J P Marshall

C J Milli

É Choquet

C Del Burgo

G M Kennedy

L Matrà

S Ertel

A Boccaletti

COMPREHENSIVE ANALYSIS OF HD 105, A YOUNG SOLAR SYSTEM ANALOG

Keywords: stars, circumstellar matter -stars, planetary systems -stars, individual (HD 105)

HD 105 is a nearby, pre-main sequence G0 star hosting a moderately bright debris disc (L dust /L ∼ 2.6 × 10 -4 ). HD 105 and its surroundings might therefore be considered an analogue of the young Solar System. We refine the stellar parameters based on an improved Gaia parallax distance, identify it as a pre-main sequence star with an age of 50 ± 16 Myr. The circumstellar disc was marginally resolved by Herschel/PACS imaging at far-infrared wavelengths. Here we present an archival ALMA observation at 1.3 mm, revealing the extent and orientation of the disc. We also present HST/NICMOS and VLT/SPHERE near-infrared images, where we recover the disc in scattered light at the ≥ 5-σ level. This was achieved by employing a novel annular averaging technique, and is the first time this has been achieved for a disc in scattered light. Simultaneous modelling of the available photometry, disc architecture, and detection in scattered light allow better determination of the disc's architecture, and dust grain minimum size, composition, and albedo. We measure the dust albedo to lie between 0.19 and 0.06, the lower value being consistent with Edgeworth-Kuiper belt objects.

INTRODUCTION

Through the results of the Kepler transit-based exoplanet survey, we are now aware that exoplanets are near-ubiquitous (Coughlin et al. 2016;Foreman-Mackey et al. 2016). Circumstellar debris, most commonly identified through excess emission at infrared wavelengths (Matthews et al. 2014), is less frequently found. However, this is more severely biased and limited by current instrumental sensitivity. A detection rate of 1 Computational Engineering and Science Research Centre, University of Southern Queensland, Toowoomba, QLD 4350, Australia 2 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA 3 Hubble Fellow 20 % has been recorded for cool discs around nearby FGKtype stars (Eiroa et al. 2013;Montesinos et al. 2016), with a slightly higher detection rate around A-type stars (Thureau et al. 2014). A tentative trend for a higher incidence of debris discs is seen around stars with sub-Solar metallicities hosting low mass planets (Wyatt et al. 2012;Marshall et al. 2014a).

For high mass planets, a recent study of disc-host systems has identified a tentative correlation between the presence of > 5 M Jup planets and debris discs (Meshkat et al. 2017). A study of exoplanet host stars and debris discs reveals a trend between these components of planetary systems being seen together (Matthews et al. 2014). However, no evidence of correlations between these properties has been seen in larger stellar samples, whilst potentially attributable to the paucity of information on both faint debris discs and low mass planets around nearby stars, the presence of real correlations cannot be ruled out due to sample construction (Moro-Martín et al. 2015).

A critical component in developing an understanding of the diversity of architectures exhibited by known planetary systems is to obtain multi-wavelength resolved images of their dusty debris discs (e.g. Ertel et al. 2014;Marshall et al. 2014bMarshall et al. , 2016;;Hengst et al. 2017). Using a sample of 34 resolved debris discs a relationship between stellar luminosity and dust properties has been identified, showing that the dust grains around higher luminosity stars are closer to emitting like blackbodies (Pawellek et al. 2014). Fitting that observed relationship with a range of dust material compositions further identified that a mixture of astronomical silicate and water ice provided the best fit to the distribution of spatially resolved discs (Pawellek & Krivov 2015). The presence of icy material in cool debris discs has also been inferred through modelling of continuum emission of several spatially resolved debris discs, supporting the adoption of icy materials in the analysis of debris dust (Lebreton et al. 2012;Morales et al. 2016). More recently, a relationship between stellar luminosity and disc radius has been measured for a similar sized sample of debris discs imaged at sub-millimetre wavelengths (Matrà et al. 2018a). However, no evidence was found for the trend of disc radius relative to blackbody radius identified at far-infrared wavelengths. This may be attributed to the low spatial resolution and bias toward higher luminosity stars of the far-infrared sample making it less representative overall.

Analysis of the infrared continuum emission from debris dust can only take our understanding so far. To obtain insight into the dust grain composition and structure (porosity) we require measurement of the silicate features at mid-infrared wavelengths (e.g. Beichman et al. 2006;Chen et al. 2006;Lawler et al. 2009;Johnson et al. 2012;Mittal et al. 2015), and/or measurement of scattered light from the disc (either total intensity or polarised light). A review of recent advancements in the interpretation of scattered light imaging of debris discs is presented in Hughes et al. (2018). In the case of silicate features, a minority of debris disc host stars exhibit features in their mid-infrared spectra (Beichman et al. 2006;Chen et al. 2014), suggesting the constituent dust grains are large (> 10 µm) and/or cold. In the latter case, the scattered light brightness is poorly correlated with expectations from the disc brightness in continuum emission (e.g. Schneider et al. 2014) but, for those discs that have been imaged in scattered light, determination of the dust optical and scattering properties has been possible (e.g. [START_REF] Graham | [END_REF]Krist et al. 2010;Rodigas et al. 2012Rodigas et al. , 2014;;Soummer et al. 2014;[START_REF] Choquet | [END_REF]Schneider et al. 2016). Complementary to the properties of solid material in debris discs, the recent detections of a gaseous component to some of these systems [START_REF] Gray | The observation and analysis of stellar photospheres[END_REF] provides insight into the volatile content of the dust parent bodies (e.g. Matrà et al. 2017), and its origins (Kral et al. 2017).

For Sun-like stars, from which we may draw parallels with the evolution of the Solar System, there exist only a few cases with comprehensive multi-wavelength data sets to support such detailed analyses, e.g. HD 15115 (Debes et al. 2008;MacGregor et al. 2015), HD 61005 (Hines et al. 2007;Maness et al. 2009;Buenzli et al. 2010;Olofsson et al. 2016), HD 107146 (Ertel et al. 2011;Ricci et al. 2015;Marino et al. 2018), HD 207129 (Krist et al. 2010;Marshall et al. 2011;Löhne et al. 2012), HIP 17439 (Ertel et al. 2014;Schüppler et al. 2014), andHD 10647 (Liseau et al. 2010;Schüppler et al. 2016). The addition of more targets to this list, particularly within the narrow range of stellar spectral types representative of Solar analogues (i.e. early G-type main sequence stars cf a broad range of F6 to K0 spectral types), is therefore important to understand the diversity of potential outcomes for planet formation processes. The relative novelty of the Solar System may therefore be determined by comparison to the range of properties exhibited by analogous debris discs systems.

HD 105 is a young, Sun-like star at a distance of 40 pc that hosts a moderately bright debris disc (L dust /L ∼ 2.6 × 10 -4 ). HD 105 is a member of the Tucana-Horologium association and has a well established age of 28 ± 4 Myr (Torres et al. 2000;Zuckerman & Webb 2000). Its debris disc was reported as being marginally resolved in far-infrared Herschel/PACS imaging observations and found to have a radius of ∼ 50 au [START_REF] Donaldson | [END_REF]. With a spectral type of G0, HD 105 might therefore be considered an analogue of the young Solar System.

Here we model the continuum emission and structure of HD 105's debris disc using a combination of far-infrared and sub-millimetre photometry obtained from a new reduction of archival Herschel/PACS and SPIRE images, and a spatially resolved archival image of the disc at millimetre wavelengths from ALMA. We complement this approach with analysis of near-infrared images from HST/NICMOS and VLT/SPHERE in order to constrain the dust optical properties through scattered light.

In Sect. 2, we present the imaging and photometric data compiled to model both the stellar and disc components of this system. In Sect. 3 we lay out our approach to analysing the combined data set and state our findings. Sect. 4 the results are put in context, and their impact on our understanding of this system. Finally, in Sect. 5, we summarise our findings and detail the conclusions of this work.

OBSERVATIONS

We have compiled broad band photometric data from optical to millimetre wavelengths to construct the SED of HD 105. These previous observations, combined with the ALMA, VLT/SPHERE, and HST/NICMOS imaging data presented here, facilitate comprehensive modelling of the HD 105 system. A summary of the photometry used in the modelling process is presented in Table 1.

At optical wavelengths we use the Strömgren data from Paunzen (2015), Johnson BV and Cousins I data from (Mermilliod et al. 1997), near-infrared 2MASS JHK s (Cutri et al. 2003), and WISE W1 and W2 fluxes (Wright et al. 2010) to scale the model stellar photosphere and gauge its contribution to the total emission.

In the mid-infrared, the WISE W3 (12 µm) and W4 (22 µm) fluxes are complemented by AKARI/IRC 9 µm data point (Ishihara et al. 2010), a Spitzer/MIPS measurement at 24 µm, and Spitzer/IRS photometry at 13 µm and 31 µm. The Spitzer/IRS spectrum was obtained from CAS-SIS 4 (Lebouteiller et al. 2011). As the target exhibits no evidence of mid-infrared excess, the spectrum was scaled to the photospheric model using a least-squares fit weighted by the measurement uncertainties at wavelengths < 15 µm. Target fluxes were then extracted in two windows centred at 13 and 31 µm and 4 µm in width using the error-weighted average of values within the bin.

At far-infrared wavelengths, we combine ISO/ISOPHOT measurements at 60 and 90 µm (Spangler et al. 2001) and a Spitzer/MIPS measurement at 70 µm (Carpenter et al. 2009) with Herschel/PACS measurements at 70, 100, and 160 µm. The spatial resolution of ISO is much poorer (1.5 beam FWHM) than Spitzer or Herschel (6-18 beam FWHM); inclusion of background contamination and errors in the zero level calibration elevating the ISO-measured fluxes are possibilities [START_REF] Héraudeau | Exploiting the ISO Data Archive[END_REF][START_REF] Del Burgo | Exploiting the ISO Data Archive[END_REF]. However, the target is well isolated in the Herschel maps and there is good agreement between measurements by all three facilities at similar wavelengths so we therefore do not consider it to have a significant impact on the shape of the SED.

The Herschel5 observations were first presented in [START_REF] Donaldson | [END_REF]. Here we re-reduced these data to obtain revised values for the source fluxes based on updates to the instrument calibration made available in the intervening time. The data reduction and analysis were carried out in the same manner as described in Eiroa et al. (2013). The PACS data were reduced in the Herschel Interactive Processing Environment6 (HIPE, Ott 2010) using the standard pipeline processing scripts (HIPE version 15, PACS calibration 78). Target fluxes were measured in each mosaic using an aperture of 15 radius and a sky annulus of 30 to 40 . Appropriate aperture and colour corrections (determined by a blackbody fit to the aperture corrected measurements), based on values tabulated in Balog et al. (2014), were applied to the measurements before modelling.

We also present sub-millimetre measurements taken from Herschel/SPIRE (Programme ot2_aroberge_3, PI: A. Roberge) and an APEX/LABOCA measurement at 880 µm (Nilsson et al. 2010). The SPIRE observations were again reduced in HIPE version 15, using SPIRE calibration 14_3. The target fluxes in the three SPIRE maps were measured using the sourceExtractorSussextractor routine. The millimetre SED is further constrained by ALMA photometry at 1300 µm (Programme 2012.1.00437.S, PI: D. Rodriguez) and an ATCA measurement at 9 mm taken from Marshall et al. (2017).

