO adsorption and modeling. It is also well known [52] that zeolites bridged acid sites strength increases when the Si/Al ratio increases. As confirmed by infrared and NMR experiments, the OHSod, which are spectators in FAU-Y becomes accessible to probe molecules in USY via the proton transfer mechanism demonstrated above. Indeed, the presence of silanols next to Brønsted acid sites in the USY zeolite provides several advantages. It increases the accessibility of USY vs FAU-Y, toward sterically hindered molecules, as it was observed for Pyridine and 2,6-Di-tertbutylpyridine. Even if a bridging hydroxyl from the sodalite cage points to the supercage, it will remain embedded in the rigid zeolite framework, rendering molecule's protonation more difficult and therefore slower. On the other hand, a silanol, in which the hydroxyl oxygen is bound to only one T-atom, is more mobile than the oxygen from a bridging hydroxyl, which is fixed with covalent bond between two T-atoms of the zeolite framework (this can be confirmed by the higher value of the calculated bending mode of the oxygen from bridging hydroxyl, 250 cm -1 , compared to the value for the oxygen from silanol hydroxyl, 157 cm -1 ). Such a hydroxyl is, therefore, easier to access by a basic guest. This is illustrated by comparing the computed distance between the DTBPy nitrogen and the oxygen from the proton donating hydroxyl: 312 pm for adsorption on a 'classical" bridging hydroxyl in the Y zeolite supercage vs 274 pm in the defect structure of USY, i.e., about 40 pm closer (Figures S4E,F). This favorable interaction with an acid site is important not only to protonate basic probe molecules but also for reactants (Figures 2 and3) during the catalytic reaction. The NMR study clearly showed that contrariwise to FAU-Y, USY display extremely high acidic sites (H4 sites) that might correspond to isolated Brønsted sites (AlOHSi). [50] These acid sites are absent in FAU-Y. The effect of shortening (internal) the diffusion pathlengths, i.e., the additional mesoporosity introduced during stabilization, is also present, but it is not the major factor for this high performance. To exclude this possibility a homemade nanosized zeolite, nano-Y (70 nm particle size, referred to Y-70 in the original report), [53] with a similar framework (Si/Al = 1.7) composition is tested in a model catalytic reaction (isooctane cracking) under identical conditions. The isooctane conversion on USY is 7-10 times higher than its Y parent, while on nanosized Y the conversion is only twice higher. (Figure S11) Thus, as expected, the exceptional catalytic activity of USY zeolite is therefore not due to a shortened diffusion pathlength of the reactant. It results from the combined activity of three types of active sites different in nature, in strength, and location. At the same time, the catalytic activity of FAU-Y is only due to the activity of its accessible OHSuper.

Conclusions

We show that in the USY zeolite, three different types of active sitesparticipatein the protonation of guest molecules (basic probes and reactants) and are responsible for its superior catalytic cracking activity. During the steaming of zeolite Y to produce USY, mesopores are created by partial framework destruction, including a fraction of the sodalite cages. The missing T atoms are replaced by silanol nests acting as gatekeepers preventing access of molecules, even CO, inside the sodalite cages. As a result, a direct proton transfer from a Brønsted site in the sodalite cage cannot reach a basic probe molecule or a reactant. A combined IR -NMR spectroscopy coupled with DFT molecular modeling reveals a new, as yet unreported, proton transfer mechanism in USY involving a Brønsted acid site located in a sodalite cage (OHSod). So far, reputedinaccessible acid sites, located in sodalite cage partially opened, now contribute to protonate molecules confined in the supercage. This explains the superior catalytic cracking activity of USY via a proton transfer process to the silanols gatekeepers. These findings allow a better understanding of the modus operandi of the all-important USY-based catalysts, an already key catalyst in oil refining and petrochemistry. This will lead, in turn, to a more efficient fine-tuning of the catalytic properties of the FAU structure. As this structure is only a member of the family of cage-containing zeolites (LTL-, LTA-, CHA-, RHO…) these could also benefit from this new option to boost or better adjust their catalytic or other performances.

Figure 1 .

 1 Figure 1. IR spectra in the OH window of Y (A) and USY (B) zeolites after (a) activation at 723 K and (b) CO adsorption at 100 K. IR spectra of Y (C) and USY (D) after (a) activation at 723 K, (b) adsorption of Py and evacuation at 423 K under secondary vacuum, and (c) adsorption of DTBPy and evacuation at 423 K under secondary vacuum.

