Darya Antonova

  Первая каракалпакская студия в Москве / Р.Ж. Уснатов. -Текст : непосредственный // Театр. Живопись. Кино. Музыка -2016. -№ 4. -С.119-127. 6. Mangiante J.M., Parpette C. Le Français sur Objectif Universitaire / J.M. Mangiante, C. Parpette. -Текст : непосредственный // Grenoble: PUG, 2011. -252 p.

	также академическим языком [6, c.9], с которым они столкнутся в учебные пособия представлены). Соответственно, чтобы обеспечить докторской диссертации. На наш взгляд, именно целостный и
	Антонова Дарья Александровна, Магистр педагогических наук, должную лингвометодическую подготовку иноязычных студентов вузов университете. структурированный курс французского как языка искусства и языка
	докторант лаборатории DILTEC ED622, Французская идея объединить иноязычных студентов для обучения искусств, необходимо разработать для них соответствующие учебные академического является незаменимым компонентом подготовки
	преподаватель университета Новая Сорбонна, Париж, Франция их языку и их интеграции в университетское пространство не нова. Можно материалы, на базе которых преподаватели смогли бы вести занятия. новоприбывших студентов к учебе во Франции и позволяет свести к
	с уверенностью сказать, что пионером в этом вопросе был СССР: с начала Примечательно, что в вопросе составления учебных пособий именно минимуму риски, связанные с адаптацией и социализацией в новой
	ЯЗЫКОВАЯ ИНТЕГРАЦИЯ ИНОСТРАННЫХ СТУДЕНТОВ В 30-х гг. в институтах театра и кино существовали «национальные студии» российские методисты оказались на шаг впереди своих французских университетской среде.
	ХУДОЖЕСТВЕННЫЕ ВУЗЫ ФРАНЦИИ [5, c.119] для подготовки актеров -нерусскоязычных студентов из других коллег. Учебник В.С. Круговец «Французский язык для изучающих
	Во Франции интерес к обмену культурным и художественным советских республик -и для их обучения русскому языку как языку культуру и искусства» [3] можно назвать одной из первых интересных Литература
	опытом послужил толчком к росту мобильности студентов в искусства и языку для специальных и академических целей. В марте 2020 попыток в данной области. По словам автора, книга предназначена «для 1. ГИТИС Российский институт театрального искусства | Приглашаем на
	государственных и частных художественных школах и университетах. года ГИТИС организовал фотовыставку [1], которая была посвящена студентов вузов искусств, а также магистров, аспирантов и слушателей виртуальную экспозицию «Национальные студии ГИТИСa». 2020. -URL:
	Большинство иностранных абитуриентов, желающих поступить в своим первым национальным студиям. На выставке были представлены https://www.gitis.net/actors/item/vystavka-7 (дата обращения: 18.10.2020) -Текст : специальных курсов по французскому языку, деятельность которых электронный.
	художественные вузы Франции, имеют уровень французского языка А2-B2 документы об истории создания «прототипов» интернациональных связана с искусством и дизайном» [3, c.2]. Пособие содержит довольно
	подготовительных классов и преподавания искусствоведческого языка. разнообразные тексты для чтения для уровня А2-В1, а также
	В ряде художественных вузов Франции, однако, наблюдается коммуникативные задачи для развития устной и письменной речи, однако
	следующий парадокс. Анкетирование, проведенное нами в рамках такого компонента как аудирование в учебнике нет. Тематику некоторых
	диссертационного текстов также стоит пересмотреть (например, биография Эдит Пиаф [3, исследования, показало, что, несмотря на
	пособие для вузов / В.С. Круговец. -Москва: Юрайт (Серия: Университеты России), громкоговорящие названия подготовительных курсов для иностранцев -c.24]). 2018. -243 с. -Текст : непосредственный.
	«Французский язык + искусство», «Французский язык + дизайн» и т.д. -Мы же, благодаря анкетированию преподавателей, определились с 4. Совет Европы | CEFR, уровни и профили -URL:
	сами преподаватели в большинстве своем не воспринимают обучение тематикой и типологией аутентичных текстов. Так, для того, чтобы
	Чтобы решить данную проблему, во многих частных или французскому языку как интегрированный компонент, неразрывно студенты овладевали терминологией и умели пользоваться французским
	государственных высших школах искусств (например, École Supérieure связанный с преподаванием дисциплин искусства. В результате вместо языком как инструментом для решения профессиональных задач [2], мы
	d'Art Dunkerque-Tourcoing, École IntuitLab, 3iS -Institut International de овладения языком для специальных целей студенты изучают язык отдали предпочтение видеоурокам по использованию программ Adobe
	l'Image et du Son и др.) существуют подготовительные курсы, повседневный, что влечет за собой острую нехватку методической и Photoshop, Blender и т.д. Оказалось также, что сами «продукты искусства»,
	предназначенные только для иностранных студентов, не имеющих лингвистической подготовки к последующему слушанию лекций на такие как реклама, 3D анимация, фильмы и видео «making of», могут быть
	достаточного уровня французского языка для поступления в вуз на первый французском языке, выполнению академических письменных заданий, использованы в качестве «пусковых механизмов», на базе которых можно
	курс бакалавриата. Обучение, как правило, длится один академический устной защите творческих портфолио и проектов. Мы довольно быстро организовывать профессиональные дискуссии, способствующие развитию
	год, в течение которого, помимо подготовки к вступительным конкурсам обнаружили корень этой проблемы: полноценных учебно-методических навыков понимания и интеракции.
	по основным предметам, иностранные студенты совершенствуют свое пособий по французскому языку для изучающих искусства нет (хотя для Осознанный отбор аутентичных текстов и создание к ним
	владение французским как языком художественной специализации, а студентов-юристов, экономистов, инженеров, медиков и т.д. подобные коммуникативных упражнений составляют конечную цель нашей

[4], что подтверждается дипломами DELF и DALF или сертификатами TCF или TEF. Независимо от уровня, будущие студенты часто испытывают обоснованный страх перед поступлением в университет. Поскольку если общий французский язык изучался абитуриентами, о чем свидетельствует наличие одного из вышеупомянутых дипломов, то французский как язык искусства и академический язык с его специальной лексикой и дискурсивными особенностями никем не преподавался и не оценивался. 2. Давыденко Ю.И. Деятельностный подход при обучении английскому языку // Ю.И. Давыденко. -Текст : электронный // Труды БГТУ. Серия 6: История, философия. -2016. -№5 (187). -URL: https://cyberleninka.ru/article/n/deyatelnostnyy-podhod-priobuchenii-angliyskomu-yazyku (дата обращения: 18.10.2020). 3. Круговец В. С. Французский язык для изучающих культуру и искусства : учеб. https://www.coe.int/ru/web/lang-migrants/cefr-and-profiles (дата обращения: 18.10.2020) -Текст : электронный. 5. Уснатов Р. Ж.


Marianne Beaumont

Irène Ventrillard

Daniele Romanini
email: daniel.romanini@univ-grenoble-alpes.fr

Optical cavity spectroscopy using heterodyne detection with optical feedback laser frequency locking

We demonstrate an accurate high sensitivity method for cavity spectroscopy. We measure the frequency intervals of transverse electromagnetic modes relative to a fundamental mode in a high finesse optical resonator, and attribute their mode numbers unambiguously. A laser is frequency locked to a fundamental TEM 00 cavity mode by optical feedback, and phase modulation is used to obtain frequency side bands, which may come to resonance with other transverse cavity modes as the radio-frequency of the modulation is tuned. At these resonances, transmission of the side bands is sensitively detected by heterodyning with the carrier. We also analyse the transverse spatial profile of the heterodyne signal for identification of mode numbers.

The adjustment of the Gaussian cavity model to the measured frequency intervals yields accurate values of cavity length, mirrors radius of curvature and mirrors ellipticity, to the ppm level.

Introduction

Optical resonant cavities are widely used in different domains of optics, such as interferometry [START_REF] Magaña-Sandoval | Sensing optical cavity mismatch with a mode-converter and quadrant photodiode[END_REF], spectroscopy [START_REF] Romanini | Introduction to Cavity Enhanced Absorption Spectroscopy[END_REF], or time-frequency metrology [START_REF] Dahmani | Frequency stabilization of semiconductor lasers by resonant optical feedback[END_REF][START_REF] Häfner | 8 × 10 -17 fractional laser frequency instability with a long room-temperature cavity[END_REF], since their properties are useful to increase sensitivity and/or spectral resolution. In particular, frequency locking of a laser to a cavity resonance is an effective means to stabilize its emission frequency and reduce its emission linewidth [START_REF] Drever | Laser phase and frequency stabilization using an optical resonator[END_REF]. Among other means, optical feedback (OF) may be put to profit for this, especially for distributed feedback (DFB) laser diodes [START_REF] Laurent | Frequency noise analysis of optically self-locked diode lasers[END_REF]. OF fast and efficient locking of a semiconductor laser to an optical cavity has been exploited to perform absorption spectroscopy of gas samples with high sensitivity for trace analysis applications [START_REF] Richard | Nitric Oxide Analysis Down to ppt Levels by Optical-Feedback Cavity-Enhanced Absorption Spectroscopy[END_REF][START_REF] Morville | Cavity Enhanced Absorption Spectroscopy with Optical Feedback[END_REF]. It was also exploited to obtain other types of sensitive measurements in optics such as birefringence effect [START_REF] Durand | Shot-noise-limited measurement of sub-parts-per-trillion birefringence phase shift in a high-finesse cavity[END_REF] or polarisation mode splitting [START_REF] Djevahirdjian | Gouy phase shift measurement in a high-finesse cavity by optical feedback frequency locking[END_REF].