METHODOLOGY AND RESULTS

Here we undertake a determination of the stellar parameters by fitting a grid of stellar atmosphere models to archival high resolution spectra. We also inferred the same stellar parameters as well as the mass and age from stellar evolution models combined with a Bayesian approach. We then present a self-consistent analysis of both the millimetre and scattered light imaging data to determine the disc architecture and dust albedo, respectively. The disc structure is applied as a constraint in modelling of the disc spectral energy distribution (SED) in combination with photometry from archival sources. Gaia Collaboration et al. (2016, 2018);3. Mermilliod et al. (1997); 4. This work; 5. Tsantaki et al. (2014) (G = 7.3833 ± 0.0007) are taken from the Gaia DR catalogue (Gaia Collaboration et al. 2016;Lindegren et al. 2016;Gaia Collaboration et al. 2018). The stellar physical properties were derived by modelling of available high resolution spectra along with optical and near-infrared photometry.

The stellar parameters of HD 105, i.e. luminosity, radius, effective temperature, surface gravity, age, and mass, were derived by using the absolute G magnitude, parallax, B -V colour, and [Fe/H] as input parameters using the Bayesian approach applied in del Burgo & Allende Prieto (2016, 2018). We derived an age of 50 ± 16 Myr, which is in agreement with that one more constrained assuming membership of Tucana Horologium,of 30 Myr (del Burgo et al. in preparation). The colour B -V = 0.600 ± 0.003 is taken from Mermilliod et al. (1997), and metallicity [Fe/H] = 0.02 ± 0.04 from Tsantaki et al. (2014). The observed G magnitude was converted to absolute magnitude M G using the Gaia DR2 parallax of π = 25.75 ± 0.06 mas (Gaia Collaboration et al. 2018). We find that HD 105 is a pre-main-sequence star. In order to derive the stellar parameters we downloaded and arranged a grid of PARSEC isochrones (version 1.2S, Bressan et al. 2012;Chen et al. 2014Chen et al. , 2015;;Tang et al. 2014), with steps in age of 5 per cent, steps in mass of 0.1 per cent, and steps of 0.02 dex in [Fe/H]. The lower and upper limits of these parameters are well beyond 6-σ around the solution, see Table 2. Synthetic photometry estimates were obtained using filter curves from Evans et al. (2018) for the Gaia G band, and Maíz Apellániz (2006) for the B and V bands. For a more detailed description see del Burgo & Allende Prieto (2018).

In addition, high resolution FEROS and HARPS spectra were also used to obtain the stellar effective temperature T eff , metallicity [Fe/H], projected rotational velocity ν sin i, and surface gravity log g. We performed a comparison with BT-Settl stellar atmosphere models (Allard et al. 2012). The models in the grid were modified in order to be compared with the FEROS spectrum similar to the approach of del [START_REF] Del Burgo | Exploiting the ISO Data Archive[END_REF]. First, the synthetic spectra were transformed to take into account the stellar projected rotational velocity (ν sin i) using the formalism of [START_REF] Gray | The observation and analysis of stellar photospheres[END_REF], with a limb darkening parameter equal to 0.8. These spectra were convolved with a Gaussian that mimics the instrumental profile along the dispersion axis. The resulting spectra were rebinned to the same resolution and grid of the observed spectrum. This was corrected from the velocity shift from a cross-correlation analysis with the models. All modelled and observed spectra were normalized before performing the comparison.

We obtain values from the FEROS spectrum (R = 48, 000) of T eff = 6000 ± 50 K, log g = 4.50 ± 0.25, and ν sin i = 17.5 ± 0.5 km/s. Values derived from the higher resolution HARPS spectrum (R = 115, 000) are consistent with those of FEROS, with the FEROS values being determined at higher signal-to-noise. These values are consistent with those derived from the aforementioned stellar atmosphere models.

We have a good agreement between the stellar evolution analysis and the stellar atmosphere model fitting for the parameters in common. The values obtained through this analysis are likewise consistent with available values from the literature e.g. Valenti & Fischer (2005), Tsantaki et al. (2014).

The Hα and CaII H&K lines in the FEROS spectrum are presented in Fig. 1. The median FEROS spectrum, calculated from the available spectra (in black) is plotted together with our best fitting model. Some regions of the Hα segment of the spectrum have been blanked out due to contamination from telluric lines at those wavelengths (see Fig. 1,top). Despite the low ν sin i (consistent with a more pole-on viewing angle), we see clear evidence of CaII H&K emission in the spectrum (see Fig. 1,bottom). This is consistent with activity in the stellar chromosphere, an indicator of stellar youth.

We also examined the high resolution spectra for evidence of gas revealed by the presence of circumstellar absorption or emission lines. The origin of such gas may be primordial, or secondary from e.g. photodesorption or cometary activity. We note that the FEROS fibre input has a 2 footprint on the sky that could include light scattered from material in the circumstellar disc, depending on the seeing quality. No evidence of such features is seen. No evidence of gas features in either absorption (implying cool gas) or emission (implying hot gas) were identified in the spectra. ), providing a velocity resolution of 0.64 kms -1 . In combination, the four channels provide 6.9 GHz bandwidth to study the continuum emission. The on-source integration time for HD 105 was 2238 s. Neptune was used as the flux calibrator, J0006-0623 was the bandpass calibrator, and J0012-3954 was the phase calibrator. Calibration and reduction of the ALMA observation were carried out in CASA 4.1 using the provided scripts. Image reconstruction was carried out using the clean task combining all four spectral windows for the greatest signal-tonoise. We reconstruct the image using natural weighting. With natural weighting the continuum image r.m.s. noise is 26.8 µJy beam -1 . The dirty beam has an ellipsoidal FWHM 0. 95 × 0. 67 at a position angle of 88 • , equivalent to a spatial resolution of 34 × 26 au.

Disc architecture

The disc is visible as an annular structure centred on the stellar position, but the low signal-to-noise of the observations means that it is not detected with high significance (> 3-σ) at all position angles. The ALMA image and the best fit annular disc model are presented in Fig. 2.

We model the architecture of HD 105's disc as a single annulus with a semi-major axis R and width ∆R oriented at a position angle θ and inclination i. The disc surface brightness exponent α is assumed to be constant, and a vertical opening angle of 10 • , similar to the Solar system, was assumed. A grid of models is generated spanning reasonable values for each of these parameters. To determine the best-fit properties for the disc we subtract each disc model, convolved with the dirty ALMA beam (FWHM ∼ 0.9 × 0.7 ), from the observed image and seek to minimise the residuals within the region of the model-subtracted image where there is significant emission in the observed image.

The best-fit architecture derived from the model fitting is dominated by the disc ansae, which are the only regions of the disc detected at high signal-to-noise (> 3-σ). The disc semimajor axis is 2. 16 ± 0. 13 (85 ± 5 au), with an inclination of 50 ± 5 • with respect to the line of sight at a position angle of 15 ± 5 • . The width of the disc annulus is 0. 75 ± 0. 25 (30 ± 10 au). The belt is perhaps marginally resolved in the ALMA image, but the quality of the data does not allow us to conclude that, so we hereafter assume the belt is unresolved. The best-fit disc architecture obtained from this approach is summarised in Table 3.

The total flux of the disc in the sub-millimetre, 1.45 ± 0.28 mJy, may be used to calculate the mass of dust grains, assuming that the disc is fully optically thin at 1.3 mm. For a disc observed at a frequency ν with a flux F ν the dust mass, M dust , is given by

M dust = F ν d 2 /κ ν B ν,Tdust (1)
where d is the stellar distance, κ ν is the dust opacity, T dust is the dust temperature, and B ν,Tdust is the Planck function (Zuckerman & Becklin 1993;Draine 2006). Assuming the dust opacity κ is 1.7 gcm -2 (Beckwith et al. 1990;Pollack et al. 1994;Draine 2006), that is commonly assumed for debris dust (e.g. Panić et al. 2013;Holland et al. 2017), we calculate a mass of 0.035 +0.013 -0.009 M ⊕ for the dust. This value does not include uncertainty on the parameter κ, which leads to an uncertainty in M dust of a factor 3 to 5.

Non-detection of cold CO emission

Molecular carbon monoxide (CO) has been identified in around a dozen debris disc systems (e.g. [START_REF] Gray | The observation and analysis of stellar photospheres[END_REF]Moór et al. 2017), believed in most cases to be the result of secondary production after liberation from exocomets (Kral et al. 2017). Most of the stars identified with gas in their debris discs are young (t age < 30 Myr) A-type stars, with relatively bright debris discs (L d /L ∼ 10 -3 ). Here we search the ALMA observation for emission from the disc associated with the CO (2-1) transition at 230.538 GHz following the method employed to extract CO emission at low signalto-noise in ALMA data as presented in Matrà et al. (2015); Marino et al. (2016Marino et al. ( , 2017)).

To begin we produced a continuum subtracted measurement set from the HD 105 data using uvcontsub. The continuum subtracted data set is then cleaned to produce an image cube with the same weighting and pixel scale as the continuum image. The cube spans barycentric velocities between -25 and 25 kms -1 , with a step size of 0.5 kms -1 . The cube is corrected for the effect of the primary beam using the impbcor task before further analysis. No significant CO emission is observed in the velocity-integrated, continuum-subtracted image produced by integrating the cube.

We then proceed by implementing the spectro-spatial filtering technique presented in Matrà et al. (2017). Any circumstellar CO is assumed to originate from collisions between planetesimals and therefore reside in the same region around the star as the debris dust detected in the continuum image. The CO is further assumed to be situated in a vertically flat disc, orbiting with Keplerian velocity around a star of 1.114 M . We then determine the projected radial velocity for each position in the disc image given the extent and inclination of the disc for cases of the disc rotating toward or away from us. Using the predicted velocity field the spectrum at each position is then shifted to match the stellar velocity (i.e. spectral filtering). We then integrate over all positions where continuum emission is detected to avoid contributions to the CO spectrum from locations and velocities where no circumstellar CO is expected (i.e. spatial filtering). The spectrospatial filtering yields a 3-σ upper limit of 8.5 mJy kms -1 to CO emission.

We use a non-Local Thermodynamic Equilibrium CO excitation code to calculate a CO mass upper limit, following Matrà et al. (2018b). This code includes fluorescent excitation from the central star at ultraviolet wavelengths. Depending on the excitation conditions we determine a CO mass upper limit ranging from 2.5×10 -7 to 2.5×10 -6 M ⊕ . This translates to upper limits on the CO and CO 2 fraction (by mass) in exocomets between < 88 % and < 46 % for the respective mass upper limits. This limit is consistent with the range of values for other exocometary discs, and Solar system bodies (see e.g. Matrà et al. 2017).

3.3. Detection of scattered light HD 105 has been observed by both HST/NICMOS and VLT/SPHERE. The disc was not directly visible in either data set. However, we recover the disc at the correct angular separation and orientation using an angular averaging technique to aggregate the disc emission to a detectable level. From this measurement of the disc scattered light brightness we determine the albedo of the dust grains at near-infrared wavelengths.

