Eduard R Muslimov

Gennady G Valyavin

Sergei N Fabrika

Nadezhda K Pavlycheva

Design and optimization of a dispersive unit based on cascaded volume phase holographic gratings

Keywords: volume-phase holographic grating, diffraction efficiency, spectrograph, conical diffraction

We describe a dispersive unit consisting of cascaded volume-phase holographic gratings for spectroscopic applications. Each of the gratings provides high diffractive efficiency in a relatively narrow wavelength range and transmits the rest of the radiation to the 0 th order of diffraction. The spectral lines formed by different gratings are centered in the longitudal direction and separated in the transverse direction due to tilt of the gratings around two axes. We consider a technique of design and optimization of such a scheme. It allows to define modulation of index of refraction and thickness of the holographic layer for each of the gratings as well as their fringes frequencies and inclination angles. At the first stage the gratings parameters are found approximately using analytical expressions of Kogelnik's coupled wave theory. Then each of the grating starting from the longwave sub-range is optimized separately by using of numerical optimization procedure and rigorous coupled wave analysis to achieve a high diffraction efficiency profile with a steep shortwave edge. In parallel such targets as ray aiming and linear dispersion maintenance are controlled by means of ray tracing. We demonstrate this technique on example of a small-sized spectrograph for astronomical applications. It works in the range of 500-650 nm and uses three gratings covering 50 nm each. It has spectral resolution of 6130-12548.Obtaining of the asymmetrical efficiency curve is shown with use of dichromated gelatin and a photopolymer. Change of the curve shape allows to increase filling coefficient for the target sub-range up to 2.3 times.

INTRODUCTION

We consider an optical scheme of spectrograph based on a cascade of volume-phase holographic (VPH) gratings. The concept of such an instrument uses spectral selectivity and high diffraction efficiency which are inherent to the VPH's [START_REF] Caulfield | Handbook of optical holography[END_REF] . Each of the gratings in the cascaded dispersive unit provides high diffractive efficiency in a relatively narrow wavelength range and almost doesn't affect the beams propagation outside of this region. If a few of such gratings are mounted one after another with proper inclination angles in both of tangential and sagittal planes, it appears to be possible to form a spectral image consisting of several lines and characterized by a relatively high dispersion and high illuminance. Two versions of such a spectrograph were considered by the authors before [START_REF] Muslimov | Moderate-resolution holographic spectrograph[END_REF][START_REF] Muslimov | Design and modeling of a moderate-resolution astronomic spectrograph with volumephase holographic gratings[END_REF] .

One of possible problems with this type of design consists of partial overlapping and energy losses on the edges of the spectral sub-ranges. We must note here, that precise computation of the losses represents a separate modelling problem and should be considered separately. However, the losses can be minimized due to optimization of the diffraction efficiency (DE) curve for each grating. Specifically, in the present work we study possibilities to obtain a DE dependence with sharp edges and/or asymmetric profile, which could allow to decrease the losses and overlapping. The computation practice indicates, that DE curves with these features can be observed in cases when a conical diffraction is taken into account. Also, the properties of the holographic layer and the substrate have an influence on the actual DE curve. So we propose a design algorithm, in which optimization of the optical scheme geometry and the VPH gratings properties goes in parallel.

The paper is organized as follows. In section 2 we describe the algorithm used for design and optimization of the cascaded dispersive unit. In section 3 a certain example of the spectrograph optical scheme is shown and its' geometry and image quality are analyzed. Section 4 is devoted to the VPH gratings design analysis: we consider initial conditions and target functions used for the gratings parameters optimization, and demonstrate results obtained with different holographic materials. Section 5 contains discussion of the obtained results and proposals for their further application.

DESIGN ALGORITHM

Previously optimization of the design in terms of image quality and resolution and that in terms of DE and throughput were performed separately. Groves frequency and angle of incidence in the tangential plane were varied to get the desired values of dispersion and provide longitudinal centering of the spectral images, while the gratings inclination angles in the sagittal plane were used for transverse separation of the image lines. On the other hand, such parameters as thickness of the holographic layer, refraction index modulation and slanting angle of the fringes were used to optimize the DE curve. But if additional targets like steepness of the DE curve are controlled, this separation is not possible anymore. Though the risk of overlapping exists only for grating, which are mounted in the cascade after the one after consideration. It is known from practice [START_REF] Saunders | Very fast transmissive spectrograph designs for highly multiplexed fiber spectroscopy[END_REF] that blueshift of a DE curve vertex is easier to obtain than its' redshift. So we should mount the longwave or 'red' grating on the first place in the cascade and then subsequently control steepness of the DE curve's 'blue' edge at each grating. With this assumption the optimization algorithm takes the form presented in Table 1.

Lenses design parameters -

As one can see from this table as the optimization moves along the optical path, the free parameters move from the DE targets to the optical scheme design targets. We must note here that the incident ray is defined by two spherical anglesangle with the normal (θ) and angle between the incidence plane and the plane orthogonal to the fringes (δ). We assume that the linear magnification between the collimator and camera is close to the unity and the entrance slit height is small enough to neglect it.

Further we consider in details optical design of a certain spectrograph scheme developed with use of this algorithm.

OPTICAL DESIGN ANALYSIS

Optical scheme

The spectrograph works in the longwave visible domain 500-650 nm and uses 3 VPH gratings covering the sub-ranges of 500-550, 550-600 and 600-650 nm, respectively. The entrance beam has F/# of 4. It's collimated by a triplet-type lens

Image quality

For estimation of the image quality and resolution achieved in the scheme we use the spectrograph instrument function (Fig. 2.) calculated for the entrance slit width of 25 µm. The instrument functions FWHM varies from 25.6 to 46.2µm across the spectrum. If we account for the reciprocal linear dispersion, which is 1.90-1.95 nm/mm for different wavelengths, we obtain spectral resolution equal to 6130-8651, 6776-11077 and 9491-12548 for the sub-ranges of 500-550, 550-600 and 600-650 nm, respectively.

VPH GRATINGS DESIGN ANALYSIS

Assumptions

We assume that the gratings are VPH elements recorded on a sensitive layer embedded between substrate and cover plate made of BK7 glass. We account for the dispersive properties of the materials: Sellmeier formula [START_REF] Tatian | Fitting refractive-index data with the Sellmeier dispersion formula[END_REF] is used to model the glass and Schott formula [START_REF] Bach | The Properties of Optical Glass[END_REF] for the holographic layer.

When optimizing the holographic layer parameters, we do not use the refractive index modulation value directly. Instead we operate with value of exposure and apply a model of holographic material reaction. We consider mainly dichromated gelatin (DCG), the most widespread material for fabrication of VPH's. Its' properties are studied in details, so relying on the existing data [START_REF] Newell | Optical holography in dichromated gelatin[END_REF] we model the DCG reaction in the following form:

( ),

1 1 d c e a n b E + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - ⋅ = Δ - λ (1)
This equation indicates that the ascendant branch of the exposure reaction curve can be described by an exponent. In addition, the index modulation (Δn 1 ) depends linearly on the wavelength (λ). The coefficients values found for DCG are a=1.078, b=800, c= -0.00417and d=0.01534.

Also for comparison we consider one of the modern holographic materials, namely the Bayfol™ photopolymer. Unfortunately, there is no data on change of its' dispersion with exposure, so we have to neglect it (i.e. c=0, d=1). The exposure reaction curve was studied [START_REF] Bruder | Precision holographic optical elements in Bayfol HX photopolymer[END_REF][START_REF] Zanutta | Photopolymer-based volume phase holographic grating for astronomical instrumentations[END_REF] and if we consider only its' first part free of saturation, it's possible to use equation ( 1) with a=0.023; and b=0.087. Both of the models are presented graphically on Fig. 3. Note that the exposure is given in different units, because of the difference in the initial data and the recording conditions. In contrast with the approach used before 3 , we put a special accent on steepness of the DE curve edge. So the target function used for optimization contains the curve inclination angle and DE value on a specified wavelength.

λ η λ η λ η + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = i tar x f (2)
Here x i are the free parameters given in Table 1 for each case, η is DE value on the specified wavelengths. For the simplicity reasons the λ 1-3 were taken 5 nm away from the sub-range borders (see Fig. 4) To assess the effect achieved due to optimization we integrate the DE curves over the working subrange and over the subranges of the gratings standing after it in the cascade. We obtain the following coefficients, which are approximately proportional to flux directed to the working sub-range and the losses, respectively. They correspond to the red and blue zones on Fig. To provide a simple and visual estimation of the achieved effect we also use the following coefficient, corresponding to filling of the rectangular areas (see the highlighted regions on Fig. 4)

, ) ( 1 ) 500 ( 1 min max min in ex fil K K K - - ⋅ - - ⋅ = λ λ λ (4)

Starting point

To define the initial values of VPH parameters we use analytical equations of Kogelnik's coupled wave theory [START_REF] Kogelnik | Coupled Wave Theory for Thick Hologram Gratings[END_REF] . The DE curves obtained with these equations are shown on Fig. 5. The parameters values obtained after optimization are given in Table 3 in the end of this section together with those for other cases. It is easy to see that the DE curves have

Optimized design for gratings recorded in DCG

Using the starting point described above we perform optimization taking into account conical diffraction, change of the material dispersion with exposure. In this case rigorous coupled wave analysis (RCWA) method, implemented in reticolo [START_REF] Chateau | Algorithm for the rigorous coupled-wave analysis of grating diffraction[END_REF][START_REF] Lalanne | High-order effective-medium theory of subwavelength gratings in classical mounting: application to volume holograms[END_REF] software is used. The resulting DE curves are shown on Fig. 6, while the optimized values are summarized in Table 2. The plot demonstrates that using the conical diffraction one can obtain an asymmetric DE profile with blueshifted maximum and a steep shortwave edge. This effect is clearly visible for the 'red' grating while for the shorter wavelengths it gradually decreases. However, change of the curve shape also includes growth of its' secondary maxima, which can increase undesired overlapping of the sub-ranges and losses.