For some applications, consideration of the transverse modes may be of importance because their frequency splitting and probability of injection depend on the geometry of the cavity and on the alignment and focussing of the input beam, respectively. Heterodyne detection of transverse modes was especially exploited in gravitational interferometers, as a probe of the correct alignment [START_REF] Magaña-Sandoval | Sensing optical cavity mismatch with a mode-converter and quadrant photodiode[END_REF][START_REF] Mueller | Determination and optimization of mode matching into optical cavities by heterodyne detection[END_REF] and to verify the geometrical parameters of the cavity, such as mirror separation and curvature radius, as well as to reveal the presence of astigmatism due to mirror deformation [START_REF] Stochino | Technique for in situ measurement of free spectral range and transverse mode spacing of optical cavities[END_REF].

In this work, we demonstrate accurate measurements of the frequency separation of the transverse electromagnetic (TEM 𝑚𝑛 ) resonances of a high finesse optical cavity with a technique relying on OF frequency locking and heterodyne detection. A diode laser is frequency locked to a TEM 00 cavity mode, and we use phase modulation to obtain frequency side bands, which may come to resonance with other transverse cavity modes as the radio-frequency (RF) of the modulation is tuned. In order to observe the occurrence of these resonances we use heterodyne detection. In other words, we detect the RF beating signal between the laser locking mode and the injected side band at cavity transmission by using a fast photodiode. Specifically, the laser-cavity frequency locking is obtained by optical feedback (as done in [START_REF] Djevahirdjian | Gouy phase shift measurement in a high-finesse cavity by optical feedback frequency locking[END_REF]), but other locking schemes have been used, such as the well-known Pound-Drever-Hall method [START_REF] Drever | Laser phase and frequency stabilization using an optical resonator[END_REF], in other approaching setups [START_REF] Magaña-Sandoval | Sensing optical cavity mismatch with a mode-converter and quadrant photodiode[END_REF][START_REF] Mueller | Determination and optimization of mode matching into optical cavities by heterodyne detection[END_REF][START_REF] Stochino | Technique for in situ measurement of free spectral range and transverse mode spacing of optical cavities[END_REF]. It should be noted that OF locking is particularly adapted to a V-shaped cavity geometry, but it can also be implemented with linear cavities [START_REF] Durand | Shot-noise-limited measurement of sub-parts-per-trillion birefringence phase shift in a high-finesse cavity[END_REF][START_REF] Manfred | Optical-feedback cavity-enhanced absorption spectroscopy in a linear cavity: model and experiments[END_REF][START_REF] Rakhmanov | Characterization of the LIGO 4 km Fabry-Perot cavities via their high-frequency dynamic responses to length and laser frequency variations[END_REF]. Moreover, contrary to the Pound-Drever-Hall method, OF locking works well with semiconductor lasers possessing high phase noise and correspondingly large linewidth. Indeed, OF locking induces a spectacular narrowing in DFB diode lasers emission, to below the resonator line width [START_REF] Laurent | Frequency noise analysis of optically self-locked diode lasers[END_REF][START_REF] Casado | High stability in near-infrared spectroscopy: part 1, adapting clock techniques to optical feedback[END_REF][START_REF] Morville | Fast, low-noise, mode-by-mode, cavity-enhanced absorption spectroscopy by diode-laser self-locking[END_REF].

Comparison of the beat note measurements with a model allows to retrieve the length of the cavity down to a few microns accuracy, and the mean radius of curvature of the mirrors down to ten microns accuracy along orthogonal transverse directions, allowing to assess mirrors ellipticity. These parameters are deduced after identification of the transverse indices 𝑚 and 𝑛 of the resonances, from intensity measurement of the transverse mode profiles. Frequency deviation from the fundamental TEM 00 mode and spectral widths of the resonances are also accurately measured, allowing assessment of the surface homogeneity of mirror losses.

Experimental setup

Our experimental setup is outlined in fig. 1. A distributed feedback diode laser (DFB), emitting at 1.65 𝜇m, is optically locked to a fundamental transverse TEM 00 mode of a V-shaped cavity.

The optical feedback from the optical cavity induces strong laser line-width narrowing, delivers intense and steady cavity transmission signal, and locks the diode laser frequency to the TEM 00 mode [START_REF] Laurent | Frequency noise analysis of optically self-locked diode lasers[END_REF][START_REF] Morville | Fast, low-noise, mode-by-mode, cavity-enhanced absorption spectroscopy by diode-laser self-locking[END_REF]. A detailed description and discussion of this type of OF locking setup can be found e.g. in [START_REF] Kerstel | A water isotope (2H, 17O, and 18O) spectrometer based on optical feedback cavity-enhanced absorption for in situ airborne applications[END_REF]. This section will provide specific modifications from the standard setup, implemented in this work for the cavity spectroscopy by heterodyne detection. The optical cavity (in blue) is a V shaped cavity made of 3 identical mirrors, with an arm length of roughly 49 cm and a small angle of incidence at the folding mirror of 0.44°. Mirror radius of curvature is specified to 100 cm. The laser is locked to a TEM 00 mode by detection of the transmitted signal ("PD lock") and by phase control of the optical feedback (OF) acting on a piezoelectric (PZT) mounted mirror. An electro-optic modulator (EOM) allows to create side bands at a radio-frequency (RF) digitally controlled. A second photodiode monitors the RF heterodyne signal. A masking blade, mounted in a translation stage, allows to cut off part of the cavity output beam before heterodyne detection.

The OF rate is controlled by an attenuator composed here of two successive polarisers, the second one having a fixed orientation in the plane of the V-cavity (𝑥-axis in fig. 1). The OF phase is controlled to a fraction of a wavelength by acting on a PZT-mounted mirror before the cavity [START_REF] Kerstel | A water isotope (2H, 17O, and 18O) spectrometer based on optical feedback cavity-enhanced absorption for in situ airborne applications[END_REF]. A photodiode (100 kHz bandpass) monitoring one of the two cavity outputs delivers the cavity transmission signal used for OF phase control. In the present implementation, the error signal for OF phase stabilization is obtained by frequency scanning the laser diode across a single TEM 00 cavity mode at a repetition rate of 10 Hz in order to derive the asymmetry of the mode profile. This allows to obtain heterodyne signal during the few ms when the laser is locked to the center of the resonance.

The laser beam is phase modulated with an EOM (electro-optic modulator) in order to create frequency side bands. The EOM extraordinary axis is aligned with the beam polarisation in order to create pure phase modulation and insignificant beam ellipticity. The beam polarisation is chosen to be in the plane of the cavity (𝑥-axis) to ensure that only horizontal polarisation cavity modes are excited thus avoiding complications associated with polarization modes and their splitting in such a cavity (due to off normal reflection on the folding mirror). The EOM is fed with a sine signal produced by a RF synthesizer (Rohde&Schwarz SMB100A) digitally controlled. We keep a low power on the side bands compared to the carrier (about 1 %), thus no RF amplifier is needed. This is largely sufficient to obtain strong heterodyne signals. As the RF modulation is swept, one or the other of the side bands can resonate with transverse cavity modes (TEM 𝑚𝑛 ) and be transmitted by the cavity together with the carrier which is always present due to the OF locking. To increase the injection of different transverse modes, the beam is not perfectly mode-matched to the cavity. One should note that OF locking to the TEM 00 mode is sufficiently robust to tolerate higher side band levels (up to several percent), and that even with a non optimal mode matching.

The optical V-cavity is made of three mirrors coming from the same batch, spaced by about 49 cm (fig. 1), with radius of curvatures specified to 1 m. This means we are close to a half-confocal configuration (which would correspond to cavity length of 50 cm), with transverse modes frequencies falling roughly at one-third (for the modes with 𝑚 + 𝑛 = 1) and two-thirds (modes with 𝑚 + 𝑛 = 2) of the cavity Free Spectral Range (FSR, which is about 150 MHz) [START_REF] Lehmann | The superposition principle and cavity ring-down spectroscopy[END_REF]. Hence, to inject TEM 𝑚𝑛 modes while the laser is locked to a TEM 00 mode, it is sufficient to sweep the RF frequency of a few MHz around 50 MHz. The modes whose frequencies are around FSR/3 are injected by the upper side band, while the lower side band injects modes whose frequencies are around 2FSR/3. As the RF is swept, the beat note is recorded on a high bandwidth photodiode placed at the second cavity output ("heterodyne PD" in fig. 1, 150 MHz bandwidth). Its signal is sampled by a high speed acquisition card (Gage Razor Compuscope, 200 MHz acquisition frequency) installed in a personal computer. Fourier transform is then computed to obtain the beat note amplitude at the RF frequency. A Labview software allows to control the RF frequency and to perform data acquisition and processing in order to record the frequency profiles of the transverse mode as a function of the RF frequency (fig. 2). The transverse mode spectral width and frequency interval from the TEM 00 can then be accurately determined by a Lorentzian fit.

In fig. 2, transmission spectra of five detected transverse modes are shown, whose frequencies fall around 50 MHz from the TEM 00 mode as expected. The small divergence of our setup from the half-confocal configuration is enough to ensure that both side bands are not injected in the cavity at once, as the frequencies of the groups of modes differ of a few MHz from FSR/3 and 2FSR/3. Furthermore, because of mirrors ellipticity, modes of same order 𝑚 + 𝑛 are not degenerated. Their frequency separation (∼ 0.5 MHz) is much larger than their spectral width (∼ 20 kHz), therefore only one side band is transmitted at a time and allows to detect individually each mode.