HST/NICMOS

HD 105 was observed with the Hubble Space Telescope (HST) NICMOS instrument at two epochs using the coronagraphic mode (mask radius 0. 3) of the mid-resolution NIC2 channel (0. 07565 pixel -1 ). The first data were acquired on UT-1998-11-03 as part of program GTO-7226 (PI: E. Becklin), a survey looking for giant planets around young nearby stars (Lowrance et al. 2005), using the F160W filter of bandpass very comparable to the H band (pivot wavelength 1.600 µm, 98%-integrated bandwidth 0.410 µm). The second dataset was obtained on UT-2006-07-23 as part of program GO-10527 (PI: D. Hines), a survey looking for debris discs around stars with infrared excess detected with Spitzer as part of the FEPS program [START_REF] Héraudeau | Exploiting the ISO Data Archive[END_REF]. The data were obtained with the F110W filter, which is twice as extended toward shorter wavelengths as the J band (pivot wavelength 1.116 µm, 98%-integrated bandwidth 0.584 µm).

The F160W data were acquired at two different spacecraft orientations, obtaining a total of 6 exposures with HST rolled by ∼ 30 • in the middle of the sequence, to enable subtraction of the PSF with Roll Differential Imaging (Lowrance et al. 1999). The total exposure time is 1344 s for this dataset. The F110W dataset includes 7 exposures all obtained with the same telescope orientation, for a total exposure time of 2016 s.

We reprocessed both datasets as part of the Archival Legacy Investigations for Circumstellar Environments (ALICE) program (PI: R. Soummer), a consistent re-analysis of the NIC-MOS coronagraphic archive using modern PSF subtraction techniques [START_REF] Choquet | Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave[END_REF]Hagan et al. 2018). Assembling PSF libraries from multiple reference stars in the whole archive and using the PCA-based KLIP algorithm for PSF subtraction (Soummer et al. 2012), this program demonstrated a gain in point source sensitivity by a factor of 20 at 1 from the star over classical reference star differential imaging (Debes et al. 2016), and enabled the detection of 11 faint debris discs from NICMOS archival data (Soummer et al. 2014;[START_REF] Choquet | [END_REF]Choquet et al. , 2017Choquet et al. , 2018)).

For HD 105, we used the reprocessed data from the ALICE public database8 for the F110W dataset, and an unpublished ALICE reduction for the F160W dataset with an 11 × 11 field of view, more favorable to disc detection with sensitivity limits better than the data published in the ALICE database. PSF subtraction for the F160W dataset was achieved using the 69 first eigen-modes of a library assembling 125 images from reference stars exclusively, and excluded a central zone within a radius of 12 pixels. The F110W dataset was PSFsubtracted with the 211 first eigen-modes of a 640-image library from reference stars only, excluding a central area of radius 5 pixels. The final images were obtained by rotating all exposures to North pointing up, co-adding and scaling them to surface brightness units based on NIC2's plate-scale and its calibrated photometric factors (F ν = 1.21121 µJy s DN -1 in F110W, F ν = 1.49585 µJy s DN -1 in F160W). We are able to recover the disc scattered light in the F160W filter image in the disc's ansae at low signal-to-noise, as shown in Fig. 3.

To obtain a more significant measurement of the disc surface brightness than can be obtained from the reduced images, we make use of the high spatial resolution of the scattered light image and measure the surface brightness radial profiles. We computed the surface brightness averaged in 5-pixel wide (0.38 ) elliptical apertures of increasing semimajor axes a with semi major axis a increasing with steps of 1 pixel (76 mas) and semi-minor axis b = a cos(50 • ), oriented 15 • east-of-north, to trace the disc deprojected average radial profile according to the ALMA detection geometry. We estimate the corresponding uncertainties assuming uncorrelated noise, as σ/ √ (N pix ) with σ the standard-deviation in each aperture and N pix the number of pixel in the aperture. Since only reference stars were used for PSF subtraction, these measurements are not affected by disc self-subtraction. They are however subject to over-subtraction by the PCA algorithm along with PSF features (Soummer et al. 2012;Pueyo 2016). We corrected our average profiles and uncertainties by estimating the algorithm throughput with analytical forward modeling. From this exercise we find a significant peak with a surface brightness of 15 ± 1 µJy/arcsec 2 at a separation of around 2.2 from the star, consistent with the ALMA image. The binned radial profiles are presented in Fig. 3.

VLT/SPHERE

HD 105 was observed with the Spectro-Polarimeter Highcontrast Exoplanet REsearch (SPHERE, [START_REF] Beuzit | Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series[END_REF] on the night of October, 2nd 2015. These observations used the Infra-Red Dual-beam Imager and Spectrograph (IRDIS, [START_REF] Dohlen | Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series[END_REF] of SPHERE to obtain high-contrast and high-angular resolution images of the circumstellar environment of HD 105, in the broad-band H filter (centred at 1.625 µm, width 291 nm) with the apodized Lyot coronagraph of diameter 185mas. They are part of the SPHERE High-Angular Resolution Debris Disc Survey 9 (SHARDDS, Milli et al. in prep.). SHARDDS is an open-time program on SPHERE to perform the first comprehensive near-infrared survey of all bright, nearby debris discs (infrared excess greater than 10 -4 ), yet undetected in scattered light, within 100 pc. It already led to detections of several discs and a brown dwarf companion: HD 114082 (Wahhaj et al. 2016), 49 Ceti (Choquet et al. 2017) and HD 206893 (Milli et al. 2017a). HD 105 was observed for a total of 50 min on-source, among which the total exposure time is 2432 s, in pupil-stabilized mode to allow Angular Differential Imaging (Marois et al. 2006). The source was observed just before meridian crossing, which resulted in a parallactic angle rotation of 35 • .

The data were sky-subtracted, flat-fielded and bad pixel corrected using the official SPHERE Data Reduction and Handling pipeline [START_REF] Pavlov | Astronomical Data Analysis Software and Systems[END_REF]) in order to make a temporal cube of 606 frames. Each coronagraphic frame was re-centred using the set of four satellite spots imprinted in the image during the centring sequence. This sequence is obtained by applying a waffle pattern to the deformable mirror and was done prior to and after the deep science coronagraphic observations. Then, different data reduction techniques were applied to subtract the stellar halo and try to detect the scattered light of the disc.

At the expected semi-major axis of the disc of 2.16 , the technique that yields the best sensitivity for point source detection is the classical Angular Differential Imaging (cADI, Marois et al. 2006) technique. However for an azimuthally extended structure such as the HD 105 disc, the throughput of the algorithm is very low because of self-subtraction (Milli et al. 2012) and we estimated it to be a few percent. The 5-σ sensitivity of the ADI reduction is 100µJy at 2.2 after 9 ESO program ID 096.C-0388(A) throughput correction, and we are therefore unable to confirm the HST/NICMOS detection with this data reduction technique.

Therefore, we also applied the Reference Star Differential Imaging (R[S]DI) technique, commonly used in spacebased observations (Schneider et al. 1999;Lafrenière et al. 2009;Soummer et al. 2011), and more recently applied to ground-based observations [START_REF] Gerard | Adaptive Optics Systems V[END_REF]. We implemented this technique in the SHARDDS program by building a library of all the coronagraphic images obtained in the program. We then selected 610 images from 22 stars most correlated with the ones from HD 105 (excluding the HD 105 images themselves) and used this set of references to subtract the stellar halo, using a custom PCA algorithm, keeping 100 modes. The disc is not directly visible in the VLT/SPHERE image.

The image is binned by a factor 4, giving a new pixel size of 49 mas, corresponding to the measured FWHM in the noncoronagraphic image. In the image reduced with PCA-RDI, the flux is integrated in elliptical apertures having an aspect ratio matching the inclination of the disc as seen by ALMA, and a radial width of 10 binned pixels or 490 mas. This ellipse is increased iteratively by step of 1 binned pixels or 49 mas, and at each iteration, the signal is computed as the sum of the flux over this elliptical aperture. It is corrected for the PCA throughput and converted in µJy/arcsec 2 using the binned pixel size of 49 mas and the star flux as measured in the non coronagraphic image. The noise is given by the standard deviation of the pixels over the same elliptical annulus.

This averaging technique increases the depth to which we can detect disc scattered light emission using geometrical prior from the resolved mm detection. We find a peak in emission at 5.9 µJy/arcsec 2 for a semi-major axis of 2.3 , consistent with the disc semi-major axis, as shown in Fig. 4. This is a factor of ∼4 fainter than the disc detected by HST/NICMOS; examination of the separate channels reveals that the right channel is consistent with the NICMOSdetected signal in both semi-major axis and brightness, but the left channel is not. Both channels are consistent in the presence of emission between 1.5 and 3.5 , consistent with HD 105's disc. The origin of the discrepancy between the HST/NICMOS and VLT/SPHERE detections potentially lies in the estimation of noise, which has been assumed Gaussian here, but it can be seen in the reduced image that the noise is non-Gaussian.

Combined modelling

The scattered light measurements from HST/NICMOS and VLT/SPHERE can be used to calculate the albedo of the dust grains. Assuming optically thin dust, and using the approximation from Weinberger et al. (1999), the maximum allowed optical depth τ times the albedo ω is 4πφ 2 S/F where S is the surface brightness of the disc in mJy/arcsec 2 , F the total star flux in mJy and φ the separation of the scatterers. Approximating the optical depth by τ = 2 f φ cos(i)/[dφ(1 -ω)] where dφ is the disc width in arcsec and f the fractional luminosity, we obtain:

S = f Fω (2π × φ × dφ × cos(i))(1 -ω) (2)
Using a disc semi-major axis φ = 2.16 with a width dφ = 0.75 at an inclination i = 50 • we obtain an albedo of 0.15 based on the NICMOS surface brightness (15 µJy/arcsec 2 ), ). The emission peaks at 5.9 µJy/arcsec 2 at a separation of 2.3 , denoted by the black solid line in the figure; the black dashed line denotes the 5-σ surface brightness limit. The red and blue lines denote the measured brightness and 5-σ limits for the left and right channels, respectively. We note that the two channels are very different, but are consistent in they reveal significant emission between 1.5 to 3.0 from the star on the same orientation as the disc in the ALMA image. See text for details. or 0.06 based on the SPHERE measurement (5.9 µJy/arcsec 2 ). The apparent surface brightness can be significantly enhanced in case of strong forward scattering (Hedman & Stark 2015;Milli et al. 2017b;Olofsson et al. 2016) but given the inclination of the HD 105 disc scattering angles below 40 • are not probed.

With the radial averaging, we have been able to measure dust albedoes a factor of 3 to 10 deeper than expected. The detection of HD 105 in scattered light made here points the way to recovery of similarly faint discs in scattered light where the disc geometry is already known from spatially resolved continuum imaging. Deeper near-infrared or optical images are required to spatially resolve the scattered light disc, in particular to determine the degree of asymmetry in the dust scattering.

3.4. Constraints on the dust composition Using the mean surface brightness from the scattered light detections we can use the inferred albedoes to constrain the dust grain composition. We calculate the scattering albedo ω = F scat /(F scat + F therm ), where F scat is the scattered light fractional luminosity in the F160W filter, and F therm is the dust continuum fractional luminosity. Following Choquet et al. (2018), we assume that the debris disc to be composed entirely of dust grains with a single size and of a single composition. It is further assumed that the same dust grains responsible for the scattered light are also responsible for the thermal emission so the value of F therm is therefore held fixed for all the tested combinations of grain size and compositions, i.e. L dust /L = 2.6 × 10 -4 . For consistency, we keep the dust mass of the disc fixed to that calculated for HD 105 based on its ALMA mm flux.