Figure 2 .

 2 Figure 2. Local structure showing the proton transfer from bridging OH group to 2,6-di-tertbutylpyridyne via one silanol: initial unprotonated state (A), protonated by the silanol (B), proton from the bridging OH group transferred to restore the silanol group (C).Fragments of the optimized structures shown in the left column are presented as structural formula in the right column.

Figure 3 .

 3 Figure 3. Proton transfer mechanism in a USY sodalite cage: A) FAU framework viewed along [111], B) Sodalite building unit (for clarity only the oxygen atoms in the highlighted hexagonal facet are represented) with the proton transfer via a silanol nest.

Figure 4 .

 4 Figure 4. (A)1H-1H SQ-DQ BABA spectrum of USY zeolite dried at 200°C for 12 h. The dotted horizontal lines highlight a correlation, verified by the extraction of corresponding slices of the direct dimension. 1H-1H magnetization exchange spectra of USY zeolite dried at 200°C for 12 h, with mixing times of (B) 5 ms, (C) 50 ms, and (D) 500 ms.

Table 1 .

 1 Physico-chemical properties of employed zeolite Y and USY samples.

	Zeolite	U.C.	Si/Al	[b]	VMicro (cm	3 /g)	[c]	SBET (m	2 /g)	[c]	SMeso (m 2 /g)	Brønsted sites (mmol/g)	Lewis	sites
		(Å) [a]										[c]	[d]	(mmol/g) [d]
	Y	24.7	2.3		0.34			754			60	1.30	0.04
	USY	24.2	28		0.27			805			261	0.24	0.08
	[a]refined unit cell parameter;[b]determined by deconvolution of	29 Si MAS NMR spectra;[c]determined by nitrogen sorption analyses;[d]determined
	by pyridine adsorption ɛ Brønsted at 1550 cm -1 =1.35 cm.µmol -1 , ɛ Lewis at 1450 cm -1 =1.5 cm.µmol -1 .( see supporting information for nitrogen sorption
	analyses and pore size distributions).

Table 2 .

 2 Framework Al ( 29 Si NMR spectroscopy) and acid site concentrations (IR spectroscopy) for Y and USY zeolites.

	Zeolite	Al	IV	OHSuper		OHSod		OH	CO		OH	PY	OH	2,6DTBPy
		mmol/g [a]	mmol/g	[b]	mmol/g	[b]	mmol/g	[c]	mmol/g	[d]	mmol/g [d]
	Y	3.56	1.30		2.10		1.10		1.30	1.20
	USY	0.52	0.14		0.18		0.10		0.24	0.26
	[a] Tetrahedral (framework) aluminum calculated from the Si/Al ratio determined by	29

Table 3 .

 3 Cross-correlation (in ppm) in the

		1 H	1 H magnetization exchange experiment, with corresponding hydrogen species, for 5 ms, 50 ms, and
	500 ms mixing times.
	Mixing time (ms)	Cross-correlations / ppm	Corresponding Hydrogens	Hydrogen species
	5	No correlation		-	-
	50	(1.9;4.5)		H1 / H3	SiOH / OHSod
		(1.9;4.5)		H1 / H3	SiOH / OHSod
	500	(1.9;6.5)		H1 / H4	SiOH / hydrogen-bonded acidic OHSuper

Acknowledgments

The authors acknowledge the financial support from the Industrial Chair ANR-TOTAL "NanoClean Energy" (ANR-17-CHIN-0005-01) and FEDER 18P01675. HAA is grateful for support from Bulgarian ScienceFund (contract KP-06-DO02/2), GNV acknowledges the support of the project EXTREME, funded by the Bulgarian Ministry of Education and Science, D01-76/30.03.2021, programme "European Scientific Networks". Keywords:Zeolite • Faujasite • Supercage • Brønsted acid sites • Sodalite

Author contributions

V.V. and L.L. led the overall design and direction of the project. L.L. performed the catalytic and IR experiments; C.K. performed the NMR studies; H.A.A. and G.N.V. performed the molecular modelingand participated in the analysis of the results. J-P.G., S.M. and N.N. participated in the analysis and discussion of experimental results. L.L., J-P.G. and V.V. prepared the manuscript with help from all authors.