It is interesting to note that the heterodyne signal is different from zero only when a side band is transmitted, contrary to normal transmission measurement. Such zero background measurement is then not affected by source amplitude fluctuations which would degrade the sensitivity. In addition, even though the side band has low power and is additionally poorly coupled to the cavity transverse modes (given the non optimal mode-matching), heterodyne detection provides an important optical amplification effect before detection. This is due to the beat signal being the product of the side band field with the strong carrier field. The noise level of our measurement (fig. 2) is estimated to correspond to a side band detection limit of 60 fW. Detected side band power levels range from 1 pW to 90 pW, while the carrier power at cavity output is 8 µW. This was obtained with acquisition at 200 Msamples/s, acquired during 2.6 ms, leading to a Fourier resolution of 380 Hz. Side bands have been detected with a power ratio at cavity output with the carrier of around 10 -6 .

Mode frequencies are seen to shift (up to 100 kHz) from one acquisition to the other because the three sets of data in fig. 2 have been recorded at several days of interval. Indeed, the cavity is not thermally regulated. If direct impact of thermal cavity length change on mode frequencies is negligible for room temperature of 1 °C, mode separation is very sensitive to changes of the mirrors surface curvature (also affected by drifting mechanical stress). We see from the model (introduced in the following) that a change of 0.15 % on the radius of curvature of the mirrors shifts the transverse modes by more than 100 kHz.

The last key element in the setup is the masking blade placed before the heterodyne photodiode (fig. 1), that allows to partially cut the transmitted beam. This is necessary as the TEM cavity modes form a basis of orthogonal functions [START_REF] Yariv | Quantum Electronics[END_REF], thus the unbounded overlap integral of any two modes of different orders 𝑚, 𝑛 is null when taken over any transverse surface. This spatial orthogonality must be removed in order to observe a beat note when a side band is resonant with a transverse mode. Various methods have been previously used. For example, Mueller et al. [START_REF] Mueller | Determination and optimization of mode matching into optical cavities by heterodyne detection[END_REF] used a ring-segment photodiode whose individual segments are smaller than the transverse extent of the beam, resulting in a non-zero signal for each segment. Here, we proceed by partially masking the output beam from the cavity (as Stochino et al. [START_REF] Stochino | Technique for in situ measurement of free spectral range and transverse mode spacing of optical cavities[END_REF]), using an opaque blade placed in front of the heterodyne detection photodiode (fig. 1). As shown in fig. 2 and as discussed in section 4, according to the angle of the edge of the blade in the (𝑥𝑦) plane some modes are detected or not. The blade is mounted on a motorized translation stage in order to study the beat note amplitude as a function of the masked area. However, if the beam is not perfectly focussed on the photodiode, then it is partially cut and the beat note is non null even without using the masking blade. To ensure to have a zero background, we adjusted the alignment of the beam on the photodiode to have a zero beat note with the full beam. This is further discussed in section 4.

Transverse modes frequencies analysis

The frequencies of TEM 𝑚𝑛 modes are computed using the standard ABCD matrices formalism for Gaussian beam propagation to describe the beam path in the cavity as a single round-trip matrix ( [START_REF] Yariv | Quantum Electronics[END_REF], chap.7.2). Then, the recursive condition ( [START_REF] Yariv | Quantum Electronics[END_REF], eq. 7.3-1) on the complex phase of the beam allows to find the resonant frequencies. The cavity round-trip matrix contains a reduced number of effective parameters (curvature radii, cavity angle and cavity length) on which the transverse mode frequencies depend. To account for ellipticity, two round-trip matrices are computed, one on each transverse axis 𝑥 and 𝑦. The effect of the non-zero angle of incidence at the folding mirror (𝜃) and mirror ellipticity cumulate, as the angle of reflection on the folding mirror are equivalent to a modification of the radius of curvature of cos(𝜃) in the horizontal plane and 1/cos(𝜃) in the vertical plane. We then fixed the angle to 𝜃 = 0.44°(deduced from the mechanical design), so that the ellipticity calculated below is intrinsic to the mirrors.

It is interesting to note that we can adopt the simplifying assumption that the 3 cavity mirrors are identical, with the same radius of curvature (𝑅 𝑥 in the horizontal and 𝑅 𝑦 in the vertical directions). Actually, each mirror of a cavity is certainly affected by specific distortions of the curvature radius in any transverse direction, due to manufacturing imprecision and mounting mechanical stress. However, if we can consider these distortions as small perturbations relative to the radius specified by the manufacturer, we can show that, at the first order, the round-trip Gaussian propagation matrix for each transverse direction can be rewritten with all the mirror radii being fixed at the nominal value modified by only one and the same perturbation parameter.

A proof of this is given in the Appendix. In the case of mirrors with the same specifications as in our setup, we can use just two adjustable parameters for the perturbed radii in the 𝑥 and 𝑦 directions. We will then have to confirm that the difference of the values giving the best match of the experimental resonance frequencies are actually close to the nominal radius given by the manufacturer.

We then adjusted the values of the mean radii of the mirrors on axes 𝑥 and 𝑦, and the value of the cavity length, so that the calculated frequencies of the transverse modes correspond as closely as possible to the measured frequencies. This was achieved by manual adjustment without setting up a specific fitting routine, although for future experiments using this configuration, this should achieve better accuracy and precision for the cavity parameters, and be much more effective in the case of a repeatable and systematic study.

The comparison between the calculated frequencies and the measured ones is presented in table 1. We reached a good agreement, to less than 10 kHz, which is smaller than the day to day variability of the measurements (long term drifts can lead to differences of over 100 kHz, as seen in fig. 2). Experimental accuracy is limited by the temporal resolution of the acquisition and the SNR of the mode detection, and is therefore improved for more strongly injected modes. For all modes, accuracy is under the kHz, well below the spectral width of the transverse modes (about 20 kHz).

Model fitting gives a cavity arm length of 49.1910 cm ± 5 µm (corresponding to an FSR of 152.3924 MHz). This value is consistent with macroscopic measurements, but is far more accurate than any dimensional metrology. The mean radius of curvature of the mirrors obtained from the model is 100.6096 cm ± 10 µm, and the ellipticity (difference between the radius of curvature on the 𝑥-axis and the 𝑦-axis) is 0.9180 cm ± 3 µm. This is within the manufacturer's specifications (1 m with 1 % accuracy), but the ellipticity is quite large, close to 1 % of the mirror curvature. Uncertainties on cavity parameters are estimated by modifying each parameter independently until the difference of all calculated mode frequencies diverges by more than 10 -3 . This method does not take into account the correlation between the various parameters. This method hence yields values at the ppm level for our setup.

We were able to detect 5 transverse modes. The injection of these modes depends on mode matching, offset and angle relative to the cavity propagation axis [START_REF] Romanini | Modelling the excitation field of an optical resonator[END_REF] modes would then be possible by modifying the beam alignment. In our case, we used the 5 detected modes to deduce the cavity parameters. This would be possible with a minimum of one mode on each transverse axis, although with fewer modes the accuracy would be degraded. On the other hand, if more modes can be detected, accuracy will be improved, enabling the model to be constrained more effectively.

The heterodyne beat note acquisition shown in fig. 2 allows to measure directly the spectral width of the transverse modes. Indeed, the laser carrier frequency being locked by optical feedback to the TEM 00 mode, its spectral width falls far below the width of the TEM 00 mode [START_REF] Laurent | Frequency noise analysis of optically self-locked diode lasers[END_REF][START_REF] Casado | High stability in near-infrared spectroscopy: part 1, adapting clock techniques to optical feedback[END_REF][START_REF] Morville | Fast, low-noise, mode-by-mode, cavity-enhanced absorption spectroscopy by diode-laser self-locking[END_REF]. In addition, the locked frequency is stabilized to the centre of the TEM 00 mode to a small fraction of the mode linewidth. Hence the side band is then a direct probe to the width of the transverse mode.

In addition, the cavity is flushed with dry nitrogen to prevent any absorption loss by molecular absorption lines. This insures that the observed spectral widths correspond to the intrinsic cavity losses for each transverse mode. The values of the full width at half maximum (FWHM) are

given in table 1. The FWHM of TEM 00 mode can be deduced from the measurement of the cavity ring-down time thanks to the following relation ( [START_REF] Morville | Cavity Enhanced Absorption Spectroscopy with Optical Feedback[END_REF], eq.1.3, modified to account for a V cavity):

𝜏 𝑟 𝑑 = 1 2𝜋Δ𝜈 = 𝐿 2𝑐 √ R 1 -R (1)
where 𝜏 𝑟 𝑑 is the ring-down time of the mode, Δ𝜈 is the mode width, 𝐿 is the total cavity length (here 2 * 49.191 cm), R is the reflectivity of the mirrors and 𝑐 the speed of light. We can see that the ring-down time and the mode width are both related to the cavity losses through the reflectivity coefficient R. The measured ring-down time for TEM00 mode is 𝜏 𝑟 𝑑 = 11.0 µs, corresponding to a finesse 𝐹 ∼ 10570, and a mode width of 14.5 kHz. The measured FWHM of transverse modes are then in the same range as the TEM 00 's FWHM.

However it must be noted that the actual transverse mode widths are different and vary as seen in table 1 from 10.1 to 34.7 kHz, respectively for the TEM 01 and TEM 20 modes.

These differences can be explained by the transverse spatial distribution of the field on the mirrors, which is specific to each transverse mode. Indeed, if the mirrors have defects that lie within the spatial lobes of a mode, the effective reflectivity of the mirror will be modified for that mode and, consequently, the width of that mode (see equation 𝜏 𝑟 𝑑 ). In our case, we can hypothesize that there is a defect on a mirror that falls on the 𝑥-axis, as TEM modes along the 𝑥-axis have wider FWHMs than others. Although we have not investigated this aspect further, it is clear that the spectral width of transverse modes can be useful for characterizing mirror defects.

Transverse modes spatial profile

As TEM 𝑚𝑛 modes are all orthogonal to each other, the product of the fields of a TEM 00 and of a TEM 𝑚𝑛 mode, when integrated over the PD transverse plane, must yield a zero beat note. This is true for any transverse mode, and not just for antisymmetric modes, unlike what seems to be asserted in [START_REF] Stochino | Technique for in situ measurement of free spectral range and transverse mode spacing of optical cavities[END_REF].