For each combination of grain size and composition we calculate the scattered light brightness of the disc at the observed orientation and inclination of HD 105, convolving the disc's scattered light SED with the F160W bandpass to determine the scattered light fractional brightness F scat , and hence the scattering albedo ω. The scattered light modelling is carried out using the 3-D radiative transfer code HYPERION (Robitaille 2011). We test grain sizes from 0.1 to 500 µm, and dust compositions of pure astronomical silicate, and water ice:astronomical silicate mixtures in the ratios 10:90, 30:70, and 50:50. Optical constants for the materials were taken from Draine (2003) (astronomical silicate) and Li & Greenberg (1998) (water ice). Calculation of the optical constants for composite materials was done using the Bruggeman effecive medium theory (Bruggeman 1935).

We find that dust grain sizes of a few microns, and either pure silicate or moderately icy compositions, are consistent with the dust albedo derived from the NICMOS-observed mean disc surface brightness.

Revised SED model

Figure 5. Plot of dust grain size vs. scattering albedo illustrating the constraints on dust grain size and composition possible with the new scattered light detection of HD 105's disc. The horizontal black lines denote the expected dust albedo required to reproduce the mean surface brightness from NICMOS (solid, ω = 0.15) and SPHERE (dashed, ω = 0.06). The coloured lines denote the scattering albedoes for discs composed of dust grains with a single size and composition. We find that silicate (or slightly icy) dust grains around a few µm in size are consistent with the NICMOS-derived scattering albedo; larger grains (≥ 10 µm) are required to reproduce the SPHEREderived albedo, but this would be inconsistent with the typical grain size of dust around Sun-like stars of a few to ten times the blow-out radius.

Here we combine the results previously determined from each dataset in order to model the full system. We use the disc architecture, as determined from the ALMA image, and the dust scattering albedo measurement from the scattered light images, as constraints in a radiative transfer model of the debris disc. We model the disc as being an annulus lying 70 to 100 au from the star with uniform radial surface brightness, as per the ALMA image. The disc is composed of compact, spherical dust grains described by a power law size distribution (dn ∝ a -q da) with exponent q spanning a size range from a min to a max . The minimum dust grain size is a free parameter, but the maximum grain size is fixed at 10 mm to best replicate the millimetre wavelength photometry. The dust composition is assumed to be pure astronomical silicate (Draine 2003).

The resultant disc SED based on the best-fit model parameters is presented in Fig 6, and a summary of the best-fit disc properties (including the disc extent as fixed input) is provided in Table 4. The stellar photosphere component (dashed black line) is represented by a Castelli-Kurucz model (Castelli & Kurucz 2004) of appropriate to the stellar spectral type (log g = 4.5 cm/s 2 , T eff = 6000 K, [Fe/H] = 0.02) scaled to the optical and near-infrared fluxes (green data points). Blue and red data points denote Spitzer/IRS and Herschel/PACS fluxes, respectively. Grey data points are the excess (i.e. total -star) fluxes for wavelengths > 20µm. The individual models comprising the model grid used fit the observations is represented by the set of solid light grey lines, with the best-fit disc model denoted by the dashed dark grey line. The total model of the system (star+disc) is denoted by the solid black line.

DISCUSSION

There is a growing body of work examining the resolved extent of debris discs as a function of their host stars' luminosities (e.g., [START_REF] Booth | [END_REF]Morales et al. 2013;Pawellek et al. 2014;Pawellek & Krivov 2015). Generally, the resolved extent of debris discs around lower luminosity stars are found to be greater than would be predicted from their blackbody temperature when compared to debris discs around higher luminosity stars. We calculate the parameter Γ (R disc /R bb ) for HD 105's disc using the relationship between stellar luminosity and disc temperature (radius) derived in Pawellek & Krivov (2015), tacitly assuming a dust composition of ice and astronomical silicate. We find Γ = 2.43 ± 0.42, whereas a value of 4.94 ± 0.42 would have been predicted from the relationship shown in Pawellek & Krivov (2015). The dust grains that constitute the disc would thus be larger than we expect from Pawellek & Krivov (2015).

However, the minimum grain size inferred from our revised SED model is 3.25 +3.00 -1.00 µm, consistent with the expectations of Pawellek & Krivov (2015). We also infer a size distribution exponent of q = 3.55 +0.30 -0.15 . Our fit to the SED in the submillimetre is dominated by the far-infrared photometry, particularly the PACS 160 µm. In combination, the best-fit grain size and slopes parameters result in a steeper millimetre SED than inferred from the 9-mm ATCA data point. A relatively bright point source resides at 10 from HD 105 in the ALMA image. Whilst this could be responsible for contaminating the APEX/LABOCA 880 µm data point, the ATCA beam was 5×4 , so it would be well separated from the disc at 9 mm. Alternatively, stellar chromospheric emission could be con-tributing to the 9-mm flux measurement; enhanced emission (above that predicted from Rayleigh-Jean extrapolation) has been observed at millimetre wavelengths for several stars (e.g. MacGregor et al. 2013;Liseau et al. 2015;Chavez-Dagostino et al. 2016). We might also infer that the dust grains have enhanced emissivity at sub-millimetre wavelengths, an inferrence supported by the relatively shallow millimetre spectral slope index β = 0.3. Increasing the emissivity of dust grains at millimetre wavelengths can be achieved in many ways, including increasing the porosity or adopting a core-mantle model.

As an additional data point in the menagerie of resolved debris discs around Sun-like stars, we can examine the properties of the system in contrast to discs around stars of similar spectral type. The Jena catalogue of resolved debris discs10 has 16 debris disc host stars that fit this criterion, taken from various literature sources (Krist et al. 2010;Ertel et al. 2011;Krist et al. 2012;Eiroa et al. 2013;Marshall et al. 2014b;Soummer et al. 2014;[START_REF] Kennedy | [END_REF]Dodson-Robinson et al. 2016;[START_REF] Choquet | [END_REF]Lieman-Sifry et al. 2016;Moór et al. 2016;Morales et al. 2016). Amongst this cohort, the properties of HD 105's disc are fairly typical, having a radius of ∼ 90 au (70 to 150 au), a typical dust temperature around 50 K (44 to 112 K), and a fractional luminosity of a few ×10 -4 (1×10 -5 to 5×10 -3 ). However, the constituents of this data set are heterogeneous, the discs having been resolved through various methods (optical, far-infrared, and millimetre) and the host stars having a wide variety of ages.

Through angular averaging we have managed to obtain a detection of the disc in scattered light. The angular averaging method, used here for the first time, facilitated a detection of disc scattered light and a deeper probe of the dust albedo than obtainable from standard analysis of the imaging observations through injection of a disc model; application of the angular averaging method to similar cases of continuum bright discs that remain undetected in scattered light are expected to yield further detections. The architecture of HD 105's disc bears striking resemblance to those of both HD 207129 (Krist et al. 2010;Marshall et al. 2011;Löhne et al. 2012) and HD 377 (Steele et al. 2016;[START_REF] Choquet | [END_REF]). The albedo determined from the scattered light brightness calculated here is comparable to that found for dust grains in imaged discs of comparable fractional luminosity around nominally similar stars e.g. HD 207129 (albedo ∼ 0.05 Krist et al. 2010), and the properties of planetesimals in the Solar system.

CONCLUSIONS

We have presented an analysis combining the interpretation of high resolution stellar spectra alongside new and archival scattered light, far-infrared, and (sub-)millimetre imaging observations of HD 105 and its circumstellar disc. In combination, these observations have provided the basis for a more detailed interpretation of the system, including its architecture and evolutionary state.

From a Bayesian approach applied to stellar evolution models we confirm its youth, with an age of 50 ± 16 Myr. Additionally, we found activity indicators which are consistent with being at a pre-main sequence evolutionary stage. The disc's planetesimal belt is spatially resolved at millimetre wavelengths. The best-fitting architecture is found to be a single annulus with a semi-major axis of 85 au and a width of 30 au at a moderate inclination of 50 • . The disc is more compact than would be predicted from application of simple stellar luminosity-disc radius relationships, suggesting that the dust grains comprising the disc are either large, or efficient emitters at (sub-)millimetre wavelengths (e.g. through increased porosity del Burgo et al. 2003).

There is no evidence of any offset between the planetesimal belt and the stellar position or any evidence of nonaxisymmetric structure in the disc, either of which might imply the presence of a low mass companion interacting with the disc. No evidence of significant CO 2-1 line emission was found from the area of the disc, consistent with recent discoveries of gas in young (< 60 Myr) debris discs being found in discs around A-type stars.

Using an angular averaging technique to measure the scattered light radial profile, we have obtained a first detection of HD 105's debris disc in scattered light, determining a value for the dust albedo between 0.15 and 0.06. The application of this new technique to similar extant data sets, where the spatial extent of the disc is known from continuum emission but scattered light imaging has proven fruitless, should easily yield additional detections of debris discs in scattered light. This will provide an expanded pool of systems from which inferences can be drawn regarding their scattered light and thermal emission properties.

Comparison of the dust albedo inferred from the scattered light with simple grain models and simplifying assumptions with water ice/astronomical silicate compositions allows us to infer that the dust grains are consistent with either pure astronomical silicate such as is typically assumed for debris dust, or a moderately icy composition as might be expected for material lying well beyond the snow-line of the system and consistent with SED modelling of a number of other debris discs.

Figure 1 .

 1 Figure1. Segments of the FEROS spectrum covering the Hα (top) and CaII H&K (bottom). The median stellar spectrum from X FEROS observations is presented in black, whilst the best-fitting model is presented in red. Some regions have been blanked out due to contamination from telluric lines. The residual (observed-model) spectrum is shown for clarity in the Hα panel.

3. 2 .

 2 ALMA millimetre imaging data An ALMA band 6 (1.3 mm) observation for HD 105 was downloaded from the ESO ALMA Science Archive 7 . The observation was originally carried out as part of project 2012.1.00437.S (PI: D. Rodriguez) during Cycle 1. The spectral setup consists of four windows. Three windows were set

Figure 2 .

 2 Figure 2. ALMA 1.3 mm (Band 6) naturally weighted, dirty image of HD 105, the best fit annular model for the disc, and the residuals in the observed image after subtraction of a model convolved with the dirty beam (from left to right). Orientation is north up, east left. The colour scale for the observations and residuals is linear between -3-σ and +3-σ. The ALMA beam is represented by the black ellipse in the bottom left of the observation. Contours are at -2, 2, 4, and 6-σ, with the broken contour denoting negative values.

Figure 3 .Figure 4 .