MD Géraud Forestier
email: geraud.forestier@chu-limoges.fr

MD Antoine Dusonchet

MD François Lun

MD-PhD Grégoire Boulouis

Sudden coma onset following simultaneous bilateral carotid occlusion

Keywords:

Introduction:

Acute onset coma can have a cerebro-vascular etiology, being a common clinical feature of patients with ischemic strokes of the posterior circulation when involving both thalami, and/or the basilar perforators. Conversely, in the anterior circulation, internal carotid artery occlusion as only been reported in rare instances to be a cause for coma [START_REF] Jadhav | Management of Bilateral Carotid Occlusive Disease[END_REF][START_REF] Aburahma | Bilateral internal carotid artery occlusion: natural history and surgical alternatives[END_REF]. Mechanical thrombectomy (MT) has emerged as a first line strategy for emergent cerebral large vessel occlusion, with a compelling efficacy in reducing handicap and mortality, (3) in the anterior circulation.

Description of the case:

We report the case of a septuagenarian man with a prior history of cardiovascular events (minor stroke, coronary syndrome, atrial fibrillation and hypertension) admitted to the emergency department for the sudden onset of altered vigilance (Glasgow Coma Scale Y3V1M4), with reduced movements in the 4 limbs and right facial paresis. Emergent Brain MRI revealed bilateral internal carotid artery (ICA) occlusion and a recent infarct in the superficial territory of left middle cerebral artery (Figure 1A). No thrombolysis was initiated as there was a doubt whether the patient was taking an anticoagulant treatment, and the patient was immediately rereferred to a comprehensive stroke center for MT.

The initial angiographic runs confirmed the occlusion of the terminal segment of both ICA (Figure 1B,C). MT was successfully achieved using a stent retriever and aspiration catheter (combined technique) in 45 minutes with complete bilateral revascularization (Figure 1D,E,F Emergent bilateral anterior cerebral occlusions can be revealed with unilateral symptoms caused by embolic event and can in rare instances lead to sudden onset of coma [START_REF] Pop | Endovascular Treatment in Two Cases of Bilateral Ischemic Stroke[END_REF][START_REF] Ota | A Case of Acute Simultaneous Bilateral Internal Carotid Artery Occlusion Treated by Thrombectomy[END_REF]. They are associated with severe clinical outcomes in the absence of early recanalization. In the presented case, successful reperfusion was swiftly achieved and translated to a favourable clinical outcome. This case highlights both the need for vascular imaging for the etiological diagnosis of sudden coma, and the potential for recovery after adequate revascularization of multiple simultaneous cerebral arteries occlusions. Acute simultaneous bilateral carotid occlusion should not delay patient management and may not be a good argument to deny MT if parenchymal imaging confirms a favorable imaging profile.

FIGURE'S TITLE: Not applicable

FIGURE'S LEGEND:

Before thrombectomy: Brain MR angiography and diffusion weighted imaging (inset) demonstrating no signal in both carotids and restricted diffusion in the left middle cerebral artery territory (A). Digital subtraction angiography (B, C) shows bilateral occlusion of internal carotids (arrowheads).

After mechanical thrombectomy: Antegrade flow is restored in both carotid arteries (E, F). Diffusion weighted image demonstrates limited infarct progression on the left, and no infarct on the right (D).

  ) corresponding to modified Treatment In Cerebral Ischemia (mTICI) 3. No hemorrhagic transformation occurred, and patient evolved favorably with spectacular motor recovery. He was able to ambulate independently and presented a residual right brachiofacial paresis at discharge, with a National Institutes of Health Stroke Scale (NIHSS) score of 2.

data are made available upon reasonable request to the corresponding author. There analyses and the images; full access to all of the data; and the right to publish any and all data separate and apart from any sponsor.


Thuy Trang Bui

Hai Thuong Cao

Thao Xuan Pham

Hung Mac

Rene Grée

A New Approach for the Synthesis of Sotolon in Racemic and Enantioenriched Forms

Keywords: Sotolon, Isomerization-Mannich, Catalysis, Nickel, α-aminoester

Sotolon (3-hydroxy-4,5-dimethyl-2(5H)-furanone) has been synthezised both in racemic and enantioenriched forms by a short sequence involving intramolecular tandem isomerization -aldolisation and tandem izomerization/Mann ich reactions as key steps. Optically active Sotolon has been obtained by using (S)-N-tert-butane sulfinimine as chiral starting material.