When the RF frequency matches the separation of a TEM 𝑚𝑛 mode from the TEM 00 laser lock mode, the beat note amplitude between this mode and the TEM 00 depends on the position of the edge of the masking blade relative to the beam centre and on its angle relative to the transverse 𝑥, 𝑦 axes. We have made measurements with the edge of the blade along the 𝑥-axis moving it vertically, with the edge on the 𝑦-axis moving it horizontally, and at an angle of 45°. In the following, when we will say that the blade is along a given axis, we will mean that its edge is along that axis. It should be noted that transverse modes were not initially along the principal axes of the setup: they had an angle of more than 30°with the plane of the cavity. The orientation of transverse mode profiles depends on the ellipticity of the optical system [START_REF] Arnaud | Gaussian Light Beams with General Astigmatism[END_REF]. The fact that it is not along the principal axes implies that mirrors ellipticities have a stronger effect than the ellipticity induced by the cavity angle. For convenience, we made sure that the transverse mode profiles are aligned with the principal axes 𝑥 and 𝑦 of the cavity. To do this, we locked the laser on transverse modes by slightly degrading injection alignment and reducing OF, in order to monitor the transmitted beam on a camera. Then we adjusted cavity mirror mounts stress to reduce mirror ellipticity until observing modes aligned along the principal axes.

The expected beat note dependence on position and blade angle can be calculated as the product between the two modes integrated over the region of transverse space that is not masked by the blade and therefore produces a signal on the photodiode. We express this as follows : 𝐼 𝑏𝑒𝑎𝑡 is the beat note amplitude, 𝐸 00 and 𝐸 𝑚𝑛 are the modes fields (the full expression of the modes is found in [START_REF] Romanini | Introduction to Cavity Enhanced Absorption Spectroscopy[END_REF], eq.1), and they depend on the frequency 𝜈, the blade edge position in the transverse plane 𝐷 (with origin at the beam centre), and on the position of the blade translation plane along the propagation axis 𝑧, while the angle of the blade is expressed by different parametrization of the integration limits:

𝑥-axis : 𝐼 𝑏𝑒𝑎𝑡 ,𝑚𝑛 (𝐷, 𝜈, 𝑧) = 2 • ∫ ∞ 𝐷 ∫ ∞ -∞ 𝐸 00 (𝜈, 𝑥, 𝑦, 𝑧) • 𝐸 𝑚𝑛 (𝜈, 𝑥, 𝑦, 𝑧) 𝑑𝑦 𝑑𝑥 (2)
𝑦-axis :

𝐼 𝑏𝑒𝑎𝑡 ,𝑚𝑛 (𝐷, 𝜈, 𝑧) = 2 • ∫ ∞ -∞ ∫ ∞ 𝐷 𝐸 00 (𝜈, 𝑥, 𝑦, 𝑧) • 𝐸 𝑚𝑛 (𝜈, 𝑥, 𝑦, 𝑧) 𝑑𝑦 𝑑𝑥 (3) 45
• : 𝐼 𝑏𝑒𝑎𝑡 ,𝑚𝑛 (𝐷, 𝜈, 𝑧) = 2 • ∫ ∞ -∞ ∫ ∞ 𝑥-𝐷 𝐸 00 (𝜈, 𝑥, 𝑦, 𝑧) • 𝐸 𝑚𝑛 (𝜈, 𝑥, 𝑦, 𝑧) 𝑑𝑦 𝑑𝑥 (4)
From these integrals, it is possible to choose the optimum blade position to obtain the maximum signal when scanning the RF side band. Simulated beat traces are shown in fig. 3 for three transverse modes TEM 01 , TEM 02 and TEM 11 , each beating with a TEM 00 mode. The beat notes obtained with modes of order 𝑚 + 𝑛 = 2 (TEM 02 and TEM 11 ) show a characteristic double hump, while the TEM 01 mode shows only a single maximum. The characteristic mode shapes result from the presence of Hermite polynomials in the definition of Hermite-Gauss modes. The effect of the blade angle allows the direction of the transverse mode to be identified: for example, the TEM 02 mode gives a null profile for the beat note when the blade is along the 𝑥 axis, while it gives a non-zero profile when the blade is along the 𝑦 axis, and the opposite is obtained for the TEM 20 mode. On the other hand, the TEM 11 mode gives null beat note profiles when the blade edge is along both axes, and the blade must be tilted, for instance at 45°, for the beat note to become visible.

In fig. 4, we show the beat note recorded at a fixed RF frequency (corresponding to the RF peak of each transverse mode), as a function of the position of the blade. This was done with the blade at a 45°angle, in order to record all the modes at once. The red dashed lines indicate the position of the blade for the red trace in fig. 2 : this highlights the fact that to see all 5 modes, the blade must also be positioned out of the beam centre, otherwise modes such as 𝑚 + 𝑛 = 2 are undetectable. Double humps can be seen for the triplet of 𝑚 + 𝑛 = 2 modes and a single hump for the doublet of 𝑚 + 𝑛 = 1 modes. We did the same with the blade along the 𝑥 and 𝑦 axes, we found as expected that some modes are not visible, allowing us to assign the indices 𝑚 and 𝑛 to the corresponding modes without any possible confusion. The extinction can be seen in fig. 2, where when the blade is along the 𝑥-axis (yellow line) we see only the TEM 20 and TEM 10 modes, when it is along the 𝑦-axis (blue line) we see only the TEM 02 and TEM 01 modes, and for the blade at 45°(red line) we see all five modes.

If transverse modes of higher order are injected in the cavity (depending on cavity parameters and mode matching), the use of a masking blade will work all the same to identify their indices 𝑚 and 𝑛.

Conclusion

In this work we demonstrate high precision cavity spectroscopy by combining optical feedback locking of a laser diode to a cavity mode, with heterodyne detection at cavity transmission after imposing frequency side bands to the laser. Optical heterodyne detection requires to remove mode orthogonality by partially masking the transmitted beam before detection. When scanning the side band frequency, the heterodyne amplitude provides transverse mode excitation spectra relative to the locked cavity mode with high sensitivity and a zero background. This yields the relative frequencies of transverse modes even if they are weakly injected. Moreover, while keeping one side band resonant with a transverse mode, scanning the blade position yields transverse intensity profiles corresponding to the spatial profile of that mode, allowing direct identification of its transverse order. The frequency measurements that we obtain are then used to adjust a cavity model that yields the geometrical cavity parameters with high precision (on the ppm levels in our case). The relative amplitude of the heterodyne signal according to the modes is not studied further in this work, but it could be used to obtain information about the injection coefficients of transverse modes, which depend on the alignment and mode matching [START_REF] Romanini | Modelling the excitation field of an optical resonator[END_REF].

Finally, the spectral width of transverse modes is also accurately measured, which correspond directly to cavity losses. We have observed that the spectral widths are quite different according to the transverse mode. This clearly implies that our mirrors have non uniform surface quality (assuming our cleaning procedure removed all impurities). Measurements of the width of even more transverse modes with higher orders should in principle allow to build a map of surface losses.

While other laser locking techniques can be used, optical feedback was found to provide robust and reproducible results and to work well with standard DFB lasers in the near infrared region.

In our setup, monitoring transverse mode frequencies is quite slow (a few minutes to scan 5 MHz and infer the frequencies) as it requires to sweep the RF frequency between each acquisition.The acquisition time can be significantly reduced with consistent experimental modification, for example with several single RF modulations corresponding to transverse mode frequencies able to inject all transverse modes at once.

Our method allows to detect side band power down to fW levels even with a small carrier power transmitted through the cavity (0.4 ‰ of the laser beam initial power). Increasing the power of the carrier may lead to detection of even lower side bands, but this must be done without increasing the OF. This is possible by replacing the 2 polarisers (which attenuate the beam in both directions) by an adjustable optical isolator, as it would allow the full laser power to be delivered to the cavity, with only a small fraction coming back for OF.

Appendix

We are going here to demonstrate that in the limit of small deformations of the mirrors surfaces, one can use a single distortion parameter for each cavity transverse axis 𝑥 and 𝑦 to account for the deformations of all mirrors composing an optical cavity. This is valid independently of the shape of the cavity (linear, V shaped, bow tie, ring ...). The first hypothesis is that the cavity admits two orthogonal axes, thus being well described by Hermite-Gauss modes relative to orthogonal 𝑥 and 𝑦 transverse axes. In this case, one can write a round-trip Gaussian propagation matrix for each axis.