 34 Figure 3. Left: HST/NICMOS F160W image of HD 105, smoothed by convolution with a model PSF. Significant emission from the disc is recovered at the ansae. The '+' symbol denotes the stellar position, the circular dashed line denotes the numerical mask, and the dashed elliptical line denotes the extent and orientation of the debris disc from the ALMA image. Orientation is north up, east left. Middle: Signal-to-noise map of the HST/NICMOS image also smoothed by convolution with a model PSF. The noise was estimated from from the pixel-wise standard deviation of reference stars PSF-subtracted with the same parameters as for HD 105(Choquet et al. 2018). Right: Results of elliptical averaging of the HST/NICMOS F110W and F160W images, revealing significant emission at the orientation and semi-major axis of HD 105's disc as imaged by ALMA (i = 50 • , θ = 15 • ). The signal peaks in the F110W filter around 10µJy/arcsec 2 at a separation of 2.2 , and the signal in the F160W filter peaks at 15µJy/arcsec 2 at a similar separation. The solid blue (red) line denotes the extended emission in F110W (F160W) filter, whilst the dashed red (blue) line denotes the 5-σ surface brightness limit in the respective filters.

Figure 6 .

 6 Figure6. SED of HD 105, from power law modelling. Green dots denote optical and infrared photometry used to scale the stellar photosphere model. Blue lines and dots denote Spitzer/IRS data. Orange dots denote Herschel/PACS data. White dots denote ancilliary photometry from various literature sources (see Table1). Uncertainties are 1σ. Dark grey data points denote excess values. The greyed-out region denotes the envelope of disc models produced from the grid parameter space. The dashed line denotes the individual stellar photosphere and disc components, whilst the solid line is the total emission.

Table 2

 2 Summary of stellar parameters.

	Parameter	Value	Ref.
	Right Ascension (hms)	00 05 52.54
	Declination (dms)	-41 45 11.0
	Proper motions (mas/yr)	97.96, -76.51
	Distance (pc)	38.85 ± 0.08
	V (mag)	7.513 ± 0.005
	B -V (mag)	0.600 ± 0.003
	Spectral type	G0
	Luminosity (L )	1.216 ± 0.005
	Radius (R )	1.009 ± 0.003
	Mass (M )	1.116 ± 0.012
	Temperature (K)	6034 ± 8
	Surface gravity, log g	4.478 ± 0.006
	Metallicity, [Fe/H]	0.02 ± 0.04
	ν sin i (km/s)	17.5 ± 0.5
	Age (Myr)	50 ± 16
	References. -1. van Leeuwen (2007); 2. Lindegren et al. (2016);

Table 3

 3 Architecture of HD 105's disc determined from the ALMA image.

	Parameter	Range	Values Spacing	Value
	Inclination ( • ) Position Angle ( • )	0 -90 0 -90	19 19	Linear Linear	50 ± 5 15 ± 5
	Semi-major axis (au) 50 -150	41	Linear	85 ± 5
	Semi-major axis ( )				2.16 ± 0.13
	Belt width (au)	10 -50	21	Linear	30 ± 10
	Belt width ( )				0.75 ± 0.25
	S.B. exponent, α	0.0	1	Fixed	0.0
	Total disc flux (mJy)	1.0 -4.0	301	Linear	1.45 ± 0.28
	Dust mass (M ⊕ )				0.035 +0.013 -0.009
	up to measure the continuum, each with 128 channels over
	2 GHz bandwidth. The fourth covered the 12 CO (2-1) line at
	230.538 GHz and sampled its 0.94 GHz bandwith with 3840
	channels (0.32 kms -1

Table 4

 4 Modelling results from power law fit to disc emission.

	Parameter	Range	Values Spacing	Value
	Semi-major axis (au)		1	Fixed	85
	Belt width (au)		1	Fixed	30
	α	0.0	1	Fixed	0.0
	a min (µm) amax (µm)	0.5 -20.5 10 4	41 1	Linear Fixed	3.25 +3.00 -1.00 10 4
	q Composition	2.50 -4.50 -	11 1	Linear -	3.55 +0.30 -0.15 Astro. sil.

The Cornell Atlas of Spitzer/IRS Sources (CASSIS) is a product of the Infrared Science Center at Cornell University, supported by NASA and JPL.

Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.

6 HIPE is a joint development by the Herschel Science Ground Segment Consortium, consisting of ESA, the NASA Herschel Science Center, and the HIFI, PACS and SPIRE consortia.

http://almascience.eso.org/aq/

https://archive.stsci.edu/prepds/alice/

http://www.astro.uni-jena.de/index.php/theory/catalog-of-resolveddebris-disks.html

We thank the anonymous referee for their constructive criticism that improved the manuscript.

Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.

This paper makes use of the following ALMA data: ADS/JAO.ALMA#2012.1.00437.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan) and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.

This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.

This research has made use of NASA's Astrophysics Data System.

This paper has made use of the Python packages ASTROPY (Astropy Collaboration et al. 2013, 2018), SCIPY (Jones et al. 2001-), MATPLOTLIB (Hunter 2007), and HYPERION (Robitaille 2011).

JPM acknowledges research support by the Ministry of Science and Technology of Taiwan under grants MOST104-2628-M-001-004-MY3 and MOST107-2119-M-001-031-MY3, and Academia Sinica under grant AS-IA-106-M03.

EC acknowledges support from NASA through Hubble Fellowship grant HF2-51355 awarded by STScI, which is operated by AURA, Inc. for NASA under contract NAS5-26555, for research carried out at the Jet Propulsion Laboratory, California Institute of Technology. This work is based on data reprocessed as part of the ALICE program, which was supported by NASA through grants HST-AR-12652 (PI: R. Soummer), HST-GO-11136 (PI: D. Golimowski), HST-GO-13855 (PI: E. Choquet), HST-GO-13331 (PI: L. Pueyo), and STScI Director's Discretionary Research funds.

CdB acknowledges that this work has been supported by Mexican CONACyT research grant CB-2012-183007 GMK is supported by the Royal Society as a Royal Society University Research Fellow.

LM acknowledges support from the Smithsonian Institution as a Submillimeter Array (SMA) Fellow.


Lihong Liu
email: lihong.liu@etu.unistra.fr

Thierry Engel

Manuel Flury
email: manuel.flury@insa-strasbourg.fr

Marco De Visser

Faceted structure: a design for desired illumination and manufacture using 3D printing

A new optical component of transmissive faceted structure that allows the reshaping of collimated white light beam is depicted and realized. After being illuminated, each facet element in the structure can slightly tilt along its own axes to deflect the incident light. The calculation of the tilt angles is made by a numerical approach, which is the inverse solution of the analytical relationships between the two-dimensional tilt angles of the facet element and the local coordinates of the outgoing ray on the target plan. For a predefined illumination pattern on a fixed screen, the numerical computation of the 2D tilt angle matrix of the faceted structure is described. In the meantime, a Monte Carlo ray tracing program is created to calculate and verify the irradiance map, all compared with well-recognized commercial illumination software programs. Afterwards, the component is fabricated using an innovative additive technology, inspired by 3D printing and proposed by Luximprint.

Introduction

The shaping of white light beam with high quality is important in non-imaging optics. Various kinds of optical elements have been proposed to reshape light with broad spectrum. Lightpipe is used for example in the micro display and is focused on the efficiency and uniformity of the light on the output plan [START_REF] Bokor | Anamorphic, adiabatic beam shaping of diffuse light using a tapered reflective tube[END_REF][START_REF] Fournier | Method to improve spatial uniformity with lightpipes[END_REF][START_REF] Van Derlofske | Analytical model of flux propagation in light-pipe systems[END_REF][START_REF] Fournier | Optimization of freeform lightpipes for light-emitting-diode projectors[END_REF][START_REF] Moreno | Output irradiance of tapered lightpipes[END_REF][START_REF] Cheng | Illuminance formation and color difference of mixed-color light emitting diodes in a rectangular light pipe: an analytical approach[END_REF][START_REF] Chen | Light pipe line beam shaper[END_REF]. In the areas of solar energy, non-imaging Fresnel lens have been designed as the concentrators [START_REF] O'gallagher | Nonimaging solar concentrator with near uniform irradiance for photovoltaic arrays[END_REF][START_REF] Kritchman | Efficient Fresnel lens for solar concentration[END_REF][START_REF] Leutz | Design of a nonimaging Fresnel lens for solar concentrators[END_REF][START_REF] Xie | Concentrated solar energy applications using Fresnel lenses: A review[END_REF][START_REF] Ryu | Concept and design of modular Fresnel lenses for concentration solar PV system[END_REF]. Freeform optics for beam shaping involves a series of calculation and optimization methods [START_REF] Miñano | Free-form optics for illumination[END_REF][START_REF] Miñano | High-efficiency free-form condenser overcoming rotational symmetry limitations[END_REF][START_REF] Michaelis | Cartesian oval representation of freeform optics in illumination systems[END_REF][START_REF] Ries | Tailored freeform optical surfaces[END_REF][START_REF] Hicks | Controlling a ray bundle with a free-form reflector[END_REF]. But the fabrication of the surface profile of the free-form is still very complicated and expensive. Some other solutions are also possible in the field of faceted reflector. But the overall geometrical structure is often rotational symmetric, and arbitrary intensity profile has not been reported yet [START_REF] David | Faceted reflector design for uniform illumination[END_REF]. In computer graphics domain, Weyrich et al. create the idea of "microfacet height field" by series of algorithms to obtain a custom reflectance [START_REF] Weyrich | Fabricating microgeometry for custom surface reflectance[END_REF]. Independently, we see some faceted structures in micro dimensions called mirror cells arrays fabricated by two photon polymerizations in the recent literature [START_REF] Asoubar | Customized homogenization and shaping of LED light by micro cells arrays[END_REF][START_REF] Harnisch | Optimization of hybrid polymer materials for 2PP and fabrication of individually designed hybrid microoptical elements thereof[END_REF].

We propose another original approach based on geometrical optics to allow white light beam shaping and to achieve arbitrary intensity profile on a fixed plan. The optical configuration contains a white light source, the transmissive faceted structure, and the detector. The faceted structure is composed of a matrix of square facets distributed in two directions. The geometrical dimensions of the facets are large with respect to the wavelength in this work.

The analytical mapping from the tilt angles of the facet element to the ray coordinates on the target plan is calculated. With the reverse solution of the analytical mapping, the 2D tilt angles matrix of the faceted structure can be computed numerically, when a desired illumination pattern is demanded. Then, using a house-made Monte Carlo raytracing program [START_REF] Patow | A survey of inverse surface design from light transport behavior specification[END_REF], the irradiance map can be calculated directly in Matlab (local illumination). Afterward, the calculated faceted model is transferred to the optical software Zemax OpticStudio 16 or LightTools 8.2 from Synopsis (global illumination), which confirm the light redistribution results on the detector. The local illumination result in Matlab and the global illumination result in illumination software programs are numerically compared with each other. The realization of the designed component with an innovative technology of 3D Printing proposed by Luximprint [START_REF] Luximprint | Additive Fabrication Services for Functional and Decorative Optical Plastics[END_REF] is also demonstrated.

Optical configuration

The optical configuration is composed of a source of visible light (S), the transmissive faceted structure (FS) and the detector (D), given in Fig. 1. In our design, the incident beam from the source (S) is collimated and uniform. The transmissive faceted structure employs a matrix of N Fx × N Fy facets. FS has a flat front surface and a faceted back one. The incident rays, which are normal to the front surface, are refracted by the output facets whose angles are modified along the axis x and y to get the desired illumination on D. Here, a complete coordinate system is defined: the global coordinates are (x, y, z), the local coordinates on the detector are (X, Y). O (0, 0, 0) is the center of the faceted structure. D o is the center of the detector, with the global coordinates (0, 0, d FD ) and the local coordinates noted (0, 0). d FD is the distance between the rear base plan of FS and D.