INTRODUCTION

3-Hydroxy-4,5-dimethyl-2(5H)-furanone (Sotolon) is a powerful flavor compound found in various foods and spices, as well as in beverages including aged beers, wines and sake. It has an extremely powerful aroma with a typical smell of fenugreek, maple syrup, caramel and burnt sugar at low concentration. [1,[START_REF] Heil | Sotolon has been also found to be responsible for the characteristic odor of maple syrup urine disease: Podebrad[END_REF] Both the olfactory properties and the odour intensity are very different for the two enantiomers: in dry white wines for instance, the perception threshold of (R)-sotolon was determined to be 89 μg/l, whereas the threshold of (S)-sotolon was significantly 100 times lower with 0.8μg/l. [3] It has been proposed that, in wine, Sotolon was produced by the oxidative degradation of ascorbic acid, [4] whereas in aged sake, this molecule was formed by condensation of αketobutyrate and acetaldehyde, both being acid decomposition products of threonine. [5] Blank et al. have formulated a pathway for the formation of Sotolon via thermally induced oxidation of 4-hydroxy isoleucine and its lactones with α-dicarbonyl. [6] The results showed that methylglyoxal was the best dicarbonyl compound at a 1:10 molar ratio and aminolactone was a better precursor than its aminoacid 4-hydroxyl isoleucine. Very recently, Lanfermann et al. gave support for this pathway in cultures of Laetiporus sulphureus by using labelled derivatives as precursors. [7] 2. MATERIALS AND METHODS All reagents were obtained commercially and used without further purification. All reactions have been carried out under a nitrogen atmosphere and dry conditions. The solvents used were freshly distilled under anhydrous conditions, unless otherwise specified. The reaction mixtures have been magnetically stirred with teflon stirring bars, and the temperatures were measured externally. Reactions that require anhydrous conditions have been carried out by using oven dried (120°C, 24 h) glassware. Yields refer to chromatographically and spectroscopically