We start by considering that each of the 3 mirrors (identified in the following by an integer 𝑚) is affected by different perturbations of the curvature in each transverse direction, due to manufacturing imprecision and mounting mechanical stress. The matrix for cavity mirror 𝑚 along 𝑥 (to fix things) will then be :

𝑀 𝑚 = 1 0 -2 𝑅 𝑚 +𝜖 𝑚 1 (5)
where 𝜖 𝑚 << 𝑅 𝑚 is the small deformation parameter for the mirror curvature radius 𝑅 𝑚 along 𝑥 (this would be the value given by the manufacturer). The mirror matrix for Gaussian beam propagation along 𝑥 will be the ordered chain product of this matrix with propagation matrices for the free space in between cavity mirrors and similar matrices as eq. ( 5) for other cavity mirrors, each with a given curvature radius and deformation parameter. For example for a V shaped cavity (of equal arms of length 𝑑) this round trip matrix will be :

𝑀 = 1 𝑑 0 1 • 1 0 -2 𝑅 2 +𝜖 2 1 • 1 𝑑 0 1 • 1 0 -2 𝑅 3 +𝜖 3 1 • 1 𝑑 0 1 • 1 0 -2 𝑅 2 +𝜖 2 1 • 1 𝑑 0 1 • 1 0 -2 𝑅 1 +𝜖 1 1 (6)
We then recognize that the perturbed mirror matrix can be written, at the first order in 𝜖/𝑅 as the product of the unperturbed matrix times a perturbation matrix, as follows :

𝑀 𝑚 = 1 0 -2 𝑅 𝑚 +𝜖 𝑚 1 ≈ 1 0 -2 𝑅 𝑚 1 • 1 0 2𝜖 𝑚 𝑅 2 𝑚 1 ( 7
)
and it is easy to check that these two matrices commute. In addition, we can consider that the perturbation matrix commutes with the free space propagation matrix :

1 𝑑 0 1 • 1 0 2𝜖 𝑚 𝑅 2 𝑚 1 = 1 + 𝑑•2𝜖 𝑚 𝑅 2 𝑚 𝑑 2𝜖 𝑚 𝑅 2 𝑚 1 ≈ 1 𝑑 2𝜖 𝑚 𝑅 2 𝑚 1 (8)
1 0

2𝜖 𝑚 𝑅 2 𝑚 1 • 1 𝑑 0 1 = 1 𝑑 2𝜖 𝑚 𝑅 2 𝑚 1 + 𝑑•2𝜖 𝑚 𝑅 2 𝑚 ≈ 1 𝑑 2𝜖 𝑚 𝑅 2 𝑚 1 (9)
hence 1 𝑑 0 1

• 1 0 2𝜖 𝑚 𝑅 2 𝑚 1 ≈ 1 0 2𝜖 𝑚 𝑅 2 𝑚 1 • 1 𝑑 0 1 ( 10
)
where in the diagonal elements on the resultant matrix, the term 𝑑.2𝜖 𝑅 2 was neglected as compared to 1 (first order in 𝜖/𝑅 and also because in general as 𝑑/𝑅 < 1 imposed by the cavity stability condition).

Thanks to these matrix commutation approximations, the perturbation matrices for all the mirrors composing the cavity can be displaced inside the round-trip matrix chain product and can be moved close together. Their product can be rewritten as a product of the same number of identical perturbation matrices :

𝑁 𝑚=1 1 0 2𝜖 𝑚 𝑅 2 𝑚 1 = 1 0 𝑁 𝑚=1 2𝜖 𝑚 𝑅 2 𝑚 1 = 1 0 𝑁 • 𝛼 1 = 𝑁 𝑚=1 1 0 𝛼 1 ( 11
)
where we introduce the global perturbation parameter 𝛼 << 𝑅 𝑚 (for all 𝑚) given by the sum of 2𝜖 𝑚 /𝑅 2 𝑚 over the number of cavity mirrors 𝑁. These perturbation matrices can be replaced back together with the mirror matrices which now result to be affected by the same perturbation parameter.

Disclosures. The authors declare no conflicts of interest.

Fig. 1 .

 1 Fig.1. Diagram of the experimental setup. The optical cavity (in blue) is a V shaped cavity made of 3 identical mirrors, with an arm length of roughly 49 cm and a small angle of incidence at the folding mirror of 0.44°. Mirror radius of curvature is specified to 100 cm. The laser is locked to a TEM 00 mode by detection of the transmitted signal ("PD lock") and by phase control of the optical feedback (OF) acting on a piezoelectric (PZT) mounted mirror. An electro-optic modulator (EOM) allows to create side bands at a radio-frequency (RF) digitally controlled. A second photodiode monitors the RF heterodyne signal. A masking blade, mounted in a translation stage, allows to cut off part of the cavity output beam before heterodyne detection.

Fig. 2 .

 2 Fig. 2. Beat note amplitude at cavity output, between a side band and the carrier, as a function of the modulation frequency (RF). Measurements recorded on different days and for different blade angles used to mask part the cavity output beam and produce heterodyne signals (see text).

ig. 3 .

 3 Expected amplitude of the beat note with TEM 00 field, for several TEM 𝑚𝑛 modes, according to the position of the edge of the masking blade in the transverse plane, and for different blade angles. The beam waist corresponds to the transverse radius of the beam at the position of the blade along the propagation axis.

Fig. 4 .

 4 Fig. 4. Measured beat note amplitude of transverse modes according to the masking blade position. The blade angle is 45°. For each mode, this is recorded at a fixed modulation frequency corresponding to the resonant frequency of their respective transverse modes. The red dashed lines show the position of the blade during acquisition of the data presented in fig. 2, blade at 45°in red line.

Table 1 .

 1 Transverse mode frequencies relative to the TEM 00 mode, obtained by adjustment of the model to fit experimental values derived from the measurements plotted in red in fig.2. The given width is full width at half maximum of the measured beat note.

	. The injection of other

Data Availability Statement. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.


Lorenzo Busoni
email: lorenzo.busoni@inaf.it

Guido Agapito

Ballone Alessandro

Alfio Puglisi

Alexander Goncharov

Amedeo Petrella

Amico Di Cianno

Andrea Balestra

Andrea Baruffolo

Andrea Bianco

Alessandro Ballone

Andrea Di Dato

Angelo Valentini

Benedetta Di Francesco

Benoit Sassolas

Bernardo Salasnich

Carmelo Arcidiacono

Cédric Plantet

Christian Eredia

Daniela Fantinel

Danilo Selvestrel

Deborah Malone

Demetrio Magrin

Domenico D'auria

Edoardo Redaelli

Elena Carolo

Elia Costa

Elisa Portaluri

Enrico Cascone

Enrico Giro

Federico Battaini

Francesca Annibali

Fulvio Laudisio

Gabriele Rodeghiero

Gabriele Umbriaco

Gael Chauvin

Gianluca Di Rico

Giorgio Pariani

Giulia Carlà

Giulio Capasso

Giuseppe Cosentino

Jean Jacques Correia

Italo Foppiani

Ivan Di Antonio

Jacopo Farinato

Kalyan Kumar Radhakrishnan

Laurence Gluck

Laurent Pinard

Luca Marafatto

Marcello Agostino Scalera

Marco Gullieuszik

Marco Bonaglia

Marco Riva

Marco Xompero

Maria Bergomi

Matteo Aliverti

Matteo Genoni

Matteo Munari

Mauro Dolci

Michel Christophe

Michele Cantiello

Mirko Colapietro

Nicholas Devaney

Nicolò Azzaroli

Paolo Grani

Paolo Ciliegi

Patrick Rabou

Philippe Feautrier

Pietro Schipani

Roberto Ragazzoni

Rosanna Sordo

Runa Briguglio

Salvatore Lampitelli

Salvatore Savarese

Simone Benedetti

Simone Di Filippo

Simone Esposito

Simonetta Chinellato

Sylvain Oberti

Sylvain Rochat

Tommaso Lapucci

Ugo Di Giammatteo

Vincenzo Cianniello

Vincenzo De Caprio

Zoltan Hubert

MORFEO enters final design phase

Keywords: MORFEO, Extremely Large Telescope, European Extremely Large Telescope, Multi-Conjugate Adaptive Optics, status, preliminary design review, phase C, final design, MICADO, update

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.

INTRODUCTION

The Multi-conjugate adaptive Optics Relay For ELT Observations (MORFEO, formerly known as MAORY) [START_REF] Ciliegi | MAORY: A Multi-conjugate Adaptive Optics RelaY for ELT[END_REF][START_REF] Ciliegi | MAORY/MORFEO at ELT: general overview up to the preliminary design and a look towards the final design[END_REF] will deliver a multiconjugate adaptive optics (MCAO) correction to the first light instrument MICADO (Multi-AO Imaging Camera for Deep Observations) [START_REF] Davies | MICADO: The Multi-Adaptive Optics Camera for Deep Observations[END_REF] and to a future second port instrument of the ESO Extremely Large Telescope (ELT) [START_REF] Tamai | The ESO's ELT construction progress[END_REF]. The MCAO correction of MORFEO is provided by the fourth mirror of the ELT (M4), that is conjugated at 600 m, and by two post-focal deformable mirrors (DMs) conjugated at 6.5 and 17.5 km. The DMs compensation is based on the measurement from both three natural guide star (NGS) wavefront sensors (WFSs) and six laser guide star (LGS) WFSs. MORFEO is required to provide a uniform correction over a 1-arcmin field of view, with a Strehl Ratio of 30% (goal 50%) in K band under ESO's median atmospheric conditions and with 50% sky coverage, allowing MICADO to exploit the full resolution potential of the 39-m ELT aperture in the 0.8 -2.4 µm wavelength coverage. In particular, MICADO will take advantage of MORFEO correction in imaging and spectroscopy observing modes. For what concerns the imaging mode, two options are foreseen: a low-resolution imager, covering a wide field of view of 50.5 × 50.5 arcsec 2 at a pixel scale of 4 mas, and a high-resolution imager covering a smaller field of view of 19 × 19 arcsec 2 at a pixel scale of 1.5 mas. This will allow a significant step forward for several science cases, with a significant increase in spatial resolution with respect to HST and even JWST (pixel scale ∼ 30mas/pix). For what concerns spectroscopy, the full wavelength range is covered with two settings: a short slit for 0.84 -1.48 µm and a long slit 1.48 -2.46 µm.