The In Fig. 2, F (m, n, 0, 0) is the center of the facet F m,n , whose position remains unchanged during the angular modification. In the y-z plan [Fig. 2

β α α β ⋅ ⋅   - -⋅ + -⋅ +     ⋅ ⋅   = - -⋅ + -⋅ +     ⋅ ⋅ ⋅ ⋅   +      (1)

Analytical equations

The unit normal vector F n  of the facet element F m,n is given as:

, , 2 2 , , (tan , tan , 1)
,

1 tan tan m n m n F m n m n n β α β α - = + +  ( 2
)
where the vector F n  is pointing from the back surface towards the incident light.

The unit vector of the refracted ray t  is derived from the Snell's law [START_REF] Ries | Tailored freeform optical surfaces[END_REF], and is given by:

2 1 1 2 2 cos 1 sin , i t F n n t s n n n θ θ   = + --        (3)
where, n 1 (resp. n 2 ) denotes the refractive index of the faceted structure (resp. surrounding material). In our case, the surrounding material is air (n 2 = 1).

The unit vector of the parallel incident ray s  is given by: 0 0 , 1

s     =        (4)
where θ i (resp. θ t ) is the incident (resp. refracted) angle. The incident angle θ i is calculated according to the dot product of F n  and s  , and is given by:

2 2 , , 1 cos . 1 tan tan i F m n m n n s θ β α = -⋅ = + +   (5)
The refracted angle θ t is calculated according to the Snell's law in the trigonometric form and is given by: ( )

2 2 1 2 2 , , 1 sin 1 . 1 tan tan t m n m n n θ β α   = -     + +   (6)
The unit vector of the refracted ray ( )

, ,

x y z t t t t = 
is calculated with:

( )

, 2 1 1 , 1 tan cos 1 sin tan , m n i t F m n b t n s n n b n b β θ θ α ⋅     = ⋅ + ⋅ -- = ⋅     -      (7)
where the parameter b is given by: ( )

2 2 2 2 2 1 , , 1 , , 2 2 ,
,

1 tan tan tan tan . 1 tan tan m n m n m n m n m n m n n n b β α β α β α -+ + - + = + + (8)
To describe the trajectory of one refracted ray coming from the faceted structure to the detector, the mathematical line equation for one refracted ray in the 3D space is then given by Eq. ( 9):

( ) ( ) ( ) , , , , , , , , , , x y
z x F m n u v x t k y F m n u v y t k z F m n u v z t k  = ⋅ + ⋅  = ⋅ + ⋅   = ⋅ + ⋅        (9)
In Eq. ( 9), k is a parameter for the line function, ( )

, , , F m n u v 
is already given by Eq. ( 1).

The function of the plan of the detector is given by:

2 2 2 2 . D D D D FD l x l l y l z d - ≤ ≤   - ≤ ≤   =  (10)
The intersection between the refracted ray and the plan of the detector is the ray incident point on the detector ( )

, , , D m n u v 
. By solving the simultaneous set of Eq. ( 9) and Eq. ( 10), the value of the parameter k is calculated for ( )

, , , D m n u v  in global coordinates (x, y, z), as followed: ( ) , , , , FD z d F m n u v z k t - ⋅ =   (11)
In Fig. 1, the local coordinates (

) ( ) ( ) , , , , , , , D m n u v X D m n u v Y ⋅ ⋅ 
 on the detector equals the global coordinates

( ) ( ) ( ) , , , , , , , D m n u v x D m n u v y ⋅ ⋅ 
 in the optical system. In conclusion, when the rear face of the facet element F m,n tilts with (α m,n , β m,n ) along the axis x and the axis y, the local coordinates of the refracted ray (m, n, u, v) on detector is given by: (

x mn mn x y mn mn y D m n u v X f F m n u v x t k D m n u v Y f F m n u v y t k α β α β  ⋅ = = ⋅ + ⋅   ⋅ = = ⋅ + ⋅           (12) ) ( ) ( ) ( ) ( ) ( ) , , , , , , , , , , , . , , , , , , ,

Calculation of the local illumination result in Matlab

Section 3 gives the mathematical mapping (f x , f y ) between the 2D tilt angles (α m,n , β m,n ) and the local coordinates

( ) ( ) ( ) , , , , , , , D m n u v X D m n u v Y ⋅ ⋅ 
 on the detector. With the reverse solution ( )

1 1 , f f α β - -
, the numerical computation of the local illumination result in Matlab is now explained. Here, we give a sketch to declare the general steps in the computation of the irradiance map in Matlab, and the parallel simulation in Zemax and LightTools. In Fig. 3, the first step ( )

1 1 , f f α β - -
is the numerical calculation of the tilt angles (α m,n , β m,n ), depending on the desired illumination pattern, which is explained in section 4.1. The second step (f x , f y ) is the numerical calculation of the local illumination result in Matlab, which is explained in section 4.2. In parallel, from the reverse solution ( )

1 1 , f f α β - -
, the calculated faceted model is also transferred into Zemax and LightTools. The non-sequential raytracing is progressed in the two-optical illumination software. So, the global illumination result in Zemax and LightTools is compared with the local illumination result in Matlab.

In Matlab, the ray is launched randomly, and is refracted only once on the surface. The local illumination only considers the ray path in the direct forward sequence without the real physical effects as reflection losses or diffraction [START_REF] Patow | A survey of inverse surface design from light transport behavior specification[END_REF]. In Zemax and LightTools, the nonsequential raytracing, considers multiple reflections, inter-reflection, retro-reflection, split, etc.

Calculation of the tilt angles

After refraction from FS, the incident beam is divided into N Fx by N Fy sub-beams. Due to the facet surface flatness considered here, each sub-beam (m, n) has one predefined direction. The desired illumination pattern is prescribed by the local coordinates of the center ray (m, n, 0, 0) of each sub-beam. Here, the local coordinates are termed as

( ) ( ) ( ) , , , D D X m n Y m n . The 2D tilt
angles (α m,n , β m,n ) can be calculated in Matlab with the numerical reverse solution ( )

1 1 , f f α β - - : ( ) ( ) 1 ,
1 , ( , ), ( , ) .

( , ), ( , )

m n D D m n D D f X m n Y m n f X m n Y m n α β α β - -  =   =   (13)
The analytical approach in section 3 is developed to calculate the ray path after the inclined facets. Equation ( 12) calculates from the angles of the facets to the coordinates of the ray on the target plane, which are non-linear unordinary functions. So, inversing the equations (going from the desired ray coordinates on the target plane to retrieve the angles) is not possible with a simple analytical approach. Instead, we chose a numerical method to deal with the inverse problem in Eq. ( 13), which uses a specific function in Matlab called Vpasolve, inside the symbolic Toolbox.

Monte Carlo raytracing program

From the first step of the numerical reverse solution ( )

1 1 , f f α β - -
, the tilt angle matrix is calculated. Afterwards, the second step is developed to calculate the local illumination result in Matlab.

A Monte-Carlo method is employed using the rand function inside Matlab. N ray is the number of rays impinging on FS with a random position (x F , y F ), as follows:

( ) ( )

     = ⋅ ⋅ - ⋅           = ⋅ ⋅ - ⋅       (14)
where, rand[1, N ray ] is a 1 × N ray vector which contain random values in the interval [0, 1]. In sequence, vectors x F [1, N ray ] and y F [1, N ray ] give the x and y coordinates of N ray rays on FS.

The difference between Eq. ( 14) and Eq. ( 1) is that, for Eq. ( 1), it calculates the ray coordinates on the structure, supposing the facet index (m, n), the local index (u, v) and the tilt angle (α, β) are known in advance. Relying on Eq. ( 1), the analytical relationship between the tilt angle (α, β) and the ray local coordinates on the detector is calculated and given in Eq. [START_REF] Ryu | Concept and design of modular Fresnel lenses for concentration solar PV system[END_REF].

Equation [START_REF] Miñano | High-efficiency free-form condenser overcoming rotational symmetry limitations[END_REF] supposes the ray random positions on the faceted structure (x F , y F ) are known in advance. Afterwards, the facet index (m, n) and the local index (u, v) are derived with Eq. ( 15) and Eq. ( 16). Then, the ray local coordinates (X D , Y D ) on the detector are calculated with the forward solution (f x , f y ) in Eq. [START_REF] Ryu | Concept and design of modular Fresnel lenses for concentration solar PV system[END_REF].

In Eq. ( 15), vectors m[1, N ray ] and n[1, N ray ] denote the facet index, on which N ray rays incident, are given by:

m N N N n N N N        = ⋅               = ⋅         (15)
where "ceil" is an Matlab function, which rounds up towards the nearest integer.

In Eq. ( 16), vectors u[1, N ray ] and v[1, N ray ] calculate the ray local index on the facet element F m,n , are given by:

( ) ( ) 2 1, 1, 1, 1 2 , 2 1, 1, 1, 1 2
p u N x N m N p N p p v N y N n N p N p          = ⋅ - ⋅ + + ⋅                       = ⋅ - ⋅ + + ⋅             (16)
Based on Eq. ( 15) and ( 16), the ray local coordinates on the detector are calculated with the forward solution in Eq. [START_REF] Ryu | Concept and design of modular Fresnel lenses for concentration solar PV system[END_REF]. To display the image, the local coordinates (X D , Y D ), whose value is within [-l D /2, l D /2] are rounded to the pixel index (X p , Y p ), which are given by:

1, 1, 2 round . 1, 1, 2 round D r a y p r a y D D D D r a y p r a y D D D X N X N N l N X N Y N N l N            = +                    = +          (17)
where, the function "round" returns the value to its nearest integer. The pixel index (X p , Y p ) are integers and their values are within the interval between 1 and N D .

In Eq. ( 17), vectors X p [1, N ray ] and Y p [1, N ray ] record the pixel index of each ray on the detector. With the two vectors, the number of incident rays on every pixel of the detector is counted and recorded into a matrix with the dimension N D × N D . In the incoherent mode, the optical power on pixel p of the detector is determined by the number of rays incident on the pixel. The incoherent irradiance value on each pixel is the incident power over unit area of it. The matrix recording the number of rays on every pixel of the detector is converted to the matrix recording the incoherent irradiance value on every pixel, which is imaged to display the irradiance map in Matlab.