Comparison with gratings recorded in photopolymer

Finally, the DE optimization was performed with use of the photopolymer model. As well as in the previous cases, we provide the DE curves (Fig. 7) and optimized values of the gratings parameters (Table 2). The results for 'blue' and 'green' gratings are similar to that obtained with DCG. The DE spectral dependence for the 'red' grating is characterized by a higher asymmetry, though the maximum position is shifted and therefore the total flux directed to the spectral image will decrease. Finally, to provide estimation of the obtained effect we summarize the coefficients defined by ( 3) and ( 4) in Table 3.

The values indicate that use of an asymmetric DE curve allows to get better separation of the sub-ranges and higher irradiation in of the spectral image. The difference of the filling coefficients is as high as 2.3 times for the 'red' grating recorded in DCG and its' starting configuration with a symmetrical profile. Then for the 'green' and 'blue' gratings it becomes less notable (1.4 and 1.1 times, respectively). The results obtained with the photopolymer are more moderate. The gain obtained for the shorter wavelengths can be explained by larger achievable modulation values, while losses in the 'red' sub-range appears due to the maximum redshift. The latter feature can be compensated by revision of the target function or application of weight coefficients to it. In addition, we should mention here that such questions as concentration and developing conditions of the holographic materials are set out of consideration and used as they are given in the corresponding sources [START_REF] Newell | Optical holography in dichromated gelatin[END_REF][START_REF] Bruder | Precision holographic optical elements in Bayfol HX photopolymer[END_REF][START_REF] Zanutta | Photopolymer-based volume phase holographic grating for astronomical instrumentations[END_REF] .

CONCLUSION

We considered an algorithm of design and optimization of a spectrograph dispersive unit consisting of three cascaded volume-phase holographic gratings. The algorithm implies subsequent optimization of the gratings geometry and holographic layer parameters starting from the longwave sub-range. The gratings diffraction efficiency curves are controlled in parallel with their dispersive and imaging properties, while the free optimizations parameters are redistributed from the first group of target to the latter ones. Ray tracing is used when considering the dispersive and image properties. The DE optimization uses Kogelnik's theory for search of the starting point and RCWA method for accurate optimization.

Optical design of a small-sized spectrograph covering the range of 500-650 nm is considered as an example. It uses three VPH gratings providing uniform dispersion along the working range and allowing to reach spectral resolution from 6130 to 12548 depending on the wavelength. The gratings parameters are optimized to provide steep shortwave edges and maximum efficiency in the working sub-ranges. It is possible to obtain an asymmetric DE profile, thus increasing the spectral image illuminance and improving the channels separation. The best results were demonstrated for the 'red' grating and the filling coefficient in this case increases by factor of 2.3.

The achieved results can be of interest for further development of spectral instruments with VPH grating. Using the proposed approach one can achieve better performance in optical schemes with coupling of a few spectral or spectral and imaging channels.

The throughput of the best existing moderate and low-resolution spectrographs are 50 -60 percent (for example, FOCAS/Subaru 13 , OSIRIS/GTC 14 , RSS/SALT [START_REF] Wolf | Project status of the Robert Stobie spectrograph near infrared instrument (RSS-NIR) for SALT[END_REF] ) and the efficiencies of even the best existing echelle-spectrometers HARPS [START_REF] Mayor | Setting R. New Standards with HARPS[END_REF] and PEPSI [START_REF] Strassmeier | PEPSI:The high-resolution ichelle spectrograph and polarimeter for the Large Binocular Telescope[END_REF] still do not exceed 10 percent. Respectively, the low-moderate resolution spectrographs have spectral resolution of 1000 when only limiting optical range can be observed (at lesser resolution the whole range is covered, at higher resolution these can not cover all the visible range), the echelle-spectrographs may cover the whole range with spectral resolution from 10000 and notably higher. Respectively, in the first case one may observe faint objects, in the second, only much brighter.

This makes the use of the cascade-VPH spectrograph more effective in studies of faint, and/or low-contrast objects. The principal argument is that at the spectral resolution of 1000 or less, one may not distinguish between nebulae emission lines and all other objects, because the equivalent spectral resolution will be 300 km/s or worse. Nebula emission is overall in the universe and the nebulae have a dispersion velocity is less than 50 km/sec. The nebula emissions cover all the visible range, besides some principal lines like NaI D 12 lines must be resolved as well.

In such a case it is even impossible to distinguish between stellar winds of hot stars [START_REF] Sholukhova | New luminous blue variables in the Andromeda galaxy[END_REF] , accretion disks surrounding black holes or neutron stars [START_REF] Fabrika | Supercritical accretion disks in ultraluminous X-ray sources and SS 433[END_REF][START_REF] Poutanen | On the association of the ultraluminous X-ray sources in the Antennae galaxies with young stellar clusters[END_REF] , cataclismic variables containing accretion disks with white dwars in close binaries systems, novae, peculiar novae, supernovae [START_REF] Darnley | M31N 2008-12a -The Remarkable Recurrent Nova in M31: Panchromatic Observations of the 2015 Eruption[END_REF][START_REF] Kurtenkov | The January 2015 outburst of a red nova in M 31[END_REF][START_REF] Tartaglia | Interacting supernovae and supernova impostors. LSQ13zm: an outburst heralds the death of a massive star[END_REF] and many other targets. It is because stellar winds or accretion disks have the dispersion about or higher than 100 -200 km/sec, some narrow-line novae or supernovae have about the same values. That is crucially new to have spectral resolution higher than 6000 (50 km/sec), and therefore our solution is able to considerably contribute to modern areas of astrophysics.

Finally, the cascade-VPH effective spectrograph in combination with a 2-m class telescope can be used in studies of exoplanets. In particular, it will be very effective instrument in hunting for scatter light from hot-Jupiter exoplanets [START_REF] Valyavin | Search for signatures of reflected light from the exoplanet HD 189733b by the method of residual dynamical spectra[END_REF] , and for spectral studies of transit events from giant planets [START_REF] Grauzhanina | Spectroscopic observations of the exoplanet WASP-32b transit[END_REF] . These studies are presently among the hottest topics of modern astrophysics.

  Holography: Advances and Modern Trends V, edited by Miroslav Hrabovský, John T. Sheridan, Antonio Fimia, Proc. of SPIE Vol. 10233, 102331L • © 2017 SPIE • CCC code: 0277-786X/17/$18 • doi: 10.1117/12.2264679

  equal to170 mm. The camera represents a double Gauss-type lens with the same focal length, which focuses the three spectral lines onto the detector plane. The length of each line is 25.6-26.5 mm and their separation is 0.9-1.1 mm. The optical scheme general view and the image format are shown on Fig.1.

Figure 1 .

 1 Figure 1.Optical design of the spectrograph with a cascade of VPH gratings: a -optical scheme, b -arrangement of the spectral images on the detector.

Figure 2 .

 2 Figure 2. Instrument functions of the spectrograph calculated for the entrance slit width equal to 25 µm.

Figure 3 .

 3 Figure 3. Dependencies of the refraction index modulation on exposure and wavelength: a -DCG, b -photopolymer.

Figure 4 .

 4 Figure 4. Diffraction efficiency curve of the 'red' VPH grating and parameters used for the target function description.

  because the conical diffraction is set out of consideration by initial assumption of the theory. Subsequently, use of the target function described above leads to redshift of the curves maxima and decrease of DE in the working subrange.

Figure 5 .

 5 Figure 5. DE curves obtained with use of the Kogelnik's theory analytical equations.

Figure 6 .

 6 Figure 6. DE curves for gratings recorded in DCG obtained with use of RCWA method.

Figure 7 .

 7 Figure 7. DE curves for grating s recorded in DCG obtained with use of RCWA method.

Table 1 .

 1 Algorithm of the VPH gratings cascade optimization.

	Step

Elements under consideration Variable parameters in the optical scheme optimization Variable parameters in the DE optimization

	1	'Red' grating, camera and collimator lenses	Camera and collimator focal length (f' cam =f' col )	Angle of incidence (θ r ) Conical diffraction angle (δ r )
					Fringes frequency (N r )
					Fringes slanting angle (β r )
					Holographic	layer
					thickness (h r )
					Exposure (E r )
	2	'Green' grating		Angle of incidence (θ g )	Conical diffraction angle (δ g )
					Fringes frequency (N g )	Fringes slanting angle (β g )
					Holographic	layer
					thickness (h g )
					Exposure (E g )
	3	'Blue' grating		Angle of incidence (θ g )	Fringes slanting angle (β g )
					Fringes frequency (N g )	Holographic	layer
					Conical diffraction angle (δ g )	thickness (h g )
					Exposure (E g )
	4	Camera	and	collimator
		lenses

Table 2 .

 2 Results of the gratings design optimization for different cases

Parameter Recording in DCG, Optimization with Kogelnik's equations Recording in DCG, Optimization with RCWA Recording in photopolymer, Optimization with RCWA blue green red blue green red blue green red

	Angle of incidence θ,°	1.83 1.07	2.71	1.83	1.07	2.71	1.63	0.07	2.50
	Conical diffraction angle δ,°	-	-	-	17.57 20	0.02	17.52 19.99 0.29
	Fringes frequency N, mm -1	1684 1478	1318 1684 1478	1318 1684	1478	1317
	Fringes slanting angle β,°	15.5 14.33 12.66 15.50 15.67 13.18 15.7	16.6	13.99
	Holographic layer	14	16	22	15.6	18.5	22.9	13.1	17.2	23.0
	thickness h, µm
	Exposure E, mJ/cm 2 ( * -mW/cm 2 )	2500 2500	2500 2500 2500	2500 0.143 * 0.097 * 0.070 *

Table 3 .