A diagram of the ELT, MORFEO, MICADO, and the second port instrument is shown in Fig. 1: the ELT provides the main deformable mirror of MORFEO, M4 [START_REF] Vernet | ELT M4 -The Largest Adaptive Mirror Ever Built[END_REF], the laser launch telescopes and the prefocal station (PFS) [START_REF] Larringan | The final design for the Extremely Large Telescope prefocal stations[END_REF]; the MORFEO relay provides the corrected beam to MICADO or to a future second port instrument and consists of 2 postfocal deformable mirrors, of the LGS WFS module and of the NGS WFS module mounted on the MICADO derotator; the MORFEO Real-Time Computer (RTC) [START_REF] Baruffolo | MORFEO at ELT: preliminary design of the real-time computer[END_REF] controls the MORFEO hardware and interacts with the telescope, with M4 for the ground layer correction and with the launcher telescopes to correct the laser jitter and pointing. A 3D model of the ELT Nasmyth platform with MORFEO and MICADO is shown in Fig. 2: the MORFEO optical relay [START_REF] Pariani | MORFEO optical design and performances: status at preliminary design review[END_REF] is housed in a large mechanical structure of 7×7×4 m (see Refs. [START_REF] De | Mechanical design overview for the main structure of MAORY/MORFEO[END_REF] and [START_REF] Cianniello | Structural analyses of the MAORY/MORFEO main support structure at global level for preliminary design review[END_REF]), that will be shielded to increase thermal inertia, reduce airflow and local turbulence [START_REF] Aliverti | MAORY/MORFEO at ELT: Thermal Control System preliminary design[END_REF]. At the entrance port of MORFEO, close to the ELT focal plane, is installed the calibration unit [START_REF] Di | MAORY/MORFEO at ELT: calibration unit overview[END_REF] that can feed the WFS modules and MICADO through the relay; the LGS WFS module [START_REF] Schreiber | The MAORY laser guide star wavefront sensor: design status[END_REF] is not clearly visible in the picture, hidden by the structure. The last folding mirror and the NGS WFSs [START_REF] Bonaglia | MORFEO/MAORY low-order and reference WFS module preliminary design[END_REF] are above MICADO with the NGS WFSs that rotate together with MICADO to follow the sky rotation. The second port instrument will have to implement its own NGS WFS module.

The MORFEO project successfully closed the Preliminary Design Review (PDR) with ESO in February 2023. During the current phase, we will finalize the design that will be reviewed with ESO during the Final Design Review (FDR) foreseen by 2025. For several subsystems the final design will be developed in collaboration with external companies, which, after the successful conclusion of the FDR, will also be in charge of the manufacturing. Successively, the assembly process will be carried out at the Integration hall located at the INAF premises in Bologna.

SCIENCE

The development and implementation of the science cases are managed for the MORFEO consortium by the MORFEO Science Team, which works in close collaboration with the MICADO Science Team and operates to maximize the scientific return from the ELT-MORFEO-MICADO system. The MORFEO science team has widely explored the science cases for the MORFEO+MICADO. A preliminary collection of the cases studied is reported in the MORFEO science cases white book freely available on the MORFEO website [START_REF] Fiorentino | MAORY Science cases white book[END_REF].

The collected science cases address many of the major questions of astrophysics:

• Planetary systems, including cases in our own Solar Systems, exoplanets, and formation of planetary systems.

• Nearby stellar systems with stars and stellar systems within our own Galaxy and its satellite.

• Local Universe with science cases aimed to study the stellar content and the structure of distant stellar systems that can be at least partially resolved into individual stars. In many cases, they will fall within the range of resolved systems only thanks to the advent of MORFEO+MICADO.

• High-redshift Universe with the science cases addressing the formation of structures and cosmology using the formidable sensitivity and resolution of MORFEO+MICADO to probe the very distant Universe and, consequently, the earliest phases of galaxy formation, as well as high energy phenomena over the range of cosmic distance and times made accessible by ELT.

OPTICAL DESIGN

The present configuration has been initially proposed by Bernard Delabre, and, in a similar arrangement, by Johan Kosmalski. It is based on four powered mirrors (two aspherical and two spherical) able to deliver high quality in terms of WFE and field distortion. It is tolerant of defocus and astigmatism that can be compensated for by acting on the last mirror with power. This configuration implements the concept, firstly introduced by Andrew Rakich, of using an aspherical plate near the telescope focal surface to correct multiple object conjugate distances (see Refs. [START_REF] Rakich | A Maxwellian "ideal imager" optical relay suitable for AO applications[END_REF], [START_REF] Rakich | Aberration theory-based approaches to optical design[END_REF] and [START_REF] Rakich | Absolute instruments as laser guide star adaptive optics relays[END_REF]). The correcting plate (CPM) greatly improves the quality of the exit pupil image, of the DMs meta-pupil images and of the LGS images at different conjugation altitudes without degrading the quality for the infinite conjugated surface. We have adapted and re-optimized the general optical solution to the MORFEO specific requirements and constraints, namely, the available volume on the Nasmyth platform and the position of the MICADO entrance surface. Folding mirrors were added to place the instrument in a vertical layout and to provide switching between MICADO and the second instrument. A scheme is shown in Fig. 3 (see Refs. [START_REF] Magrin | MAORY: optical configuration and expected optical performances[END_REF] and [START_REF] Pariani | MORFEO optical design and performances: status at preliminary design review[END_REF]).

Main Path Optics description

The F/17.75 beam coming from the ELT focal surface initially passes through the CPM. Following the path, there is a first flat mirror, M6M, folding the beam down to the first aspherical concave mirror M7M. The beam is reflected up to the second aspherical concave mirror M8M, and then down again towards the first DM, M9M, having a spherical surface. This mirror is the only convex surface on the main path. After that, the beam is reflected by the second DM, M10M, which is concave and spherical. After M10M, the image of the pupil is formed, and just after this image, the science and NGS light is separated from the LGS light by means of a dichroic filter. The LGS light (589 nm) is transmitted, while science and NGS light (600 nm -2400 nm) are reflected. The reflected light then reaches M11M, a flat flip mirror that allows the selection of the MICADO path or of the second instrument path through a rotational axis. Finally, the light is reflected by a flat mirror, M12M (or M12M bis), installed over MICADO (or, as previously mentioned, some future second instrument) and comes to a focus at the gravity invariant entrance focal surface of the instruments.

Laser Guide Star Objective description

The LGS Objective (LGSO) reduces the LGS beam F/number at wavelength 589 nm to F/5 at the focal surface of the entrance of the LGS WFS module. It is made up of four silica spherical lenses and three-fold mirrors.

The first two mirrors have been introduced in order to maintain the objective within the available envelope, while the third mirror allows to feed the LGS module with a gravity invariant and telecentric focus. The LGS asterism required by MORFEO is fixed at a field angle of 45 arcsec. As the ELT elevation varies (Zenith angle in the 0-60 degrees range), the LGS launchers will maintain the LGS sources on the sodium layer at a fixed angle of 45 arcsec. Thanks to the telecentric beam delivered by the LGSO, the radial coordinate of the LGS images on the LGSO focal surface will not change as the apparent altitude (84-180 km) of the sodium layer varies. In this way, as the telescope elevation changes, the LGS module just needs to rotate around its longitudinal axis to keep the pupil orientation and move up and down to track the sodium layer.

WAVEFRONT SENSORS

The system makes use of the 6 laser guide stars provided by the ELT and of 3 natural guide stars for low order modes correction and for truth sensing. The corresponding wavefront sensors are grouped in a LGS WFS module and in a Low Order and Reference (LOR) WFS unit, field-derotated by MICADO itself.

The Laser Guide Star Wavefront Sensors

The LGS WFS module is gravity invariant and it can be divided in three main units: the wave-front sensor probes, that pick off the MORFEO beam after the LGSO (see Sec. 3.2) in an LGS focal plane and sample the ELT pupil to measure the wavefront aberrations (there are 6 such probes for 6 laser launch telescopes, see right part of Fig. 4), the support structure that holds the 6 probes and aligns them in the MORFEO beam (left part of Fig. 4) and the module control electronics, cabling, power and cooling lines.

Each WFS has a 68 × 68 lenslet array positioned after a pupil imaging collimator and a folding mirror and, then, an optical relay re-images the LGS spots onto the camera detector. As detector chip we selected the Sony IMX 425 and one of the key points that encouraged us to make this choice is the clear advantage of global shutter detector for our specific application (see Refs. [START_REF] Agapito | Rolling shutter-induced aberrations in laser guide star wavefront sensing[END_REF] and [START_REF] Agapito | MAORY/MORFEO and rolling shutter induced aberrations in laser guide star wavefront sensing[END_REF]). The pick-off mirror in each WFS is mounted on a 2-axis piezo stack and allows to recenter the pupil image onto the lenslet array to compensate for possible shifts of the pupil. The WFS design is similar to the one of HARMONI [START_REF] Costille | Design and challenges for the HARMONI laser guide star sensors[END_REF] and provides a large FoV of about 16 arcsec on 14×14 pixels per sub-aperture to minimize the truncation effect due to sodium layer extension (see Ref. [START_REF] Fusco | Key wavefront sensors features for laser-assisted tomographic adaptive optics systems on the Extremely Large Telescope[END_REF]).

The support structure has two actuators: a linear stage to compensate defocus due to the varying distance of the sodium layer (it depends on telescope elevation and sodium layer average altitude) and a de-rotator to track telescope elevation and maintain a fixed orientation with respect to the telescope pupil. More details on this module can be found in Ref. [START_REF] Schreiber | The MAORY laser guide star wavefront sensor: design status[END_REF].

The Low-Order and Reference Wavefront Sensors

MORFEO implements 3 NGS WFSs dedicated to: 1) sense the atmospheric low-order modes (tip, tilt and focus), 2) detrend the mid-high order modes sensed by the LGS WFS and affected by errors due to non-common path aberrations on the LGS path and sodium layer instabilities. These 2 types of WFS take place on top of the same optical breadboard and share the light of the same NGS (infrared and visible part respectively), thus their assembly is often called Low-Order and Reference WFS [START_REF] Bonaglia | MAORY/MORFEO Low-Order and Refernce WFS module preliminary design[END_REF]. The design of the first WFS, the low-order one, considers the usage of the FREDA camera [START_REF] Downing | Update on development of WFS cameras at ESO for the ELT[END_REF] while the design of the Reference WFS (R WFS) makes use of ALICE [START_REF] Marchetti | Final performance of the ESO's ALICE and LISA wavefront sensing cameras[END_REF]. The main design parameters of the two types of WFS are summarized in Table 1. Each unit has a trombone for focus compensation and a piezo gimbal mirror for pupil recentering, as in the LGS WFSs. The Low Order arm, sensing in H band, implements an ADC.