Design results

The local illumination result in Matlab and the global illumination result in Zemax and LightTools are presented in this section. The desired illumination pattern is a four-symmetric spaced squares (S4) on a plane, as shown in Fig. 4. FS is made of 6 × 6 facets: the size of the facet element (p x × p y ) is here 5 mm × 5 mm, to be large in comparison with the wavelength. The distance (d FD ) between the rear base plan of the faceted structure and the detector is 50 mm. The material (refractive index and transmission coefficient) of the faceted structure was defined according to the provided information from Luximprint, whose transparent 3D printing technology is adopted to realize the fabrication. The refractive index was used at the wavelength of 604 nm which corresponds to the primary wavelength of the source (Ocean Optics HL 2000) employed in our experiment. The raytracing configuration in Zemax and the 3D scheme of FS are shown in Fig. 5. In the design (Table 1), the maximum and minimum feature height (h max , h min ) of the faceted structure is critical for the fabrication to follow. It needs to consider the z coordinates of the four vertices (z1, z2, z3, z4) of each facet element F m,n . As the tilt angles of each facet is not the same, every facet element has different feature height. For the faceted structure matrix (N Fx × N Fy ), h max is the maximum feature height of the N Fx × N Fy facets, h min is the minimum feature height of the N Fx × N Fy facets. In Fig. 6, the maximum feature height (h max ) of the faceted structure is then 2.73 mm. According to Luximprint technology, the length of PMMA substrate is 2 mm. Ocean Optics HL 2000 is applied as the source for the simulation: the spectrum is given in Fig. 7. In Fig. 7, the peak intensity of Ocean Optics HL 2000 is at 604 nm. The spectrum has a strong output in the VIS-NIR band. With white light illumination, the 2D incoherent irradiance map is given in Fig. 8 with three different types of simulation. In Fig. 9, the 3D and 1D irradiance clearly reveal the high uniformity on the target plan. The high uniformity is obtained thanks to the collimated and uniform incident beam. The 2D angular modulations of the facets within such a small objective distance (50 mm) do not alter this property.

To evaluate the quality of the design, we calculate quality factors, which consist of the transmission (T), the efficiency (η), and the correlation coefficient (C). The transmission (T = P D / P i ) gives the optical power gained on the detector (P D ) to the incident power on the faceted structure (P i ). The efficiency factor (η = P T / P D ) measures the optical power in the target area (P T ) to the total power on the detector (P D ). It describes the ability of faceted structure to reshape and concentrate the light inside the targeting area. Correlation coefficient factor [START_REF] Flury | Investigation of M2 factor influence for paraxial computer-generated hologram reconstruction using a statistical method[END_REF][START_REF] Bokor | Monte Carlo method in computer holography[END_REF][START_REF] Bokor | Computational method for testing computer-generated holograms[END_REF][START_REF] Jennison | Iterative approaches to computer-generated holography[END_REF] is used here to assess the "difference" between the local illumination result in Matlab and the global illumination result in Zemax or LightTools. The function of the correlation coefficient (C MZ or C ML ), uses four sub-factors:

1 2 3 4 / / 1 2 2 2 2 2 / / 3 4 , where ( ) ( ) ( ) ( ) , , ( ) ( ) ( ) ( ) ,
.

M ZL ZL M p D p D p D D D D D D D Z L Z L M M p D p D p D p D D D D D D D D D c c C c c I p I p I p I p c c N N N N N N I p I p I p I p c c N N N N N N N N ∈ ∈ ∈ ∈ ∈ ∈ ∈ - = ⋅   ⋅ = = ⋅   ⋅ ⋅ ⋅       = - = -     ⋅ ⋅ ⋅ ⋅            (18)
where In Table 2, the main reason of the difference between the local and the global illumination is that the real physical effects like interreflection, retroreflection, are not considered in the local illumination. The correlation coefficient between the two distinct types of illumination results is above 95%, and the efficiency factors are very similar in the two cases. Both the correlation coefficient and the efficiency factor confirm the quality of the numerical method, at the same time verify the very low influence of the facet edges and the shadowing effects on the distribution of the refracted beam on the detector. In particular, the low shadowing effect is maintained in our various design examples, which is not just limited to this case.

I
For example, for the illumination pattern of the four squares with a new arrangement in the blow Fig. 10 (a), the correlation coefficient between the local illumination result in Matlab and the global illumination result in LightTools is 96.62%; the efficiency is about 99.84% with our program, and about 99.92% in LightTools. For the illumination pattern of the rectangular hole in Fig. 10 (b), the correlation coefficient between the local illumination result in Matlab and the global illumination result in LightTools is 95.24%; the efficiency is about 99.99% with our program, and about 99.93% in LightTools. For different design examples, the self-shadowing and the edges of the facets may cause severe artefacts only when the angles of the facets are large compared to the working distance. In our case, whatever the desired image, the interval of the possible angles is controlled to the minimum in the design process: this means that the self-shadowing effect is restricted to be as small as possible. Moreover, for the presented design in Fig. 8, both the chosen interval of the angles which is [-15°, 15°] and the facet element dimensions are a compromise between the shadowing effects and the fabrication specifications coming from additive manufacturing Luximprint. For the illumination pattern in Fig. 10 (a), the interval of the angles is [-17°, 17°]; for the illumination pattern in Fig. 10 (b), the interval of the angles is [-5°, 5°].

Polychromatic effect

As our target is the beam shaping of white light, the change of the refractive index due to the large spectral bandwidth will affect the design result. The spectral sensitivity of the refracted ray is calculated here to evaluate this polychromatic impact. The spectrum of Ocean optics HL 2000 is [0.26, 1] µm. The discrete refractive index information provided by Luximprint is on a smaller interval. In Fig. 11, for each sampled wavelength, the vertical axis is the maximum of ray local coordinate deviation ( ) , X Y Δ Δ on detector due to the change of refractive index ( ) ( ) ( )

1 1 1 604 n n n λ λ Δ = -
, where the refractive index at 604 nm is employed as the basis. From Fig. 11, within [0.26, 0.41] µm bandwidth, the maximum deviation is larger than one pixel (0.5mm/pixel). For Ocean Optics HL 2000 in Fig. 7, the intensity weight in [0.26, 0.41] µm is lower than 10 percent. So, the polychromatic effect within [0.26, 0.41] µm can be ignorable, which is also outside the visible region. For spectrum within [0.41, 0.87] µm, the ray coordinate deviation is lower than one pixel, which confirms the material dispersion has no impact on the design result.

Here, the chromatic dispersion effect is examined only for one fixed material, which is linked to the additive manufacturing technology. However, as we have stated previously, to limit the shadowing effect to a minimum level, the interval of the angles is controlled during the design process. At the same time, the prism dispersion effect is limited to a very low level. Chromatic effect will only be sensible if we have large interval angles. With the control of the interval of the angles, the possible material is not restricted to a specific one: it works for a variety of dispersive materials.

Except for the source polychromatic effect analysis, the impact of the source divergence and the variation of the working distance Δd FD on the quality of the illumination pattern are also evaluated. With the maintain of the quality decrease of the illumination pattern within 20 percent, the faceted structure can tolerant 2 degrees divergence of the incident beam, and 10 mm variation of the working distance. The quality decrease is measured by the four criteria: efficiency, uniformity, root means square error and correlation coefficient between the calculation result in Matlab and the simulation result in Zemax or LightTools.

Fabrication and experimental results

The fabrication of the transmissive faceted structure employed the transparent 3D printing technology from Luximprint. This optical 3D printing technology is inspired by the digital additive manufacturing process to fabricate typical array-typed component. Then, it is possible to use standard .step, .stl or .iges files to transfer the design information, which is under the requirements of the manufacturing process.

In the experimental setup, the employed source is Ocean Optics HL 2000 and is connected to an optical fiber P600-2-VIS-NIR, with a core diameter of 600 microns. Besides, the collimation of the output light is realized with a simple achromatic lens, the focal length f = 100 mm. In Fig. 12, the experiment setup in one dimension is given. In Fig. 12, a square diaphragm manufactured by 3D Printing is put after the collimator to adjust the size of the beam. Benefits from the cooperation with Luximprint, a smooth surface profile fabrication technology was chosen. A tracing paper is put after the sample, which allows the image to be recorded by a digital camera. The main advantage of the tracing paper is the possibility to observe directly the irradiance map without parallax or distortion. The distance between the sample and the tracing paper d FD is 50 mm. The used camera is a DSLR Canon Camera 1000D with a standard EF-S zoom objective 18-55 mm. The experiment results before and after the tracing paper are given in Fig. 13, and the efficiency η in the target area is around 65%. Efficiency η is calculated in Matlab platform, which uses a specific multicolor image processing.

Conclusion

This article demonstrates the design and the fabrication of a transparent faceted structure to do beam shaping of a white light source. The optical concept is based on a matrix of square facets inclined in two directions. Firstly, numerical solutions to calculate the tilt angles and the raytracing method to obtain the local illumination pattern on the target plan were explained in detail. Moreover, the global illumination results in Zemax OpticStudio 16 and LightTools 8.2 were compared with the local illumination result, which shows us the validity of our approach. The fabrication was accomplished with an innovative additive technology proposed by Luximprint with an optical transparent material: the dimensions of the facet element are 5 mm × 5 mm, and the component contains 6 × 6 facets. The experimental result of the irradiance map was recorded with a very simple optical setup and an estimation of the efficiency was also given using an image signal processing method to detect the target area. The optical concept of the faceted structure is proposed here in transmissive case. The reflective case is also conceivable with our method to take account different parameters like the off-axis angle of the source, the off-axis angle of the illumination screen, etc…

Fig. 1 .

 1 Fig. 1. General 2D view of the optical configuration with illumination source (S), transmissive faceted structure (FS), and the detector (D).

  vectors s  , F n  and D n  are normal to the plan of S, the back face of the facet element F m,n and the plane D. Here, the 2D local view of FS in the x-y plan is presented. The 2D local view of D opposite to D n  is also presented. F m,n denotes one facet element, (m, n) is the facet index: m and n is an integer in the range of [1, N Fx ] and [1, N Fy ]. The size of the facet element is p x and p y along the axis x and axis y. The size of D is l D × l D mm 2 (or N D × N D pixels). The ray mapping is important for a successful beam shaping. In Fig. 1, F (m, n, u, v) denotes the intersection point of the ray (m, n, u, v) on the back face of the facet F m,n , D (m, n, u, v) denotes the intersection point of the ray (m, n, u, v) on the detector D. (u, v) denotes the local index of the ray on the facet F m,n : the values of u and v are within [-1, 1]. Analytical equations are needed to solve the mathematical mapping between the ray intersection on the faceted structure ( ) , , , F m n u v  in global coordinates (x, y, z) and the ray intersection on the detector ( ) , , , D m n u v  in the local coordinates (X, Y). When F m,n tilts with (α m,n , β m,n ) along the global axis x and axis y, a sketch to declare the position of F (m, n, u, v) is given in Fig. 2.

Fig. 2 .

 2 Fig. 2. 2D view in the y-z plan and the x-z plan, when F m,n deflects with (α m,n , β m,n ) along the axis x and axis y. (a) The position of F(m, n, 0, v) shown in y-z plan. (b) The position of F (m, n, u, 0) in x-z plan.

  (a)], the facet tilts with α m,n along the global axis x (α m,n >0). In the x-z plan, [Fig. 2(b)], the facet tilts with β m,n along the global axis y (β m,n >0). In the 3D space, when F m,n tilts with (α m,n , β m,n ) along the global axis x and y, the global coordinates ( ) , , , F m n u v  are given by:

Fig. 3 .

 3 Fig. 3. Steps for local illumination in Matlab and global illumination in Zemax and LightTools.

Fig. 4 .

 4 Fig. 4. Desired illumination pattern S4 on a plane.

Fig. 5 .

 5 Fig. 5. The raytracing configuration in OpticStudio 16 and the 3D scheme of the faceted structure.

Fig. 6 .