 3 Estimation of the gratings performance after optimization for different cases

	Parameter	Recording in DCG,	Recording in DCG,		Recording in
		Optimization with	Optimization with	photopolymer,
		Kogelnik's equations		RCWA		Optimization with
									RCWA
		blue green red	blue green red	blue green red
	K in	31.7	27.6	35.4	32.9	34.2	43.9	36.5	35.7	26.9
	K ex	0	2.0	5.9	0	3.7	10.0	0	4.3	5.3
	K fil	2.7	4.4	9.9	2.9	6.1	23.0	3.7	6.7	6.3

Proc. of SPIE Vol. 10233 102331L-1 Downloaded From: http://proceedings.spiedigitallibrary.org/pdfaccess.ashx?url=/data/conferences/spiep/92823/ on 06/21/2017 Terms of Use: http://spiedigitallibrary.org/ss/term

Proc. of SPIE Vol. 10233 102331L-2 Downloaded From: http://proceedings.spiedigitallibrary.org/pdfaccess.ashx?url=/data/conferences/spiep/92823/ on 06/21/2017 Terms of Use: http://spiedigitallibrary.org/ss/term

Proc. of SPIE Vol. 10233 102331L-3 Downloaded From: http://proceedings.spiedigitallibrary.org/pdfaccess.ashx?url=/data/conferences/spiep/92823/ on 06/21/2017 Terms of Use: http://spiedigitallibrary.org/ss/term

Proc. of SPIE Vol. 10233 102331L-4 Downloaded From: http://proceedings.spiedigitallibrary.org/pdfaccess.ashx?url=/data/conferences/spiep/92823/ on 06/21/2017 Terms of Use: http://spiedigitallibrary.org/ss/term

Proc. of SPIE Vol. 10233 102331L-5 Downloaded From: http://proceedings.spiedigitallibrary.org/pdfaccess.ashx?url=/data/conferences/spiep/92823/ on 06/21/2017 Terms of Use: http://spiedigitallibrary.org/ss/term

Proc. of SPIE Vol. 10233 102331L-6 Downloaded From: http://proceedings.spiedigitallibrary.org/pdfaccess.ashx?url=/data/conferences/spiep/92823/ on 06/21/2017 Terms of Use: http://spiedigitallibrary.org/ss/term

Proc. of SPIE Vol. 10233 102331L-7 Downloaded From: http://proceedings.spiedigitallibrary.org/pdfaccess.ashx?url=/data/conferences/spiep/92823/ on 06/21/2017 Terms of Use: http://spiedigitallibrary.org/ss/term

Proc. of SPIE Vol. 10233 102331L-8 Downloaded From: http://proceedings.spiedigitallibrary.org/pdfaccess.ashx?url=/data/conferences/spiep/92823/ on 06/21/2017 Terms of Use: http://spiedigitallibrary.org/ss/term

Proc. of SPIE Vol. 10233 102331L-9 Downloaded From: http://proceedings.spiedigitallibrary.org/pdfaccess.ashx?url=/data/conferences/spiep/92823/ on 06/21/2017 Terms of Use: http://spiedigitallibrary.org/ss/term

Proc. of SPIE Vol. 10233 102331L-10 Downloaded From: http://proceedings.spiedigitallibrary.org/pdfaccess.ashx?url=/data/conferences/spiep/92823/ on 06/21/2017 Terms of Use: http://spiedigitallibrary.org/ss/term

ACKNOWLEDGEMENTS

This work was supported by the Russian Science Foundation (project No. 14-50-00043). Eduard Muslimov acknowledges the support of his personal post-doc contract by the European Research council through the H2020 -ERC-STG-2015 -678777 ICARUS program. The authors also thank J.-P. Hugonin and Ph. Lalanne from Institut d'Optique for provision of the reticolo software used in this work.


Eduard Muslimov
email: eduard.muslimov@lam.fr

Kjetil Dohlen

Benoit Neichel

Emmanuel Hugot

A N Tupolev-Kai

Kazan K Marx

Keywords: aspherical surfaces, laser guide star, refocusing, zoom lens

In the present paper, we consider the optical design of a zoom system for the active refocusing in laser guide star wavefront sensors. The system is designed according to the specifications coming from the Extremely Large Telescope (ELT)-HARMONI instrument, the first-light, integral field spectrograph for the European (E)-ELT. The system must provide a refocusing of the laser guide as a function of telescope pointing and large decentring of the incoming beam. The system considers four moving lens groups, each of them being a doublet with one aspherical surface. The advantages and shortcomings of such a solution in terms of the component displacements and complexity of the surfaces are described in detail. It is shown that the system can provide the median value of the residual wavefront error of 13.8-94.3 nm and the maximum value <206 nm, while the exit pupil distortion is 0.26-0.36% for each of the telescope pointing directions.

Introduction

Anisoplanatism is one of the main limitations in widefield imaging assisted by single-conjugate adaptive optics (SCAO). It is a consequence of the turbulence volume distribution above the telescope (mainly along the first 20 km above the telescope), and it has a significant impact on the images resulting from SCAO correction, severely limiting the size of the corrected field of view. The observation of larger fields of view requires tomographic measurement and correction of the atmospheric turbulence. This technique is based on measuring several guide stars spread across the field of view. The lack of guide stars outside the galactic plane limits the sky coverage of such adaptive optics (AO) systems and calls for the use of artificial guide stars.

Artificial guide stars, referred to as laser guide stars, are created by exciting the sodium layer in the upper atmosphere with a powerful laser. The excited zone in the layer re-emits the light and can be used as a guide star for adaptive optics wavefront sensors. Some limitations appear: [START_REF] Thatte | Proc. SPIE[END_REF] With the sodium layer being at a finite-altitude, the laser-assisted AO systems suffer from a cone effect, where the part of the atmosphere crossed by the laser (a cone) is different from the part of the atmosphere crossed by the scientific object (a cylinder). Tomographic reconstruction of the atmospheric volume allows to overcome this limitation. [START_REF] Tamai | Proc. SPIE 9906[END_REF] With the sodium layer being at a finite altitude, the distance between the artificial star and the telescope varies with the telescope pointing, leading to a defocus that has to be compensated by means of dynamic refocusing systems.

In this work, we present the design of an optical system that compensates the defocus in a laser guide star wavefront sensor working on an extremely large telescope. This work uses the specifications coming from the HARMONI instrumental requirements. HARMONI [START_REF] Thatte | Proc. SPIE[END_REF] is a visible and near-infrared integral field spectrograph, providing the European-extremely large telescope (E-ELT)'s core spectroscopic capability. To get the full sensitivity and spatial resolution gain, HARMONI will work at diffraction-limited scales, thus, requiring the use of AO systems.

The laser guide star wavefront sensor (LGSWFS) is placed between the telescope's pre-focal station (PFS) and the science instrument, housed within a large cryostat. When operating with the lasers, a dichroic beam splitter is inserted into the beam, reflecting the 589-nm sodium wavelength toward the LGSWFS.

The main challenge for the design of the LGSWFS for the E-ELT comes from the large variation in back focal length (BFL), reaching several meters from the viewpoint of the telescope focus. It implies that the wavefront sensor's pre-optics design be re-thought to minimize the translational movements and the size of optical components, while minimizing aberration in a view to maintain the quality of the tomographic wavefront reconstruction. The need to accept variable laser asterism diameters according to Zenith distance is another challenge, although the difficulty of this depends on the design approach chosen.

For HARMONI, several design options for the LGSWFS have been considered. The goal of the current paper is to investigate the possibility of using the zoom lens approach for the LGS projection system of a 40-m class telescope. This technology is highly developed and commonly employed, for instance, in cinematographic zoom optics. Potentially, it can provide the control of fixed image and pupil positions, as well as the pupil size. All the initial assumptions and targeted values are summarized in Table 1 [START_REF] Thatte | Proc. SPIE[END_REF][START_REF] Tamai | Proc. SPIE 9906[END_REF][START_REF] Downing | Proc. SPIE[END_REF][START_REF] Downing | Proc. SPIE[END_REF].

A number of LGS imaging systems with focus compensation were designed and implemented on existing 8-10-m class telescopes [START_REF] Ströbele | Proc. SPIE[END_REF][START_REF] Neichel | Proc. SPIE[END_REF][START_REF] Van Dam | [END_REF], and a few designs for the 40-m class telescopes are currently under development [START_REF] Fusco | Proc. SPIE[END_REF][START_REF] Joyce | Proc. SPIE[END_REF].

For the LGS channel in the GALACSI instrument mounted at the 8-m VLT telescope [START_REF] Ströbele | Proc. SPIE[END_REF], the back focal distance varies from 93 mm to 175 mm, which corresponds to the LGS distances of 160 km and 85 km, respectively. The similar value for the LGSWFS at the Gemini telescope [START_REF] Neichel | Proc. SPIE[END_REF] is 91-182 mm. In both cases, a simple zoom system can be used for the refocusing.

A similar solution was studied for an ELT-class telescope as part of the ATLAS (Advanced Tomography with Laser for AO Systems) project [START_REF] Fusco | Proc. SPIE[END_REF]. The proposed design provides a high optical quality across the FoV -from 58.9 nm to 176.7 nm in terms of RMS wavefront error. An interesting feature of this design is the component sizes -the extracting mirror aperture is 440 × 540 mm, and the first doublet has a diameter of 400 mm.

However, as the development of the ATLAS concept, the laser tomography adaptive optics for the E-ELT has been changed considerably, including different pointing angles of the LGS sources, changes in the pre-optics position with respect to the natural guide star focus. In particular, the HARMONI LGS asterism diameter is much smaller than that of the ATLAS concept, requiring the use of a dichroic beam splitter to separate the laser light from the science light [START_REF] Zieleniewski | Proc. SPIE[END_REF][START_REF] Neichel | Proc. SPIE[END_REF]. Therefore, the design solutions made for the ATLAS are not applicable to the HARMONI case.

Other groups also considered the use of zoom lenses, with designs based on different sets of assumptions [START_REF] Kosmalski | [END_REF].