( 1 H-and 13 C-NMR) homogeneous materials, and the reactions have been monitored by thin layer chromatography (TLC), carried out on 0.25 mm Merck silica gel plates (60 F254). The eluents used were mixtures of n-pentane (Pent) and ethyl acetate (EtOAc), with detection by UV light, or a p-anisaldehyde staining solution. Acros silica gel (60, particle size 0.040-0.063 mm) was used for column chromatography. Nuclear magnetic resonance (NMR) spectra have been recorded with Bruker Avance 300 spectrometers. 1 H NMR spectra: δ (H) are given in ppm relative to tetramethylsilane (TMS), using [δ(CHCl 3 ) = 7.26 ppm] as internal reference. 13 C NMR spectra: δ (C) are given in ppm relative to TMS, using [δ (CDCl 3 ) = 77.0 ppm] as internal reference. Multiplicities were designated as singlet (s), doublet (d), triplet (t), quadruplet (q), quintuplet (qt), multiplet (m) or br (broad). Mass spectral analyses have been performed at the Centre Régional de Mesures Physiques de l'Ouest (CRMPO) in Rennes (France). The optical rotation values have been measured with a Perkin-Elmer 141 Polarimeter, at 589 nm. The concentration was reported in gram per milliliter (c, g. ml -1 ). Tandem isomerization -aldolization reaction from allylic alcohol with ethylglyoxalate using Fe(CO)5 as the catalyst. Synthesis of 3 and 4. A solution of alcohol 1 (200 mg, 2.78 mmol); ethyl glyoxalate (238 mg, 2.78 mMol) and Fe(CO) 5 (37 μL, 0.278 mMol) in THF (15 mL) was irradiated with a Philip HPK125 lamp during 1h. Solvent was removed under reduced pressure and residue was filtered on silica gel with Et 2 O as eluent to afford a diasteroisomeric mixture of aldols (2 diastereoisomers: 75/25 by 1 H NMR). These products were separated by column chromatography on silica gel with Pent/EtOAc (70/30) as eluent. The organic extracts were combined, dried (Na 2 SO 4 ) and concentrated under vacuum. The crude product was purified by chromatography on SiO 2 by using as eluent a 4:6 mixture of n-pentane and EtOAc. The two diastereoisomers 3 and 4 were isolated as colorless solids in 98% overall yield (46.8 mg). Isomer syn-3: Rf (syn-3)=0.35; Mp=116-118°C. IR ν max cm -1 : 3400 (br, m), 1700 (s), 1620 (m), 1420 (m), 1380 (m), 1220 (s), 1160 (s), 930 (m), 860 (m), 780 (m). 1 H NMR (500 MHz, CDCl 3 ) δ (ppm) 4.59 (dd, J = 4.0 Hz, J = 7.4 Hz, 1H (CHOH)); 4.27 (q, J = 7.1 Hz, 2H (CH 2 CH 3 )); 2.96 (dq, J = 4.0 Hz, J = 7.4 Hz, 1H (CHCH 3 )); 2.24 (s, 3H (COCH 3 )); 1.31 (t, J = 7.1 Hz, 3H (CH 2 CH 3 )); 1.23 (d, J = 7.1 Hz, 3H (CHCH 3 )); 13 C NMR (125 MHz, CDCl 3 ) δ (ppm) 9.9, 13.7, 27.6, 49.6, 61.5, 70.6, 172.9, 208.9. Isomer anti-4: Rf (anti-4)=0.30; Mp=122-124°C. IR ν max cm -1 : 3400 (br, m), 1760 (s), 1620 (m), 1430 (m), 1380 (m), 1210 (s), 1150 (s), 930 (m), 860 (m), 780 (m). 1 H NMR (500 MHz, CDCl 3 ) δ (ppm) 4.29 (dd, J = 5.1 Hz, J = 7.2 Hz, 1H (CHOH), 4.23 (q, J = 7.1Hz, 2H (CH 2 CH 3 )); 3.03 (dq, J = 5.1 Hz, J = 7.2 Hz, 1H (CHCH 3 )); 2.20 (s, 3H (COCH 3 )), 1.28 (d, J = 7.2 Hz, 3H (CH 2 CH 3 )); 1.13 (d, J = 7.2 Hz, 3H (CHCH 3 )); 13 C NMR (125 MHz, CDCl 3 ) δ (ppm) 12.2, 13.7, 28.4, 49.8, 61.2, 72.1, 172.9, 209.7. Tandem cyclisation and reduction of the mixture syn-3 and anti-4 using NaBH4 and BnBr. NaBH 4 (456 mg, 12 mMol) was added in 3 portions, under nitrogen at 0°C, to a solution of the mixture of αhidroxy ester syn-3 and anti-4 (1.03 g, 6.0 mMol) and BnBr (2.85 mL, 24 mMol), in MeOH (30 mL). The reaction mixture was kept under magnetic stirring at 0°C until the starting material disappeared (TLC monitoring, about 1 h). After addition of water (15 mL) and extraction with EtOAc (3x15 mL), the organic phases were combined, washed with a saturated solution aqueous of NaCl, dried (Na 2 SO 4 ), and concentrated under vacuum. The crude product was purified by chromatography on SiO 2 by using a 4:6 mixture of n-pentane and EtOAc as the eluent. The 5, mixture of more than 3 diastereoisomers, was collected in 90% overall yield (702 mg). Synthesis of Sotolon in racemic form: A solution of oxalyl chloride (152 mg, 1.2 mMol) in 10 ml of freshly distilled CH 2 Cl 2 was cooled to -78 o C, and DMSO (187 mg, 2.4 mMol) was carefully added under nitrogen atmosphere. After stirring for 15 min, a solution of hydroxylactone 5 (78 mg, 0.6 mMol) in CH 2 Cl 2 (5 ml) and Et 3 N (7.0 ml) was added successively. The cooling bath was removed and the reaction mixture was allowed to warm to room temperature and stirred for 2.5 h. The solvent was removed under reduced pressure and the residue was extracted with ethyl acetate. The extract was washed with saturated aqueous Na 2 CO 3 solution, brine and dried over anhydrous Na 2 SO 4 . After removal of the solvent under reduced pressure, the crude product was purified by silica gel chromatography by using as eluent a 4:6 mixture of n-pentane and EtOAc, sotolon was isolated in 90% yield (69.1 mg). IR ν ma x cm -1 : 3350 (br, m), 1750 (s), 1710 (s), 1680 (m), 1335 (m), 1220 (s), 1160 (s), 1065 (m), 1025 (m), 920 (m), 780 (m). 1 H NMR (500 MHz, CDCl 3 ) δ (ppm) 5.43 (s, br, 1H (CHOH)); 4.45 (dq, J = 7.1 Hz, J = 1.5 Hz, 1H (CHCO)); 1.95 (d, J = 1.5 Hz, 3H (CCH 3 )); 1.40 (d, J = 7.1 Hz, 3H (CHCH 3 )). 13 C NMR (125 MHz, CDCl 3 ) δ(ppm) 14. 2, 29.7, 60.5, 123.8, 140.8, 178.6. (S)-Sotolon: Aminolactone 7 [11] (45 mg, 0.27 mMol) was dissolved in a phosphate buffer (20 mL, 0.1 mol/L, pH 5.0) at room temperature, then methylglyoxal (20 mg, 2.7 mMol) was added. The solution was boiled for 1 h. After cooling down, water (20 mL) were added. The mixture was saturated with NaCl, then the pH was adjusted to 4 with HCl (1 mol/L) and extracted with Et 2 O. The extract was dried over Na 2 SO 4 and concentrated in vacuo. The residue was purified by silica gel chromatography by using as eluent a 4:6 mixture of n-pentane and EtOAc, (S)sotolon was isolated in 32% yield (11 mg). 1 H NMR (300 MHz, CDCl 3 ) δ (ppm) 6.0 (s, br, 1H (CHOH)); 4.83 (dq, J=6.8 Hz, J=1.5 Hz, 1H(CHCO)); 1.95 (d, J=1.5 Hz, 3H (CHCH 3 )); 1.40 (d, J=6.8 Hz, 3H (CHCH 3 )). 13 C NMR (75 MHz, CDCl 3 ) δ(ppm) 12.2, 17.8, 76.4, 118.5, 139.5, 173 (c 0.02, Et 2 O). By using the most recent data on (S)-Sotolon (+15.2), [12] this translates into 47% ee. (R)-Sotolon: The same proceduce that used for synthesis of (S)-Sotolon: lactone 8 [11] (56 mg, 0.34 mMol), methylglyoxal (244 mg, 3.4 mMol), 30 ml phosphate buffer, (R)-Sotolon was obtained in 33% yield (14,4 mg). 1 H and 13 C NMR was the same of (S)-Sotolon. HRMS: Calcd for C (c = 0.18, Et 2 O). By using the most recent data on (R)-Sotolon (-23.1), [12] this translates into 29% ee.