 6 Fig. 6. The geometrical information of the faceted structure.

Fig. 7 .

 7 Fig. 7. Normalized spectral intensity of Ocean Optics 2000 in [0.26, 1] μm.

Fig. 8 .

 8 Fig. 8. Illumination pattern S4. (a) Local illumination result in Matlab. (b) Global illumination result in Zemax. (c) Global illumination result in LightTools. The 2D irradiance matrix obtained with LightTools is transferred to Matlab. The 3D irradiance map and 1D irradiance diagram (x = 10mm, y = 10 mm) are plotted.

Fig. 9 .

 9 Fig. 9. 3D and 1D irradiance of illumination pattern S4 in LightTools.

  M (p) is the incoherent irradiance per pixel from the local illumination result. I Z (p) (resp. I L (p)) is the incoherent irradiance per pixel from the global illumination result in Zemax or LightTools. I M (p) and I Z (p) (resp. I L (p)) are normalized before calculation, so that the correlation coefficient (C MZ or C ML ) is in the interval [0, 1].

Fig. 10 .

 10 Fig. 10. 2D irradiance of illumination patterns in LightTools. (a) Irradiance of four symmetric squares with new arrangement. (b) Irradiance of rectangular hole with 20 mm inner hole size. In the above Fig. 10. (a), the size of the facet element (p x × p y ) is 5 mm × 5 mm, the number of facets (N Fx × N Fy ) is 8 × 8; on the detector, the size of each square (L X × L Y ) is 5 mm × 5 mm, the distance between each square is 40 mm. In Fig. 10. (b), the size of the facet element (p x × p y ) is 1 mm × 1 mm, the number of facets (N Fx × N Fy ) is 20 × 20; on the detector, the size of the inner hole is 20 mm.For different design examples, the self-shadowing and the edges of the facets may cause severe artefacts only when the angles of the facets are large compared to the working distance. In our case, whatever the desired image, the interval of the possible angles is controlled to the minimum in the design process: this means that the self-shadowing effect is restricted to be as small as possible. Moreover, for the presented design in Fig.8, both the chosen interval of the angles which is [-15°, 15°] and the facet element dimensions are a compromise between the shadowing effects and the fabrication specifications coming from additive manufacturing Luximprint. For the illumination pattern in Fig.10(a), the interval of the angles is [-17°, 17°]; for the illumination pattern in Fig. 10 (b), the interval of the angles is [-5°, 5°].

Fig. 11 .

 11 Fig. 11. Spectral sensitivity of transmissive faceted structure to Ocean Optics HL 2000.

Fig. 12 .

 12 Fig. 12. 2D diagram of the optical system in the experiment.

Fig. 13 .

 13 Fig. 13. Experiment results. (a) Irradiance map taken by the camera before the tracing paper. (b) Irradiance map after the tracing paper.

Table 1 . Design Parameters Design

 1

parameters of faceted structure (FS) Value Number of facets N Fx × N Fy 6 × 6 Size of facets p x × p y [mm] 5 × 5 Distance d FD between FS and detector [mm] 50 Maximum feature height h max [mm] 2.73 Minimum feature height h min [mm] 0.84

Table 2 . Quality Evaluation of Illumination Pattern S4

 2

	Illumination pattern	S4
	Square size L X × L Y [mm]	5 × 5
	Distance between each square [mm]	20.5
	Detector (D) Size l D × l D [mm]	100 × 100
	Number of pixels N D × N D [mm]	200 × 200
		N ray [ × 10 6 ]	50
	Matlab (M)	Transmission T	1
		Efficiency η	99.11%
		Rays number N ray [ × 10 6 ]	10
	Zemax (Z)	Transmission T	90.97%
	Design evaluation	Efficiency η	97.65%
		Rays number N ray [ × 10 6 ]	50
	LightTools (L)	Transmission T	90.15%
		Efficiency η	96.56%
	Correlation coeff.	Between Matlab and Zemax C MZ Between Matlab and LightTools C ML	95.16% 95.20%

Vol. 1, No. 1 | 15 Sep 2018 | OSA CONTINUUM 28

Acknowledgments

The authors would like to acknowledge the financial support from University of Strasbourg for the fabrication of the element, the cooperation program with INSA Strasbourg, and the China Scholarship Council for the PhD Grant of Lihong Liu.


MD Flavia Maurice

Benedicte Gaborit
email: benedicte.gaborit@ap-hm.fr

MD Clara Vincentelli

PhD Ines Abdesselam

PhD Monique Bernard

MD Thomas Graillon

PhD Frank Kober

MD Thierry Brue

Frederic Castinetti

MD Anne Dutour

MD Bénédicte Gaborit

MD Frédéric Castinetti

Dr Benedicte Gaborit

Cushing Syndrome Is Associated With Subclinical LV Dysfunction and Increased Epicardial Adipose Tissue

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

Cushing syndrome (CS) results from chronic glucocorticoids (GC) excess exposure. This disease is associated with increased mortality and adverse cardiovascular (CV) outcomes due to visceral adiposity, insulin resistance, hypertension, and hypercoagulability. Interestingly, CV risk persists even after remission of the disease [START_REF] Clayton | Mortality in patients with Cushing's disease more than 10 years after remission: a multicentre, multinational, retrospective cohort study[END_REF][START_REF] Colao | Persistence of Increased Cardiovascular Risk in Patients with Cushing's Disease after Five Years of Successful Cure[END_REF] which remains partly unexplained.

The impact of cardiac adiposity on coronary artery disease (CAD) and left ventricular (LV) function has been suggested. Prospective studies have shown that epicardial adipose tissue (EAT) amount is an independent predictor of CAD (3). However, the impact of hypercortisolism on EAT is yet unknown, while it would be of interest as CS is classically associated with upper body adiposity. Our aim was to compare EAT volume and LV function in patients with active and in remission ACTH-dependent CS, and controls.

Thirty-eight patients with ACTH-dependent CS (19 active and 19 in remission for at least 2 years, mean 41.2± 4.2 months) and 19 controls of similar age and sex distribution were included. A magnetic resonance imaging was performed on a 3-Tesla wide-bore magnet to determine abdominal fat distribution, EAT volume, systolic and diastolic LV function and structure as previously described (4). The expected increase in hypertension, diabetes, dyslipidemia was evidenced in active CS (Table ) as well as visceral adiposity (higher VAT/sSCAT). Patients from the active and remission groups did not differ in uncontrolled disease duration and had comparable BMI at diagnosis.

Cushing syndrome induces EAT accumulation and subclinical LV dysfunction: Increased

EAT volume was found in active patients compared to controls (93.5±6.1 versus 55.2±3.6 mL, p<0.01) and remission patients (68.2±5.1 mL p<0.01). However, patients in remission had a higher EAT volume than controls (p<0.05.). Active patients compared to controls had higher LV mass index (LVMi 55.3±2.6 vs 43.2±2.3 g/m 2, p<0.01), concentric remodeling (ie indexed LV mass to end-diastolic volume ratio (EDVi) 1.10±0.07 vs 0.68±0.03g/mL, p<0.01) and impaired early LV relaxation (ie E/A ratio 1.34±0.07 versus 1.66±0.09, p=0.01), while LV ejection function did not differ. In remission patients, LVMi and E/A ratio were not different from controls but LVMi/EDVi was higher (0.81±0.04, p=0.02). Multivariate analysis (including age, sex, BMI, blood pressure, fasting plasma glucose and EAT) revealed that EAT volume was independently associated with LVMi (β=0.40, p=0.008) and LVMi/EDVi (β=0.41, p=0.011).

EAT is a perivascular depot releasing inflammatory adipokines that pass through the coronary wall and interact with endothelial cells, enhancing monocytes adhesion. EAT correlates with the extent and severity of CAD, chest pain, unstable angina and coronary flow reserve and predicts coronary events independently of CV risk factors .We hypothesize that together with other CV risk factors, increased EAT in CS could contribute to its excess cardiovascular morbidity.

EAT volume was independently correlated with LV mass index and mass to end-diastolic volume. This suggests a major role for EAT in CS induced LV dysfunction. Indeed, the EAT anatomical proximity to the myocardium free of any fascia gives it the distinctive function to act locally and directly on the myocardium. A link between EAT secreted adipokines and both contractile function and insulin signaling in cardiomyocytes was demonstrated (3). Furthermore, EAT volume and secretome have been linked to cardiac fibrosis, a substratum widely recognized to impair LV function [START_REF] Gaborit | Role of Epicardial Adipose Tissue in Health and Disease: A Matter of Fat?[END_REF].

In conclusion, we demonstrated a unique pattern of ectopic fat deposition in patients with active ACTH-dependent CS, with a relatively high epicardial fat accumulation. This increase in EAT was associated with subtle LV dysfunction. Furthermore, in remission patients, EAT volume was higher than in controls, possibly contributing to the persistent cardiovascular risk.

Table 1 :

 1 Characteristics of study population. Data presented as n (%) or mean ± SD or median [25 th -75 th percentile]. BMI: Body mass index, BP: Blood pressure, EAT: Epicardial adipose tissue, NS: Not Significant, sSCAT : superficial Subcutaneous adipose tissue, VAT: Visceral adipose tissue. Comparison between groups p<0.05 a active vs remission groups, b active group vs controls, c remission group vs controls, 3-groups comparison with one way-ANOVA and Holm-Sidak post-hoc test or Kruskal-Wallis and Dunn's as appropriate

		Active group	Remission	Control group	2 group	3-
		(n=19)	group	(n=19)	p value	group
			(n=19)			pvalue
	Gender, female/male, n	12/7	15/4	14/5	NS	NS
	Age, y	49.8±14.9	49.1±12.5	48.9±10.8	NS	NS
	Duration of uncontrolled CS,	25.6±15.7	32.6±31.4	-	NS	-
	months
	Comorbidities, n, %
	Hypertension	13 (68.4)	4 (21.0)	-	<0.05 a	-
	Diabetes mellitus or impaired	11 (57.9)	3 (15.8)	-	<0.05 a	-
	glucose tolerance
	Hypercholesterolemia	6 (31.6)	3 (15.8)	-	NS	-
	Smoking, n, %	6 (31.6)	4 (21.0)	4 (21.0)	NS	NS
	Clinical parameters
	BMI, kg/m 2	26.9±3.5	26.3±4.8	21.9±2.9	<0.05 b,c	<0.001
	Waist/hip ratio	0.97±0.07	0.94 [0.84-0.98]	0.88 [0.81-0.93]	<0.05 b
	Systolic BP, mm Hg	130±22	112±13	113 [105-132]	<0.05 a	0.014
	Diastolic BP, mm Hg	80±11	70±11	73±11	<0.05 a,b	0.018
						0.017
	Metabolic parameters
	Fasting plasma glucose,	5.67±1.24	4.42±0.69
	mmol/L
	HbA 1 c, %	6.71±1.65	5.66±0.49
	Serum triglyceride levels,		0.95 [0.72-1.57]
	mmol/L	1.36 [0.94-
	Total blood cholesterol	1.94]
	levels, mmol/L	5.62±1.21
	LDL-cholesterol levels,
	mmol/L	3.35±1.01
	HDL-cholesterol levels,
	mmol/L	1.48 [1.19-