Optical design

The optical design was made in Zemax software, and the optimization and analysis procedures mentioned hereafter rely on the tools proposed by that program. After a preliminary analysis and some trials and errors, it was found that the requirements related to the incoming beam geometry, the optical system dimensions, and its components' main parameters (see Table 1) can be met by the optical system configuration shown in Figure 1. It consists of four moving lens groups and folding mirrors. The lens displacements are required for refocusing and to maintain the aberration correction. The complexity of the lens design is defined by the following factors. On the one hand, it is necessary to account for the beam displacement and also for the high f-number for the first three components. On the other hand, while the optical system operates with monochromatic light, and the FoV is relatively small, it must account for the finite sodium layer thickness. In addition, the optimization should be performed for a number of separate configurations corresponding to different distances to the sodium layer and different points along the layer depth. Each of the components represents an air-spaced doublet made of BK7 glass with an aspherical first surface. As one can note from the diagram, all the lens groups move along a significant distance during the system operation. The total displacements for each of the lens groups are 574.9 mm, 683.5 mm, 388.4 mm and 600.3 mm, respectively. Moreover, the displacement law is obviously nonlinear, and its direction changes with refocusing. These large and complex movements are considered as the main difficulty related to the chosen optical design approach. Nevertheless, the movements are achievable either with four separate motors or with a transmission mechanism like a reduction drive with a few output spindles. Compactness in the X and Z dimensions and flexibility of the design geometry allow to place all the necessary mechanics.

The first folding mirrors from ix LGS channels make up a six-facet prism. It allows to arrange six identical channels side-by-side. The second folding mirror can be rotated to maintain the image centered with a transverse movement of the chief ray. Another rotating mirror should be placed after the image plane to center the exit pupil. The rotation angles of the mirrors are 1.113° and 0.79°, respectively.

Image and wavefront quality

The optical quality provided by the system is controlled twice, namely, in the image planes and in the exit pupil plane. The spot diagram root-mean square (RMS) radii at the 0° position are 4.0 μm, 15.1 μm and 17.7 μm for the layer's center, its lower and upper edges, respectively. The corresponding values for the 45° position are 0.6, 16.5 and 8.6 μm; for 60°, they are 6.7, 7.1 and 2.0 μm. These numbers are compatible with the Airy disk radius, which is 18.8 μm.

Further, we estimate the wavefront error through the standard Zernike coefficients. The results are shown in Figure 2: the Noll notation is used, and the given layer depth is measured along the plumb line with account for the object tilt change. The tip and tilt are excluded from these plots because they eventually do not affect the wavefront quality. The defocus appears only because of the sodium layer depth and cannot be compensated with a zoom system; so it is also excluded. In the worst case, which takes place for the 0° zenith angle, the RMS wavefront error reaches 206 nm (i.e. 0.35 waves). The corresponding values for 45° and 60° are 154 and 37 nm, respectively. We should note that the influence of the LGSWFS aberrations for the field of view edges is overestimated here. In a real system, they should be weighted according to the sodium intensity distribution model used (see Ref. [START_REF] Pfrommer | [END_REF] or [14], for example).

The maintenance of the exit pupil position and size was one of the main boundary conditions during the optimization. The quality and the stability of the pupil image are illustrated in Figure 3. The footprint diagrams in the upper row show that the pupil position is stable. However, the image is characterized by a certain ellipticity, and the ellipse dimensions change for different configurations. The typical ray displacement is shown on the magnified part. Such a displacement (or difference of pupil aberrations) was measured across a grid in each case and plotted in the lower row of Figure 3. The values on these plots correspond to the maximum absolute ray displacement (i.e. displacement vector length) from a ray corresponding to the center of the FoV. Note that the pupil is located at a constant finite distance.

The quantitative estimation of the pupil size and aberrations are summarized in Table 2. Note that the distortion values given there are relative, so they are calculated for the anamorphic pupil. We should emphasize that changes in pupil diameter between the different telescope positions do not exceed 5.6% and can, therefore, be calibrated.

Aspherical surfaces analysis

All the aspherical surfaces used in the system are rotationally symmetric. Each asphere was optimized twice, first, in a separate model where a doublet with a calculated f-number was created, second, after insertion of the doublet into the zoom lens system. Different types of equations were tested and compared in terms of optimization convergence speed. It was concluded that the use of the Zernike or Q-bfs [15] polynomials brought some advantages to the optimization process. For instance, the Q-bfs polynomials show the fastest merit decrease speed in the beginning and the smallest value in the end when applied for the design and optimization of the third aspherical doublet. We should note that there is no commonly adopted method to define the asphere manufacturability, and some of the approaches proposed [START_REF] Jeu | Proc. SPIE[END_REF][START_REF] Foreman | [END_REF] are valid only for the second-order surfaces or for surfaces without inflection points. In our case, an aspherical surface complexity is estimated through its departure from the sphere (BFS) and the departure slope converted to values directly measured with an interferometer, i.e. fringes per millimeter [18,19]. All the data on the aspheres are summarized in Table 3.

Figure 4 illustrates the profile and asphericity of the third lens, which has the most complex shape.

To conclude, all the surfaces have asphericities, which can be measured with an appropriate precision. It implies that the obtained parameters are within the limits of current technology. The corresponding equations are relatively simple and can be rewritten in a unified form if necessary.

In order to illustrate the reasons leading to these high asphericities as well as to the large component LGS at zenith

LGS at 45°LGS at 60°T movements, we consider the beam changes after the third aspherical surface (on the fifth lens). The transverse displacement is 19.9 mm for the chief ray, and this value becomes even larger if the finite thickness of the sodium layer is accounted for. Figure 5A shows the beam displacements occurring while the system is refocusing. Figure 5B the changes in Zernike coefficients with refocusing (log scale is applied for a better visibility). It is clear that the difference can be as large as a few orders of magnitude. It should be kept in mind that the chart includes unavoidable aberrations of the preceding lenses. The demonstrated beam reshaping features are inherent for LGSWFS at 40-m class telescope, and they explain the design complexities described above.

Conclusions

A zoom lens system designed for use with the LGSWFS for an E-ELT instrument like is demonstrated. This system allows to compensate the LGS image defocusing when the telescope moves from a zenith position to a 60° zenith angle and the associated change in line-of-sight distance to the sodium layer from 90 km to 180 km. The corresponding change in back-focal distance compared with the natural star focal plane is from 4.65 m to 2.32 m.

The proposed system consists of four moving lens groups and folding mirrors. Two of the mirrors move as well to perform beam centering. The presented design solution clearly demonstrates the existing trade-off between complexity and performance in optical system of this type. A few advantages of the system can be noted. 1. Only axisymmetric aspherical surfaces, spheres, and planes are used that would simplify the manufacturing of the optical components. 2. For the aspheres, the maximum displacement from the best-fitting sphere does not exceed 210 μm, and the maximum clear aperture diameter is 256 mm. Both of these values are within capacities of the current technologies. 3. The system shows a relatively high optical performance. Particularly, the intermediate images are diffraction limited, and the RMS wavefront aberrations excluding defocusing and tilt do not exceed 206 nm. 4. The system width is 370 mm for one LGS channel, compatible with the reserved space at the telescope site.

However, the system has a number of significant shortcomings. 1. The system has many moving components -four translating lens doublets and two rotating mirrors. Moreover, the maximum lens displacement is as large as 684 mm, and the movement law is complex and nonlinear. This should be considered as the main disadvantage of this design. We suppose this is inherent, to some extent, for such kind of zoom systems because there are no other possibilities to compensate for the large transverse shift of the incoming beam (accounting for the assumptions made in the beginning). 2. Beams from neighboring LGS are too close to each other, which can cause overlapping on the six-facet prism. It implies that the first air gap should be increased in a more advanced design. 3. There is no real LGS image in front of the system, which would be strongly desirable for performing inlab testing. The only solution to this problem, like for the previous point, is to increase the length of the first air gap.

To conclude, the developed system consists of relatively simple optical components but is characterized by a large and complex displacement of these components. The current baseline design for the HARMONI instrument is based on an alternative approach, which makes use of more complex optical surfaces to deal with the beam transverse shift and a mirror trombone to compensate for the defocusing. The baseline will be presented in detail elsewhere. If it is found to be technologically difficult, then, the proposed concept could be considered at a more detailed level including tolerance analysis, opto-mechanical design, and other points. However, we believe that the present design study could be useful for developers of auxiliary optics for the ELT-class telescopes.

Figure 1 :Figure 2 :

 12 Figure 1: General view of the system with components movement. (A) LGS at zenith (90-km altitude), (B) LGS at 45° (127 km altitude), (C) LGS at 60° (180-km altitude). The open gray circles indicate the mirrors' rotation axes (the second mirror is hidden to show the intermediate image plane).

Figure 3 :

 3 Figure 3: Footprint diagrams and ray displacement at the exit pupil plane. (A) LGS at zenith, (B) LGS at 45°, (C) LGS at 60°.