The three units LOR WFSs are located within the green volume on top of the MICADO assembly, as shown in Fig. 2. The three LOR WFS are identical opto-mechanical assemblies, arranged on a triangular geometry and installed on top of a dedicated support structure (dark grey color on the left in Fig. 5). Each LOR WFS is moved in a planar region of about 300 × 600 mm by a couple of orthogonal stages, called Acquisition Stages. Indeed these devices allow the LOR WFS to move across the "patrol field" and to reach the position where the light of the guide stars is imaged by the telescope. The whole opto-mechanical assembly of the LOR WFS is rigidly connected, and supported, by the MICADO rotator that allows for field de-rotation. The LOR WFS also implements a dedicated control electronics that is hosted in the co-rotating cabinets below the MICADO dewar. Details of the LOR WFS electronics design are given in Ref. [START_REF] Lapucci | MORFEO (formerly known as MAORY) LOR WFS module preliminary electronics design[END_REF].

Parameter

DEFORMABLE MIRRORS

MORFEO will be equipped with two post-focal deformable mirrors (DM). The current study implements the use of two large diameters DMs (928mm for M9M and 1225 mm for M10M) and respectively 918 and 1027 degrees of freedoms available for correction (see Fig 6). In particular, according to current technical requirements M10M will be the largest adaptive mirror with monolithic glass ever built.

They will provide fast turbulence correction (baseline 500Hz) and they will be able to achieve a correction better than 40nm RMS during bad seeing conditions (1.5 arcsec at 0.5 µm wavelength), with high reliability and minimal shape degradation up to 10% failing actuators. See Ref. [START_REF] Xompero | MORFEO (formerly known as MAORY) at ELT: preliminary design of the adaptive mirrors[END_REF] for a detailed description of the deformable mirror design.

REAL-TIME COMPUTER AND CONTROL

MORFEO RTC is based on HEART [START_REF] Dunn | Herzberg Extensible Adaptive Real-time Toolkit (HEART) for ELT[END_REF] for the hard real-time part and the ESO toolkit for the soft real-time part. A diagram of the hard RTC part is shown in Fig. 7: it receives the measurements of the 12 WFSs, computes the slopes, runs three tomographic reconstructions in parallel, apply temporal filters to the differential commands, combines the LGS and NGS commands, applies anti-wind up to avoid saturation of the DMs and computes the commands to the tip/tilt jitter actuators of the 6 laser launch telescopes. The Control approach that will be implemented by the RTC is described in Busoni et al. [START_REF] Busoni | Adaptive optics design status of MAORY, the MCAO system of European ELT[END_REF] and its key elements are:

• Tomography is split between high order modes and low order modes [START_REF] Gilles | Split atmospheric tomography using laser and natural guide stars[END_REF].

• Reconstruction is done through the Minimum Mean Square Error (MMSE) estimator (alternative approaches are evaluated, as for example the one presented in Ref. [START_REF] Stadler | Matrix-based vs. matrix-free real-time reconstruction with FEWHA for a MORFEO like setting[END_REF]).

• Noise priors, see Refs. [START_REF] Béchet | Optimal reconstruction for closed-loop ground-layer adaptive optics with elongated spots[END_REF] and [START_REF] Oberti | LGS tomography and spot truncation: tips and tricks[END_REF].

• Pseudo-open loop control (POLC) [START_REF] Brent | Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes[END_REF].

• Super-resolution, see Refs. [START_REF] Oberti | Super-resolution wavefront reconstruction[END_REF] and [START_REF] Fusco | Key wavefront sensors features for laser-assisted tomographic adaptive optics systems on the Extremely Large Telescope[END_REF].

• Tomographic truth sensing, see Refs. [START_REF] Busoni | Adaptive optics design status of MAORY, the MCAO system of European ELT[END_REF] and [START_REF] Busoni | MAORY/MORFEO at ELT: preliminary design of the adaptive optics subsystem[END_REF].

• Adaptive Vibration Cancellation, see Ref. [START_REF] Pettazzi | Improving the accuracy of interferometric measurements through adaptive vibration cancellation[END_REF].

Note that the testing activity of the MORFEO RTC tools and algorithms has already begun as shown in Refs.

[39] and [START_REF] Bellone | Development of test devices for the validation of MORFEO's reconstruction and control algorithms[END_REF].

OPERATION

The control strategy of MORFEO deals with the main closed loop operations, that include slope computation, wavefront reconstruction and control, and with auxiliary loops that update the system depending on external conditions variation in order to provide a stable correction during the observations. All this is managed by the Instrument Control System Software (ICSS) [START_REF] Salasnich | MORFEO (formerly known as MAORY) instrument control software: toward a consolidated design[END_REF]. ICSS will control almost 100 functions and will interface with RTC and two different instruments (MICADO and the future second port instrument).

The main loop algorithms are implemented in the RTC, as described in Sec. 6, and are driven by the WFSs, whose signals are used for a tomographic reconstruction that follows a split tomography approach [START_REF] Busoni | MAORY/MORFEO at ELT: preliminary design of the adaptive optics subsystem[END_REF]. In particular, Low Order (LO) and High Order (HO) modes have different and independent tomographic reconstructors and projection matrices: on the one hand, tip, tilt and platescale modes are reconstructed from NGS WFSs measurements and the correction foresees to command tip and tilt on M4, and platescale modes through a proper combination of focus and astigmatism on both M4 and a high-layer DM; on the other hand, the higher order modes are reconstructed from the LGS WFSs measurements.

Standard on sky operations will be guided by MICADO and MORFEO will provide the AO correction: all MICADO template scripts will be executed by the sequencer application that is part of the MICADO ICSS and control of AO functions from MICADO will be enabled through a dedicated API library that encapsulates the necessary tasks within a set of Python procedures. MORFEO will start its operation to be ready for the guide stars acquisition and for closed loop operations while the telescope is slewing to the desired target. The operation first foresees the closure of the LGS loop to compensate for HO modes (correction of focus will be biased due to the unknown sodium layer altitude, as well as the correction of few low order modes due to truncation effects), then the closure of the NGS loop that is based on the NGS acquisition sequence described in detail in Ref. [START_REF] Agapito | NGSs acquisition in MORFEO[END_REF]: the brightest NGS is acquired on the R WFS (having a larger field of view than the LO WFS) that measures tip, tilt and focus to center the star on the LO WFS and also to compensate for the focus bias; the star is then acquired on the LO WFS and tip, tilt and focus loops are closed; finally the other two NGSs are acquired in parallel to minimize the overall acquisition time.

For what concerns the auxiliary loops of MORFEO, they are summarized in the following list:

• Recurrent task to update the HO loop control matrices, driven by zenith angle.

• Recurrent task to update the LO loop control matrices, driven by rotation angle and LOR position.

• Recurrent task to update the number of controlled modes, driven by seeing monitor and DMs force saturation.

• Recurrent task to update slope computer parameters, driven by changes in the sodium profile and on the LOR correction level.

• Recurrent task to update vibration rejection filters.

• Full-framerate loop to compensate for LGS jitter, driven by LGS WFSs tip-tilt measurements and acting on laser launcher telescopes fast steering mirrors.

• Offload from laser launcher telescopes fast steering mirrors to laser launcher telescopes field selecting mirrors.

• Truth sensor loop to detrend LGS-induced bias (truncation and NCPAs) driven by R WFS and acting on modal offset.

• Loop to derotate the elevation from the LGS WFSs pupil, driven by the zenith angle and acting on the LGS derotator.

• Loop to keep the pupil footprint stable on the lenslet array, driven by subaperture illumination pattern and acting on the pupil tilt actuator in each WFS.

• Loop to maintain a matching between DM actuators and WFS subapertures driven by WFS measurements and acting on M4 and on post-focal DMs projection matrices.

• Recurrent task to sense derotation errors driven by LO measurements and acting on MICADO derotator.

CALIBRATION AND TEST

MORFEO is a complex systems that requires a wide range of calibrations to function as intended, and several test procedures will be performed to verify its functionality and performance. The calibrations range from the data required by the detectors, such as the conversion gain factor and bias level per readout channel, to the adjustment of geometric parameters, such as the relative displacement between the sub-aperture array and the DM actuator grid, and also include the estimation of field distortion and non-common aberrations between WFS and science path. MORFEO is thus equipped with a calibration plan that concatenates all the calibrations tasks that are required for MORFEO to perform and achieve its specifications. A significant fraction of the calibration templates aims at monitoring and trending the system status and performance (e.g. throughput, alignment quality), but the MORFEO Calibration Plan does not include templates needed to verify the system performance. Instead test activity is included in the Assembly, Integration and Test (AIT) in Europe [START_REF] Farinato | The MAORY/MORFEO MAIT strategy in Europe[END_REF] and Assembly, Integration and Verification (AIV) and Commissioning in Chile. This activity involves a set of tests to verify functionality and performance of the sub-systems and will culminate in performance tests on sky. A relevant part of calibration and test activity will be carried out in controlled environment in the Integration Hall in Bologna, and afterwards at the telescope, thanks to a subsystem that can act both as a Calibration Unit (CU) in Chile, or as Test Unit (TU) in Italy. The CU, shown in Fig. 8, provides a set of suitable light sources, both NGS and LGS, matching the ELT optical interfaces (focal ratio, exit pupil position, field curvature), to fully simulate the telescope performance and residual aberrations. It will drastically reduce the amount of required night-time for daily, periodic and maintenance operations. A description of the current design of the CU can be found in Refs. [START_REF] Di | MORFEO Calibration Unit: towards the final design[END_REF] and [START_REF] Di | MORFEO Calibration Unit: optimization of the performances for the final design[END_REF]. The configuration of the CU can be changed to operate as TU, by installing a deformable mirror at its intermediate pupil, providing the capability to operate the full MORFEO system in standalone mode with three deformable mirrors.