RESULTS AND DISCUSSION

To date, several syntheses of Sotolon and its derivatives have been reported but there are only few procedures to obtain the final product in optically active form. [8] In this paper, we describe a novel strategy toward the synthesis of Sotolon either in racemic form or as optically active derivatives. The racemic synthesis involves a tandem isomerizationaldolisation reaction from an allylic alcohol with ethyl glyoxylate to produce α-hydroxyesters (A), followed by a reduction/cyclization step to afford αhydroxylactones (B). Finally, Swern oxidation of these lactones should afford the desired racemic Sotolon (Figure 1). The second pathway involves a tandem isomerization-Mannich reaction with a chiral sulfinylimine as the electrophilic component, to produce a N-protected α-aminoester (C), followed by a reduction/cyclization to afford an aminolactone. Then, cleavage of N-protecting group can give aminolactone (D). Finally, reaction of the primary amine with an α-dicarbonyl compound should result in a Schiff base which, after double bond shift and subsequent hydrolysis, will give rise to Sotolon (Figure 1). [6,7] Figure 1 Retrosynthetic analysis for the preparation of racemic (1) and optically active Sotolon (2) Thus, for the synthesis of racemic Sotolon, the tandem isomerization-aldolisation reaction of allylic alcohol 2 with ethylglyoxylate, and using Fe(CO) 5 at 10 mol % as catalyst, afforded in excellent yield the α-hydroxy esters 3 and 4 in a 75/25 diastereoisomeric ratio (Scheme 1). The separation of these aldol products was performed easily by chromatography on silica gel. In agreement with previous results, [9] the stereochemistry of these α-hydroxyesters was established by 1 H NMR: the anti-diastereoisomer J 12 (Scheme 1) coupling constants were larger (5.1 Hz) than the syn one (4.0 Hz). However, the separation of these derivatives was not absolutely required since the two diastereoisomers could be used together directly for the next transformation.