Figure 4 :

 4 Figure 4: Third lens profile. (A) Cross-section, (B) Asphericity. The lens clear aperture is 255.2 mm.

Figure 5 :

 5 Figure 5: Change of the beam at the third aspherical surface. (A) Footprint diagram, (B) Noll Zernike coefficients.

Table 1 :

 1 Initial data and boundary conditions for the design.

	LGS
	Zenith angle (°)	0-60
	Distance (km)	80-100
	Working wavelength (nm)	589
	Telescope
	Focal length (m)	646.794
	Primary mirror diameter (m)	38
	Stop (M4) diameter (m)	2.254
	Stop inclination (°)	7.75
	Detector
	Size (pixels)	800 × 800
	Pixel size (μm)	24
	LGSWFS optics
	Design concept	Zoom system consisting of
		moving axisymmetric lenses,
		without trombone and/or
		additional collimators and
		field lenses
	Boundary conditions
	System length (m)	1.5-4
	System height (m)	<2.5
	Distance from the telescope	≥1
	focal plane (m)
	Lens groups focal length (mm)	>90
	Exit pupil diameter (mm)	19.2

Table 2 :

 2 Exit pupil stability estimation.

	Telescope

angle Maximum absolute ray displacement (μm) Maximum relative ray displacement (%) Pupil X size (mm)

		Pupil
		Y size
		(mm)
	0°49	0.26 19.18 18.11
	45°66	0.34 19.15 18.38
	60°70	0.36 19.24 18.61

Table 3 :

 3 Aspherical surface main parameters.

	Position	Equation	Max sag	Max asphericity	Clear aperture	RMS slope
	1st group	Even asphere 6th order	7.271 mm	125.1 μm	250.2 mm	6.3 fr/mm
	1st lens
	2nd group	Zernike 4th + 11th terms	1.205 mm	178.3 μm	255.2 mm	9.7 fr/mm
	1st lens
	3rd group	Q-bfs 0th-4th orders	9.377 mm	209.6 μm	93 mm	30.9 fr/mm
	1st lens
	4th group	Zernike 4th + 11th terms	4.823 mm	200 μm	145 mm	17.3 fr/mm
	1st lens
		13.5

Acknowledgments: Eduard Muslimov acknowledges the support of the European Research Council through the funding of the H2020 -ERC-STG ICARUS -678777 program. The authors also thank the HARMONI consortium for the statement of this design problem. We also thank Johan Kosmalski for the fruitful discussions and sharing data related to the pre-optics design.

Eduard Muslimov

Aix Marseille Univ, CNRS, LAM, Laboratoire d'Astrophysique de Marseille, 38 rue Frédéric Joliot-Curie, Marseille, 13388, France, eduard.muslimov@lam.fr; and Kazan National Research Technical University named after A.N. Tupolev-KAI, 10 K. Marx, Kazan, 420111, Russian Federation Eduard Muslimov is a postdoctoral researcher at LAM. He received his PhD degree in Optical Technology from the Kazan National Research Technical University in 2013. His research interests include complex optical system design, freeform optics, spectral instruments, diffractive optics, and holographic optical elements.

Kjetil Dohlen

Aix Marseille Univ, CNRS, LAM, Laboratoire d'Astrophysique de Marseille, 38 rue Frédéric Joliot-Curie, Marseille, 13388, France Kjetil Dohlen is the head of the Optics Department at LAM. He received his doctoral degree in Applied Optics from the Imperial College London in 1993. As an optical designer and a system engineer, he contributed to such projects as SPHERE instrument at VLT and SPIRE at Herschel telescope and others.

Benoit Neichel

Aix Marseille Univ, CNRS, LAM, Laboratoire d'Astrophysique de Marseille, 38 rue Frédéric Joliot-Curie, Marseille, 13388, France Benoit Neichel is a CR1 researcher at LAM. He received his PhD degree from ONERA in 2008. Then, he worked as an instrument scientist for the Gemini MCAO system (GeMS) and post-doctorant at LAM. His primary research field is adaptive optics.

Emmanuel Hugot

Aix Marseille Univ, CNRS, LAM, Laboratoire d'Astrophysique de Marseille, 38 rue Frédéric Joliot-Curie, Marseille, 13388, France Emmanuel Hugot is a French astrophysicist, head of the R&D group in optics and instrumentation at LAM. He is an ERC grantee, Co-I of two activities in the H2020 OPTICON network, CNRS bronze Medal, and MERAC prize from the European Astronomical Society. He coauthored seven patents, 16 papers, and 50 conference proceedings and, today, leads the European activity on curved detectors together with partners at CEA-LETI.


Francesca Lanata
email: francesca.lanata@ecoledubois.fr

Franck Schoefs
email: franck.schoefs@univ-nantes.fr

Probabilistic Assessment of the Efficiency of Algorithms for Crack Detection on Instrumented RC Beams: a non-Model Based Method

Keywords: Distribute fiber optic sensors, reinforced concrete, crack, Probability of Detection, Probability of False Alarm

Damage detection of instrumented structures has known large developments during the last decades. Some of the approaches are called model-based as they lie on a mechanical model of the structure and aim to model and identify the crack characteristics from monitoring. The main disadvantage of this approach for real structures is to model other effects that may change the response of the sensor: temperature, traffic loading, etc. Other approaches are called non-model based methods: they lie on the analysis of changes in the response only. Their main disadvantage is that the identification of the crack characteristics is unavailable. In parallel with the diffusion of monitoring systems, the development of decision support systems (DSS) has been started, that utilize the results of the monitoring data interpretation process for supporting decisions concerning maintenance/rehabilitation planning. The overall process, from data interpretation and damage identification to the definition of the lifetime or fragility curves and finally to decision models is however affected by several uncertainties. The talk will focus on the uncertainties of the damage identification process suggesting a probabilistic measurement of the change in the response through an approach largely used for Non Destructive Testing calibration. The reliability estimate of the damage identification algorithms will be evaluated defining the probability of detection (PoD), the probability of false alarms (PFA) and a receiver operating characteristic (ROC) curve. For a given detection threshold, the couple (PoD, PFA) defines the efficiency of a detection algorithm. The objective is to maximize the probability of detection and minimize the probability of false alarm.

The performance of the Proper Orthogonal Decomposition in detecting and localizing damage will be presented, regarding the application to long-term static monitoring data resulting from an experiment conducted in real environmental conditions on two pre-stressed concrete beam specimens. These ROC curves can be obtained by considering two or more algorithms and the same defect range, one technique and two defect ranges, one technique with two settings and the same defect range, or one technique applied in various conditions (even if the testing procedure is rigorously followed during inspection). In the following we will consider two settings (rough data and pre-processed data) and the same defect. The paper will also present a combination of several sensors measurements to illustrate the gain obtained from a redundant monitoring.

INTRODUCTION

Safety and usability of existing concrete structures actually depend on the level of degradation and obsolescence due to ageing. The time-dependent reliability problem for ageing concrete structures was discussed by several authors, among which [START_REF] Mori | Reliability-based Service-Life Assessment of Aging Concrete Structures[END_REF], [START_REF] Stewart | Time-dependent reliability of Deteriorating Reinforced Concrete Bridge Decks[END_REF] and [START_REF] Enright | Probabilistic Analysis of Resistance Degradation of Reinforced Concrete Bridge Beams under Corrosion[END_REF] have been pioneering the subject.

Reliability analysis in structural engineering has to enable quantification of uncertainties associated with loading, materials, geometry, deterioration, modelling and other factors. These are integrated into a method that estimates the probability of reaching the specified performance level during the service life of a structure [START_REF] Aktan | Performance-based engineering of constructed systems[END_REF]. Lifetime functions for concrete structures can be constructed from theoretical/experimental degradation models and, if a Structural Health Monitoring system is installed on the structure, at any required time, the actual state of degradation can also be determined by applying a damage identification process, allowing updating of the lifetime functions [START_REF] Orcesi | Optimization of Bridge Maintenance Strategies based on Structural Health Monitoring Information[END_REF]. For example, the use of monitored extreme data allows the reduction of uncertainties associated with numerical models, and the validation and updating of existing prediction models [START_REF] Frangopol | Use of monitoring extreme data for the performance prediction of structures: General approach[END_REF]. For complex structures in particular, SHM has been proved to be the only way to understand both complex interaction mechanisms, as soil-structure interaction, and inservice structural behaviour influenced by building/loading conditions and settlements that are expected to separate the current behaviour from the theoretical one [START_REF] Vallone | Monitoring of Termini Imerese harbour (Sicily, Italy) using satellite DInSAR data[END_REF] [START_REF] Schoefs | Polynomial chaos representation for identification of mechanical characteristics of instrumented structures[END_REF]. The overall process, from data interpretation and damage identification to the definition of the life-cycle curves and finally to decision models is anyway affected by several uncertainties, both stochastic and epistemic.

The SHM efficiency depends on the ability to detect the structural changes under environmental and operational conditions [START_REF] Sohn | Effects of environmental and operational variability on structural health monitoring[END_REF]. A lot of damage detection methods have been tested in the last years, on static [START_REF] Omenzetter | Identification of unusual events in multichannel bridge monitoring data[END_REF] and dynamic data [START_REF] Kim | Hybrid health monitoring of prestressed concrete girder bridges by sequential vibration-impedance approaches[END_REF], and on laboratory [START_REF] Park | Multiple crack detection of concrete structures impedance-based structural health monitoring techniques[END_REF] and in-field experiments [START_REF] Cheung | The application of statistical pattern recognition methods for damage detection to field data[END_REF] [START_REF] Benedetti | Structural health monitoring of wind towers: remote damage detection using strain sensors[END_REF]. Despite of the above interest that has been largely expressed by means of research papers in conferences and scientific journals, very few practical examples can be reported to date, the main reasons being the lack of extensive field evidence in the disclosure of structural defects and the inability to interpret the data obtained from hundreds of sensors in a timely manner. Robust and reliable methods capable of detecting, locating and estimating damage whilst being insensitive to changes in environmental and operating conditions have yet to be agreed upon.

With reference to a real case of permanent static heath monitoring systems [START_REF] Del Grosso | A Long-Term Static Monitoring Experiment on R.C. Beams using Fiber Optic Sensors[END_REF], the paper is aimed at discussing the performance of the Proper Orthogonal Decomposition as a damage detection algorithm [START_REF] Lanata | Damage detection and localization for continuous static monitoring of structures using a Proper Orthogonal Decomposition of signals[END_REF]. The damage indicators extracted from the proposed algorithm are used to set a reference period and a threshold value associated to damage presence. The uncertainties affecting the damage identification process on the assessment of the structural health state will be taken into account, in particular uncertainties due to environmental changes.