AO PERFORMANCE

In this section, we present the main results related to the AO performance of MORFEO. The performance requirements of MORFEO are expressed in terms of K band SR for a zenith angle of 30deg: the average SR in the small science field of view (FoV, r≤10 arcsec) in the first quartile of atmospheric conditions must be greater than 50%, and the average SR in the large science FoV (r≤30 arcsec) in the median atmospheric conditions over half the pointings in the sky must exceed 30%. The SRs are estimated through end-to-end simulations using PASSATA [START_REF] Agapito | PASSATA: object oriented numerical simulation software for adaptive optics[END_REF] and the sky coverage tool described in Ref. [START_REF] Plantet | Sky coverage assessment for the European ELT: a joint evaluation for MAORY/MICADO and HARMONI[END_REF].

The simulation parameters can be found in Ref. [START_REF] Agapito | MAORY AO performances[END_REF], Table 1, with a few exceptions: we updated the throughput considering 0.26 for the LGS path, 0.32 for the NGS path in the infrared and 0.22 in the visible, the LGS WFS configuration considering 68×68 sub-aperture with a FoV of 16.1 arcsec and the NGS WFS configuration considering a pixel pitch of 14 mas. The error budget is described in Tab. 2. Part of the error budget, 180nm RMS ("Extra" in the table), is not directly included in the end-to-end simulations, so it is added in the PSF calculation with random phase-screens with a f -2 power spectral density. This error includes terms such as non-common path aberration, residual aberrations in the telescope and in the post-focal relay optics due to design, manufacturing, alignment, thermo-elastic and optical effects of air in the optical path, aberration due to sodium layer variations, calibration errors, atmospheric chromaticity, and a contingency. In Fig. 9 we report the average SR as a function of the Table 2: Summary of the MORFEO error budget in nm RMS in the MICADO large FoV (r≤30 arcsec) for the 5 reference atmospheric profiles for coverage of half of the sky at the south galactic pole (the corresponding K band SR values are shown in Fig. 10). Zenithal distance is 30deg. High order is the term associated with the modes sensed by the LGS WFSs while low order is the term associated with tip and tilt sensed by NGS WFSs (it depends on the magnitude and position of the available NGSs, that is the so-called sky coverage). Extra includes errors not accounted for in the end-to-end simulations (see text for details). off-axis angle for the 5 reference atmospheric profiles in case of a good NGS asterism: the K band SR in the small science FoV for the first atmospheric quartile (Q1, red line) is greater than 58% (requirement >50%).

Performance as a function of the sky coverage at the south galactic pole is shown in Figs. 10 and 11: the K band SR in the large science FoV for the median atmospheric condition (black line) is greater than 38%. Therefore, at the current stage of development, the system is in compliance with the performance requirements.

Note that these optical perfomance bring to transformational scientific merit function. For example, in Fig. 12, we show the K-band point-source sensitivity at 5σ for different atmospheric conditions, here computed using ELT telescope and sky emissivities and the MICADO detector characteristics (and the 4mas pixelscale configuration). Additional information on the MORFEO + MICADO PSFs can be found in Ref. [START_REF] Arcidiacono | A preliminary design review study of the scientific performance of MAORY (MORFEO)[END_REF].

CONCLUSION

We reported on progress of the MORFEO multi conjugate adaptive optics system, which has reached the final design phase. We summarized its science goal, the design of its optics, wavefront sensors, deformable mirrors and real time computer, the development of calibration and test plans and we finally provided details on operation and performance and we showed that the system is in compliance with the performance requirements.

At this stage, system engineering activities [START_REF] Riva | MORFEO at ELT: system engineering activity up to preliminary design review[END_REF] will complete the flow-down of requirements to all subsystems and subsequent verification, so that all design aspects are finalized and the next phase of Manufacturing, Assembly, Integration and Verification (MAIV) can begin.

Finally, note that despite the large amount of information reported above there are other AO related activities being carried out that have not been described in depth, such as the analysis of the error terms of the astrometric observations (for example we studied the tip-tilt anisoplanatism in MCAO systems and its impact on astrometric observations with next-generation telescopes in Refs. [START_REF] Carlà | Temporal spectrum of multiconjugate adaptive optics residuals and impact of tip-tilt anisoplanatism on astrometric observations[END_REF][START_REF] Carlà | Tip-tilt anisoplanatism in MCAO-assisted astrometric observations[END_REF]) and research activities like the study on the so-called petalometer that is a sensor for differential segment pistons to deal with low wind effect [START_REF] Pourré | Low-wind-effect impact on Shack-Hartmann-based adaptive optics. Partial control solution in the context of SPHERE and GRAVITY+[END_REF] and other related errors (currently we are looking for concepts like using LIFT in the LO WFSs [START_REF] Agapito | MAORY/MORFEO and LIFT: can the low order wavefront sensors become phasing sensors?[END_REF] and CIAO-CIAO [START_REF] Carlà | CiaoCiao WFS: sensing phase discontinuities at the Extremely Large Telescope[END_REF]).

Figure 1 :

 1 Figure 1: Diagram of the ELT, MORFEO, MICADO and the second port instrument. Light is coming from ELT which also provides the source for the 6 laser guide stars (LGS) and the ground layer correction with M4. MORFEO interacts with the telescope and, in particular, with the 6 laser launch telescopes and M4 through the real-time computer (RTC). In addition, the RTC receives the measurements from the wavefront sensors (WFSs) and commands the two deformable mirrors (DMs) of MORFEO (for more details about the WFSs, DMs and RTC see Secs. 4, 5 and 6 respectively). One dichroic splits the laser light (589nm) and transmits it to the LGS WFSs. Finally, the corrected wavefront is provided to MICADO and to the NGS (LOR) WFS module (accessing 3 NGS stars in an outer annulus around the MICADO field), or to a second port thanks to a flat flip mirror, M11M (for more details about the optical design see Sec. 3).

Figure 2 :

 2 Figure 2: 3D model of the ELT Nasmyth platform with MORFEO and MICADO. The main elements of MORFEO are labeled in the image.

Figure 3 :

 3 Figure 3: 3D view of the MORFEO optical configuration. Blue rays represent the science and NGS path to MICADO, green rays represent the LGS path.

Figure 4 :

 4 Figure 4: Laser guide star wavefront sensor (LGS WFS). Left, 3D model of the assembly of the 6 units in the mechanical structure with derotator and focus actuator, right, schematic of a single unit showing the optics and detector.

Figure 5 :

 5 Figure 5: Natural guide star wavefront sensor (NGS WFS) known as LOR (Low Order and Reference) WFS. Left: 3D model of the assembly of the 3 units in the mechanical structure with the x and y stages; top right: schematic of a single unit showing the optics and detectors; bottom right: optical design of the LO and Ref WFS.

  (a) DM1 -M9M. (b) DM2 -M10M.

Figure 6 :

 6 Figure 6: 3D model of M9M and M10M.

Figure 7 :

 7 Figure 7: Real-time computer schematic.

Figure 8 :

 8 Figure 8: Calibration and Test Unit (CU and TU). On the left is a 3D model of the subsystem, on the right different views of its optical design (the folding mirror on top is not shown).

Figure 9 :

 9 Figure 9: Average SR as a function of the off-axis angle considering 5 2s-long simulations considering random realizations of each of the 5 reference atmospheric profiles considered. Zenith angle is 30deg. NGSs are bright, H=10, and at a distance of 55 arcsec. SR is estimated from PSFs in which 180nm RMS of error (from terms not considered in the simulations, see Tab. 2) are added with random phase screens with a f -2 power spectral density. The vertical bars show the standard deviation of the SRs on the 5 random realizations. Small and large MICADO FoV is r≤10 arcsec and r≤30 arcsec respectively.

Figure 10 :

 10 Figure 10: Average K-band (2200 nm) SR as a function of the sky coverage at the south galactic pole for the 5 reference atmospheric profiles. Here 180nm RMS of error comes from terms not considered in the simulations (see Tab. 2). Zenith angle is 30deg.

  (a) J band, 1250nm. (b) K band, 2200nm.

Figure 11 :

 11 Figure 11: Average full width at half maximum (FWHM) in the MICADO large FoV (r≤30 arcsec) as a function of the sky coverage at the south galactic pole considering the median atmospheric profile.The high order contribution to FWHM is estimated from PSFs in which 180nm RMS of error (from terms not considered in the simulations, see Tab. 2) are added with random phase screens with a f -2 power spectral density. Zenith angle is 30deg.

Figure 12 :

 12 Figure 12: K-band sensitivity at 5σ for a four-hour integration time for different atmospheric conditions.

Table 1 :

 1 Main parameters of the LO and Ref WFS design.

		LO WFS Ref WFS
	Number of WFS	3	3
	Wavelength band	H	R+I
	Subapertures on pupil diameter	2	8
	Pixels on subaperture side	128	30
	Pixel scale [asec/pix]	16	165
	Field-of-view side [arcsec]	2.048	4.950

  *) statistical value for half of the sky at the south galactic pole.

	Error term	Q1 Q2 median Q3	Q4
	High orders	191 226	239	258	316
	Low orders (*) 137 161	169	185	231
	Extra		180
	Total	296 331	343	365	431

(