Scheme 1. Tandem isomerization-aldolisation starting from allylic alcohol 2 and preparation of intermediates 3 and 4. The next step was the reduction of ketone to the corresponding hydroxyl group. Several agents were tried for the reaction, such as LiAlH 4 , NaBH 4 , Lselectride, Super-hydride… In every case, the reaction was not selective and afforded the desired product 5 as a complex mixture with several byproducts, such as the opened form derivatives and over reduction products of lactone 5. This result has been also observed recently by Wagner et al. [10] Scheme 2. Synthesis of racemic Sotolon The combination of NaBH 4 and BnBr was found to be the most effective agent for this reduction. The tandem reduction cyclization of the mixture of 3 and 4 with NaBH 4 in presence of BnBr in MeOH at 0°C afforded in 90% overall yield the mixture of lactones 5 (Scheme 2). Finally, the transformation of lactones 5 to Sotolon was carried out by Swern oxidation in CH 2 Cl 2 at -78 o C. This reaction afforded first an unstable intermediate 6, which isomerized immediately to racemic Sotolon (Scheme 2). The 1 H, 13 C NMR data of this product were in complete agreement with those reported in the literature. [6] For the synthesis of optically active Sotolon, (S)-Ntert-butane sulfinimine was used as an excellent chiral auxilliary. We have already reported the tandem isomerization-Mannich reaction from glyoxylate sulfinimine and allylic alcohol 2 by using the NiHCl(dppe)/MgBr 2 catalytic system, affording in excellent yield the protected β-aminoketones 7 (74%, 75/25 diastereoisomeric ratio). Separation of major syn product, followed by the reduction of this amino ketone by the NaBH 4 /BnBr combination, and cleavage of the sulfinyl group gave the two optically pure (ee>99%) aminolactones 8 and 9 in 37% and 29.6% yield respectively (Scheme 3). [11] Scheme 3. Preparation of aminolactones 8 and 9. [11] The transformation of these lactones amino hydrochlorides 8 and 9 to optically active (S)-and (R)-Sotolon was carried out with methylglyoxal in a phosphate buffer solution, following a procedure reported in the literature. [7] Reaction between each aminolactone and the dicarbonyl compound produced first the corresponding Schiff This was followed by migration of the double bond, and a final hydrolysis of this regioisomeric imine generated (S)and (R)-Sotolon in 32% and 33% yields respectively. Spectral data of these derivatives were identical to previous racemic compound. Comparison of the [α] D values with the most recent datas, [12] indicated that these derivatives were obtained in 47 % et 29 % ee's only. Since this process has been performed earlier only with racemic compounds, [7] we demonstrate here that some racemization is occurring during this step. It is interesting to remark that a slow racemization of Sotolon has been already demonstrated also in wine model solutions. [3] In conclusion, we reported herein the synthesis of Sotolon in both racemic and enantiomerically enriched forms by using tandem isomerizationaldolisation and isomerization-Mannich reactions starting from cheap and commercially available raw materials. This method can be extented to analogues of Sotolon, even in larger scale. Moreover, this hydroxyesters and -aminoesters, as well as amino lactones, which can be considered as interesting building blocks for organic and medicinal chemistry.

CONCLUSIONS

In conclusion, we reported herein the synthesis of Sotolon in both racemic and enantiomerically enriched forms by using tandem isomerizationaldolisation and isomerization-Mannich reactions starting from cheap and commercially available raw materials. This method can be extented to analogues of Sotolon, even in larger scale. Moreover, this strategy offers an efficient strategy towards new αhydroxyesters and α-aminoesters, as well as amino lactones, which can be considered as interesting building blocks for organic and medicinal chemistry.

Corresponding

Scheme 4 .

 4 Scheme 4. Synthesis of (S)-and (R)-Sotolon.

  6 H 12 NO 2 [M+H] +

	129.0868; found [M +H] + 129.0870.   20			6	.	6
	Accepted manuscript

D



  author:

			Hai Thuong Cao
			Le Quy Don Technical University
			236 Hoang Quoc Viet, Bac Tu Liem, Hanoi, Vietnam
			Email: haithuongcaok11@gmail.com; Tel: 0978945469
	A Approach for New	H	CO 2 R	tandem isomerization -aldolisation reaction			OH	tandem isomerization -Mannich reaction	H	CO 2 R
	the Synthesis of Sotolon in	O	+	OH		O	O	O	S	N	+	OH
	Racemic	and				Sotolon		tBu
	Enantioenriche
	d Forms
	Thuy Trang Bui, Hai Thuong Cao*, Thao Xuan Pham, Dinh Hung Mac, Rene Grée