First, theoretical aspects from detection theory are reminded. Then, based on the characteristics of real monitoring data, the experimental approach is presented and the performance estimate of damage detection algorithms (DDA) for a given detection threshold is discussed through the Receiver Operating Characteristic (ROC) curve [START_REF] Schoefs | Assessment of ROC Curves for Inspection of Random Fields[END_REF]. Looking for the best detection performances, the probability of detection should always take larger values than the probability of false alarm (low noise sensitivity).

In the paper, special emphasis is given to the influence of the environmental conditions on the probability of false alarm, because this latter depends on the noise level and the chosen threshold only. On the contrary, the probability of detection depends on the damage extension. The difficulties in obtaining a good performance curve in presence of strong temperature variations (noise) and small levels of damage is also discussed and the possibility of improvement of the detection process through post-processing of acquired data is illustrated. Data from a 2-year monitoring program on pre-stressed beams is considered for illustration and different ROC curves are deduced as a function of the threshold assumed for damage identification. The discussion is based on the availability of a reference period in which the structure can be considered as undamaged.

PROBABILISTIC MODELING OF DETECTION

The objective of the following sections is to present the expansion of the probabilistic modelling of inspection results through NDT tools to the probabilistic modelling of damage detection from data processing algorithms.

Probability of Detection and Probability of False Alarm

The most common concept which characterizes the performance of inspection tools is the probability of detection PoD. Let a d be the minimal defect size, called the detection threshold, under which it is assumed that no detection is possible. PoD is defined as (Eq. ( 1)):

( )

d a D P PoD ≥ = ˆ(1)
where P() represents a probability measure, D ˆ is the variable that represents the measured defect size d ˆ (response level of NDT tool i.e. 'signal+noise'). The real defect size (i.e. the real signal without noise) is D.

Assuming that the probability density functions of noise and signal amplitude are known, after fitting empirical distribution for instance, PoD and the probability of false alarm PFA have the following expressions (Eq. ( 2) and (3)):

( )

∫ +∞ ∂ = d a D d d f PoD ˆ(2) ( ) ∫ +∞ Λ ∂ = d a f PFA η η (3)
where η is the noise, D f ˆ and Λ f are respectively the probability density functions of the 'signal+noise' D ˆ (or measured defect) and the 'noise' Λ . In the following noise Λ and signal D are considered as independent random variables. Thus, PoD is a function of the detection threshold, the measured defect size and the noise while PFA depends on the detection threshold and noise only [START_REF] Schoefs | Assessment of ROC Curves for Inspection of Random Fields[END_REF]. Noise is dependent on the decision-chain "physical measurement-decision on defect measurement transfer of information", the conditions of inspection (harsh environment, surface quality, electronic noise) and the complexity of testing procedure (accessibility, mounting of the device). Figure 1 illustrates the Probability Density Function (PDF) and the area to be computed for the evaluation of PoD and PFA for a given detection threshold in the case where 'signal+noise' D ˆ and 'noise' Λ are normally distributed.

Receiver Operating Characteristic (ROC) curve

The ROC curve links the Probability of Detection and the Probability of False Alarm. For a given detection threshold, the pair (PFA, PoD) defines NDT performance. This pair can be considered as coordinates of a point in R 2 (square integrable space of real numbers) with axes representing PFA and PoD. Let us consider that a d takes values in the range ]-∞;+∞[, these points belong to a curve called Receiver Operating Characteristic (ROC) which is a parametric curve with parameter a d and defined by Eq. ( 2) and ( 3).

The ROC curve (ROC3) plotted on Figure 2 is computed with the PDF presented on The ROC curve is a fundamental characteristic of the NDT tool performance for a given PDF of a defect or a defect range . Perfect tool is represented by a ROC curve reduced to a single point whose coordinates are: (PFA, PoD) = [0, 1]. A ROC curve represents a NDT tool performance facing a given PDF of a defect or a defect range [START_REF] Rouhan | Probabilistic modelling of inspections results for offshore structures[END_REF].

Figure 2 presents 3 theoretical ROC curves, each one corresponding to a different NDT tool performance. The worst curve is ROC1, meaning that noise can be easily detected as a defect even if nothing is to be detected. This finally leads to a high number of false alarms. As a result, overall performances are poor. In contrast, the best plotted ROC curve is ROC3, which differs considerably from the previous curve. The probability of detection reaches values near to 1, with small probabilities of false alarms for high values of PoD. Overall performances are very good.

These ROC curves can be obtained by considering two techniques and the same defect range, one technique and two defect ranges, one technique with two settings and the same defect range, or one technique applied in various conditions (even if the testing procedure is rigorously followed during inspection). In the following two settings and the same defect will be considered.

A simple geometric characterization of ROC curves is the distance between the curve and the Best Performance Point (BPP) of coordinates (PFA=0, PoD=1) [START_REF] Schoefs | Multiple inspection modeling for decision making and management of jacket off-shore platforms: effect of false alarms[END_REF]. By definition, the bigger the distance, the worse is the performance. The point on the ROC curve corresponding to the lowest distance between BPP and the curve is called the performance point of the NDT tool (NDT-BPP). This distance (Euclidean measure) can thus be considered as a measure of performance. This paper defines a curve characterization by using the polar coordinates of the NDT-BPP. The NDT-BPP polar coordinates are defined by:

• the radius NDT α equals the performance index (NDT-PI) (distance between the best performance point and the ROC curve) ( [START_REF] Schoefs | Assessment of ROC Curves for Inspection of Random Fields[END_REF][19] [START_REF] Schoefs | The αδ method for modeling expert Judgment and combination of NDT tools in RBI context: application to Marine Structures[END_REF]);

• the NDT δ is the angle between axis (PFA=0) and the line (BPP, NDT-BPP).

It has been shown in [START_REF] Barnouin | Underwater inspection reliability trials for offshore structures[END_REF] that NDT α is essential to provide complete risk analysis, including consequence assessment after inspection. However, such a study is beyond the scope of this particular study so this parameter will not be analyzed here. Assessment of PoD and PFA from the knowledge of detection threshold can be directly deduced from inter-calibration of NDT tools from statistical analysis of inspection results ( [21][22]). Generally these projects are expensive, and consequently, it is sometimes necessary to choose another approach. Calculation of PFA and PoD thereby results from probabilistic modeling of the 'noise' and 'signal+noise' PDF ([17][19]). Note that other authors [START_REF] Straub | Modelling dependency in inspection performance[END_REF] have suggested other concepts like Probability of False Indication (PFI). The main disadvantage is that Probability of False Alarm and Probability of No Detection (1-PoD) are mixed and that the size of the defect and the modeling of the noise are no more present.

Statistically based ROC building for algorithm performance assessment

Here the objective is to find the best parameters for the algorithm that lead to a good assessment of defect. ROC curves are built from statistics and then the best set of parameters that lead to the detection threshold that minimizes NDT δ are deduced. Thus PoD and PFA are computed from Eq. ( 4) and ( 5) [START_REF] Kim | Hybrid health monitoring of prestressed concrete girder bridges by sequential vibration-impedance approaches[END_REF]:

( ) { } d j p m a f i A with n A Card PoD > ℑ ∈ = ≈ , ; (4)
( ) { } d j m a i B with n B Card PFA > ℑ ∈ = ≈ η ; (5)
Where Card(.) indicates the cardinal of a particular set and where

{ } m n , = ... 1, ℑ
and m n denotes the number of measurements.

PRACTICAL ASSESSMENT USING MONITORING DATA

As briefly mentioned in the introduction, despite of the great number of damage identification approaches that have been proposed in the literature, there is an increasing need to assess the performance and the validity of the selected algorithms with a comparative tool. The optimal damage identification approach should be able to detect the smaller possible damage at the right place and at the right time.

The probability of detection and the probability of false alarm are both needed to characterize, even partially, the results of a DDA. All algorithms have limitations and, in particular in complex environment and harsh conditions, their capabilities and abilities to well perform are different from those given by numerical simulations and laboratories sets. This leads to lower performances than theoretically expected.

In the following, definitions taken from the assessment of non-destructive testing performance are used. A parallel approach is used to evaluate the performance of a damage detection algorithm already applied by the Authors ( [START_REF] Del Grosso | A Long-Term Static Monitoring Experiment on R.C. Beams using Fiber Optic Sensors[END_REF][16]) on numerical simulation and experimental data. As previously mentioned, the inspection results from a NDT are herein substituted by the damage identification results from the Proper Orthogonal Decomposition.

More generally in this framework, the variables D ˆ and a d in Eq. ( 1) can represent the observed current value and the threshold value of one or more features extracted from the measurements by the particular damage identification algorithm. In this case, the threshold value a d should be defined observing the variation of the feature/s for a reference period during which the structure is supposed to be undamaged. The length of the reference period depends on the available information and on the periodicity of the time histories as shown in [START_REF] Posenato | Model Free Data Interpretation for Continuous Monitoring of Complex Structures[END_REF].

As the definition of the threshold value in the case of damage detection algorithm is not easy to achieve, a parametric study using a vector c of threshold values is suggested. The threshold can be for example defined using an adequate confidence interval assessed from the observations of the reference period. The best threshold value has to be set for each tested DDA.

When assessing the ROC curve using experimental data, discrete values for PFA and PoD are obtained for given conditions.

Presentation of the experiment and data analysis

An experimental monitoring program, carried out by the Authors in cooperation with Autostrade per l'Italia S.p.A., EPFL and SMARTEC SA, has been performed to validate different damage identification algorithms, previously validated on numerically simulated experiments, that can be applied to the SHM of reinforced and pre-stressed concrete beams under static continuous monitoring programs. The experiment has been conducted for more than two years and consists in the continuous monitoring of strains and temperatures in two pre-stressed concrete beam specimens, placed outdoor under environmental conditions and one of which has been subjected to increasing artificial damages. The strain time histories have been detected by means of fibre optic SOFO sensors (eight installed on the damage beam, four on the undamaged one). The details of the experimental program, the results of some applied damage detection algorithms and the method proposed to remove the temperature effects are presented in [START_REF] Grosso | Validation of an SHM procedure for concrete bridges based on static strain[END_REF].

In the present paper the performance of the Proper Orthogonal Decomposition as a damage detection method is investigated. More details on the algorithm can be found in [START_REF] Lanata | Damage detection and localization for continuous static monitoring of structures using a Proper Orthogonal Decomposition of signals[END_REF]. In this study POD has been proposed to extract a spatial and temporal correlation between the installed sensors, related in a certain way to the normal modes contained in the static structural response. The measurements have to be arranged to form the measurement matrix: each column represents a displacement history at a particular sensor location and each row represents the spatial distribution of the response at a given time instant. After subtracting the mean displacements of each sensor location, the covariance matrix has to be computed and the relative proper orthogonal modes can be obtained. The eigenvalues and the orthogonal eigenvectors are calculated from the covariance matrix using a window containing at each step all previous measurements.

Assessment of ROC points

The eigenvectors extracted by the Proper Orthogonal Decomposition are used as damage features in the analysis of PoD and PFA. For each structural static mode, the time variation of the relative eigenvector at each sensor location is available. In this way, the method allows to follow the time variations of the eight extracted eigenvectors, but gives also a spatial information following each eigenvector at each sensor location. As a consequence, 64 time trajectories are available to assess the ROC point.

The extracted eigenvectors need a certain period of time to get a stationary behavior to be assumed as reference state. As damages have been introduced too early during the monitoring, they are mixed to the non-stationary initial trend of the eigenvectors; for this reason, three months at the end of the monitoring have been assumed as the reference undamaged period to define the damage detectable threshold. Three thresholds have been defined using the standard deviation of measurements of the reference state: the first one corresponds to a confidence interval of 67% (mean +/-standard deviation), the second one to the 95% (mean +/-2*standard deviation) and the third one to the 99% (mean +/-3*standard deviation).

Results on noisy and pre-processed data

The computation is initially performed on rough data. The extracted eigenvectors contain the structural response and also all the environmental information. A vector of damage threshold is defined for each structural mode at each sensor location. A time window of 5 and 15 days respectively is used to discretize the monitoring period in N intervals, each one containing or not the artificial damage. The PoD and the PFA are evaluated as described in section 2.3 giving a maximum value for the PoD of 0.82 and a minimum value for the PFA of 0.18 for the first mode. No significant differences are noticed in results obtained with the two time windows. The ROC points are then obtained for each eigenvectors and plotted for the three investigated thresholds. Considering the distance NDT α between the experimental points and the best performance point with coordinate (0, 1), it is possible to have an evaluation of the optimal efficiency of the DDA under specific conditions. Figure 3 shows that the higher damage threshold leads to a shorter distance NDT α . Similar results are obtained for the other modes (Figure 4).

The same procedure is performed on pre-processed filtered data to remove the effect of temperature variations, as presented in [START_REF] Grosso | Validation of an SHM procedure for concrete bridges based on static strain[END_REF]. In this case, the PoD and the PFA give a maximum value for the PoD of 0.92 (corresponding to +12%) and a minimum value for the PFA of 0.08 (corresponding to -55%). A better performance of the DDA on pre-processed data is evident. The ROC points are finally obtained for each eigenvectors and plotted for the three investigated thresholds. Figure 5 shows again that the higher damage threshold leads to a shorter distance NDT α and that a better efficiency of the DDA is obtained. Similar results are obtained for the other modes (Figure 6).

As confirmed from the previous data interpretation, the performance of the Proper Orthogonal Decomposition improves after the data pre-processing to remove the noise due to temperature effect. A further improvement can be obtained computing the PoD and the PFA using a combination of different eigenvectors. Plots in Figures 7 and8 show the results using temptative combinations. It can be observed that the best results are obtained combining the first odd modes, 1 and 5 or 1, 3 and 5.

CONCLUSION

The paper has proposed a probabilistic modelling of damage detection from data processing algorithms based on existing probabilistic modelling of NDT techniques. The method has shown to be promising and appropriate when a significant number of sensors are available. The negative influence of the noise due to environmental variations has been shown by comparing the results of the method applied on rough and on filtered data. Significant improvements have been obtained after the removal of the temperature effect on the strain time histories, confirming that deeper studies have to be done for data pre-processing.

The parametric study on the damage level threshold has shown that the higher threshold corresponding to a confidence interval of 99% gives the best performance of the POD algorithm in damage detection. Unfortunately this result has not been observed for all modes, so that a general rule is still impossible to define.

The study of the combination of different modes improves the performance of the algorithm but the results could be significantly improved though the combination of the modes with the higher correlation coefficients. It has to be investigated yet if the combined modes are associated to the same level of the fixed damage threshold.

Interesting results are expected from the comparison of different damage detection algorithms on the same data set.

  Figure 1 corresponding to normal distributions.

Figure 1 -

 1 Figure 1 -Illustration of PoD and PFA (signal+noise and noise normally distributed) for detection threshold a d .

Figure 2 -

 2 Figure 2 -Example of ROC curves with several NDT performances.

Figure 7 -

 7 Figure 7 -Distance αNDT for different combinations of eigenvectors (modes) at each sensor location (from 1 to 4).

Figure 8 -

 8 Figure 8 -Distance αNDT for different combinations of eigenvectors (modes) at each sensor location (from 5 to 8).


Olivier Le Deuff
email: oledeuff@gmail.com

Search engine literacy

Keywords: literacy, transliteracy, search engines, search engine literacy, search engines, google

The aim of this article is to show that search engine literacy (SEL) can be connected as a transliteracy to digital literacy and information literacy. SEL allows a better understanding of the relationship between information retrieval and technical systems. We show here that teaching Search Engine Optimization (SEO) can be a part of SEL.

Introduction

If SEO (search engine optimization) is well known, the aim of his article is to show that people's search engine literacy is often very poor. As a professor, I ask my freshmen students every year if they can explain how Google works and how it ranks websites. Not one of my student can answer this question. This is particularly disturbing because my students want to work with information or communication systems. Moreover it demonstrates the failure of their previous education, which should have given them the key to understanding how the main search engines operate.

Many studies have shown that students have great confidence in the results given by Google [START_REF] Hargittai | Trust Online: Young Adults'Evaluation of Web Content[END_REF] [2], but they do not develop any critical thinking about its ranking methods. Moreover, the younger generation is not the only one which does not understand how Google works and how google makes money. In a study of seniors and digital health literacy [START_REF] Vigouroux-Zugasti | Le senior connecté[END_REF], we have shown that Google was the first tool used to access information on the web (92%). But the design of results is not totally understood by these older users, who very often cannot make the distinction between 'natural search results' and ads [START_REF] Deuff | La littératie digitale de santé, un domaine en émergence in Agostinelli[END_REF]. At the end of this study, we developed an application to help users to evaluate their level of digital health literacy1. The results show that the level of health literacy was better than the level of digital literacy because more than 70% of the respondents did not know how Google works.

The aim of this article is to consider search engine literacy (SEL°) as an essential part of transliteracy [START_REF] Thomas | Transliteracy: Crossing divides[END_REF]. In fact, search engines like Google are a new kind of media, and major players in information retrieval. Search engine literacy offers the possibility to study the operating mode of Google and to study algorithms from a different perspective from mathematical or computer science approaches. This literacy cannot be only grounded on good practices in the use of Google, like the demonstration of advanced research based on a simple search literacy [START_REF] Wilson | Search literacy: learning to search to learn[END_REF] ( [START_REF] Wilson | Search literacy: learning to search to learn[END_REF]. The project of Search engine literacy is more ambitious and calls for a new pedagogy of Google. Later in this article we report on some results of an experiment with our students during a SEO course. This literacy is a good starting point for explaining the history of the web and the development of a digital and information culture. This is a good method for understanding data literacy and the methods of data treatment. Search engine literacy makes the link between information literacy and data literacy too.

Methodology

This article wants to show the importance of SEL and the relationship with information literacy and culture of information. We want to show here the basis of SEL. This is an exploratory project based on many years of teaching SEO with students in a French i-school. We show here also a project of a French teacherlibrarian Florian Reynaud who develops à specific tool to explain search engines.

2

A new literacy?

2.1

The search engine tradition SEL is not a totally new literacy. We need to consider older tools for information retrieval before the success of search engines like Google. We think about bibliographic systems, index systems, card systems (like the Repertoire bibliographique universelle of Paul Otlet) and other incredible tools like the machine of Emanuel Goldberg [START_REF] Buckland | Emanuel Goldberg and His Knowledge Machine[END_REF]. Our ways of searching for information have always been connected with technologies. We always need to imagine sophisticated systems to retrieve information and to satisfy the extent of our information needs. Among these original devices, we think of Jean Hautfuney and the creation of the index of the Speculum Historiale written by Jean de Beauvais [START_REF] Paulmier-Foucart | Ut diligens et intelligens lector requirat : une banque de données du Speculum maius de Vincent de Beauvais », De la plume d'oie à l'ordinateur : études de philologie et de linguistique offertes à[END_REF]. That was the beginning of the tradition of indexes and tools for information retrieval (statim invenire). After bibliographies and index cards, a new kind of index was created by the Belgian Paul Otlet: the index of all printed documents [START_REF] Olivier | Utopies documentaires : de l'indexation des connaissances à l'indexation des existences[END_REF]. It was the 'Google of paper" 2 . We are not in the first age of information, and search engines are not the first generation of 'knowledge machines' [START_REF] Wilson | Search literacy: learning to search to learn[END_REF]. In our age of information, the novelty lies in the computational nature of a search engine's features, which operate on automatic indexing and automated language processing. Nevertheless, it is not impossible to develop historical knowledge of the operating mode of search engines. If learning the information search process requires knowledge of the basic uses of search engines, an ambitious literacy should, rather, involve understanding how they work. That means understanding the context that led to the emergence of Google, in particular the period when leaders like Alta Vista saw a degradation in the quality of their results. Web metadata had been polluted by spam keywords. Pornographic keywords and the name of Pamela Anderson3 were used by webmasters to increase their traffic.

The logic is therefore to find a way to grant credit to a resource listed by its URL. The choice made by Sergey Brin and Larry Page was to base their classification algorithm on the importance of references. They chose to build a citation index of the web. This logic was not new, it comes mainly from the work of scientometrics especially that of Garfield, Pritchard and Nalimov. Google belongs to a documentary tradition based on citation logic and on bibliographic methods originally focused on the study of bibliographic references from journals. However the corpus is different since the web is no longer a web of documents for researchers, but a collection of disparate websites. The simplicity and clean interface of Google conceal a complexity that is not understood by users.

Search engine literacy and the other literacies

SEL is not a totally new literacy and even less is it disconnected from other literacies that involve information skills on the web. SEL can be considered as a transliteracy that mobilises information skills, digital skills and communication skills. It is therefore not a separated literacy but rather a literacy that is potentially part of training plans either on the side of information literacy or on the side of digital literacy. It can thus be perfectly described as a transliteracy that makes the link between the two literacies. SEL is also a communication and media literacy. Indeed, we should think of the search engine as a mediation tool that influences content. The evolution of Google makes it a medium in its own right. We find ourselves here aligning with discourses around new media literacies [START_REF] Jenkins | Confronting the Challenges of Participatory Culture: Media Education for the 21st Century[END_REF].

The challenge is clearly to understand the articulations between content and media.

Search engine literacy allows the link with a citizen conception of information literacy [START_REF] Owens | Government and Libraries[END_REF], which is precisely a culture of information [START_REF] Deuff | Penser la conception citoyenne de la culture de l'information[END_REF]. This cultural approach implies going beyond good practices to develop a wider and more sustainable knowledge by teaching information and digital concepts. This means better ability to understand the tools used and their foundations. It is also a better cognition and recognition of the technique as part of culture [13][14]. This implies, therefore, the need to learn the history of the web and the evolution of search engines.

SEO as a strategy of SEL

The challenge is to consider a didactic perspective in teaching search engines, to overcome the "black box" effect, and especially to be able to analyse the results or even criticise them. Indeed, the algorithmic system of search engines relies on computational logic, rather than on logic of analysis. This explains why some research results are disturbed by the high ranking of bad websites with fake news and which are far from the historical truth. For example, a Google search about the historians of the shoah puts forward some negationist websites. This means information literacy must evolve to integrate more and more digital skills. Search engines hide a complexity with a simple interface but the aim is to open the 'black box' following the advice of Johanna Drucker about the necessity to examine the framework: A frame can extend, intensify, connect, embed, juxtapose, or otherwise modify another frame and perception. The terminology is spatial and dynamic. It describes cognitive processes, not simple actions of an autonomous user, but codependent relations of user and system. In invoking frame analysis as part of the diagrammatic model of interpretation, we have moved from a traditional discussion of graphical formats as elements of a mise en page to a sense that we are involved with a mise en scene or système. This puts us on the threshold of interface and a theory of constructivist processes that constitute the interface as a site of such cognitive relations. Interface is not a thing, but a zone of affordances organized to support and provoke activities and behaviors probabilistically. [15, p.7]

Learning SEO as a core skill for Informationschools

One of the ways to better understand the way search engines work, and especially their ranking methods, is to look at it from an SEO perspective. Students can better understand the key elements used by google (title markup, domain names, keywords), the page rank and the choices of weighting used in the algorithm.

The observations were made during a course of SEO in a small I-school in France in Bordeaux in DUT Information numérique dans les organisations 4 (digital information in organizations) where we train students for two years. SEO is just a course, which is connected with other courses like metadata history and taxonomies, and markup languages like html and xml. Search engine literacy needs to be connected to many other elements if we want to place it as core content in an I-school curriculum. We can list some essential content:

-The constitution of an index -History of indexes -Metadata history, systems and languages -Human indexing and automatic indexing -Crawl methods and bots -Web directories versus Search engines -Distinction between natural indexing and advertising indexing -Scientometrics and page rank -Algorithm and weighting of criteria -Keyword optimization -Communication strategies and SEO -Ethics and search engines -Personal Data and filter bubbles.

For three years we have experimented with having an SEO contest between groups of students as part of the course. The aim is for them to imagine they are content producers. It is also to apply the SEO methods they have learnt about, to try to place their sites or contents top of the hit list on Google. Each year, the choice is based on a request where there are almost no results so that the student productions can best be placed in the first results. The aim of the approach is certainly to be trained in SEO, but the objective is also to pursue the process of training in information retrieval by placing the students on the side of content producers. SEL makes it possible to consider both sides of the information, at the level of the search and also at the level of its production. It makes it easier for students to judge bias in search engines. The principle of motivating students through a logic of challenge or competition also shows that even if search engines avoid too much optimization, search engines' ranking methods differ from a human analysis as regards checking content quality. Therefore, dual training in information retrieval methods and SEO strategies makes it possible to go further in detecting content which is rubbish".. In French, we use 'évaluation de l'information' to describe the analysis of bad contents and the work which allows one to distinguish good contents and relevant websites.

If this type of course seems essential for an I-School education, it is advisable to extend it earlier on. Without necessarily training in SEO, we could teach how search engines work, especially Google, in an educational perspective.

Search engine pedagogy

Several hypotheses can be envisaged, and are also tested by French colleagues who are teacher-librarians (called in French professeurs-documentaliste) with their students, to show them how research results are linked to the constitution of an index 5 . Tools such as Google custom search can be used to create thematic engines, but you can also rely on specific engines. A webfinder simulation 6 is proposed by professeurdocumentaliste Florian Reynaud in order to be able to use it in class with students.

The device makes it possible to understand the steps of classification and how results are ranked. The methods of automatic processing are also explained. It enables understanding of the criteria for ranking information according to the principles of relevance (number of occurrences in connexion to the request), popularity (number of visits), and notoriety (number of links). This also helps learners to understand what an algorithm is with a weighting system.

Webfinder, a tool to understand search engines methods of ranking SEL pedagogy is an important step for information literacy which can show the complexity of the devices and goes beyond the simple satisfaction of an information need.

Conclusion

The description of a culture of information for citizens requires a better knowledge of the information retrieval processes and the tools used. SEL can't be considered as a specific literacy. It's a kind of transliteracy and it needs to be connected with information literacy. SEL show the evolution of the methods of teaching information literacy with the will to explain how search engines work and not only the best practices to use them.

Better understanding of search engines, especially through examination of the criteria used for ranking websites, is an important step in the democratization of the search engine and especially Google. It also gives users the means to know when to use Google and when to favour other search methods. Google does not want publish their algorithm because the firm pretends to protect the truth and to avoid too much SEO optimization. This position shows, too, the will to protect a business model. The democratization of Google appears difficult in this model if the users are not well trained and cannot understand the strength and the weakness of such a model. Training students can be the way to imagine that other kind of search engines could be invented in the next years. Creating an open index [START_REF] Lewandowski | Why We Need an Independent Index of the Web[END_REF] of the web could allow alternate models.

http://www.megatopie.info/testor/enquete.html?e=19

Les meta-tags hors la loi ? < http://www.sam-mag.com/archives/metatags.htm>

http://www.infonumbordeaux.fr/

ID Base. Fiches pédagogogiques à l'usage des professeurs documentalistes. <http://idbase.esmeree.fr/>

https://iddocs.fr/webfinder/

Acknowledgment

I want to thank Sheila Webber for reviewing this article. Thanks to Florian Reynaud for the screenshot of his Webfinder.


Amelia Carolina Sparavigna

On a generalized sum of the Mersenne Numbers

Keywords:

we propose the study of the Mersenne Numbers, using the same approach. About these numbers, a large literature exists (see for instance that given in [START_REF] Weisstein | Mersenne Number[END_REF]). The form of the numbers is that of a power of two minus 1. Among them we find the Mersenne primes. The numbers are named after Marin Mersenne (1588 -1648), a French Minim friar, who studied them in the early 17th century.

Mersenne numbers are:

M n =2 n -1
Let us consider them to give an example of generalized sum. We can start from the following calculus:

.

M m+n =2 m+n -1 M m+n =2 m+n -1=2 m 2 n -1-2 m +2 m -2 n +2 n -1+1=2 m ( 2 n -1)-1+ 2 m -2 n +1+2 n - 1
M m+n =(2 m -1)(2 n -1)+2 m -1+2 n - 1
Therefore, we can write the following generalized sum:

M m+n =M m ⊕M n =(2 m -1)(2 n -1)+(2 m -1)+(2 n -1)
or:

This is a generalized sum that we find in the case of the multiplicative groups (for the use of multiplicative groups in statistics and statistical mechanics see [START_REF] Sicuro | Groups, information theory, and Einstein's likelihood principle[END_REF][START_REF] Curado | A new entropy based on a grouptheoretical structure[END_REF]).

Using [START_REF] Sparavigna | Generalized Sums Based on Transcendental Functions[END_REF], for the Mersenne numbers we can imagine the following recursive relation:

M n+1 =M n ⊕M 1 =M n + M 1 + M n M 1
That is:

2 n+1 -1=(2 n -1)+(2 1 -1)+(2 n -1)(2 1 -1)=2 n + 2 n+1 -2 n -2+ 1=2 n+1 -1
The sum ( 1) is associative, so that:

M m ⊕M n ⊕M l =M m + M n + M l + M m M n + M n M l + M m M l + M m M n M l
We cannot have a group of the Mersenne numbers, without considering also the opposites of them, so that:

0=M n ⊕Opposite(M n ) Therefore: Opposite(M n )=- M n M n +1 =M -n
Explicitly:

Opposite(2 n -1)=- (2 n -1) (2 n -1)+1 = (-2 n +1) 2 n =2 -n -1
These numbers are the Mersenne numbers with a negative exponent. So we have:

M nnn =M n ⊕M -n =M n + M -n + M n M -n