Optical Design and Tolerance Analysis of

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1 Optical Design and Tolerance Analysis of a high resolution near IR spectrograph for astronomy S. Rukdee 1, L. Vanzi 1, C. Schwab 2 1. Center of Astro Engineering UC AIUC, Av. Vicuña Mackenna 4860, Macul - Santiago Chile 2. Macquarie University, Balaclava Road, North Ryde NSW, 2109, Australia Abstract The Tao Aiuc high Resolution (d) Y band Spectrograph (TARdYS) is a high resolution spectrograph which is being developed by the Center of Astro-Engineering UC-AIUC for installation at the Tokyo Atacama Observatory TAO 6.5 meter telescope. The spectrograph is optimized for the precise measurement of the radial velocity of cool stars but it will also be a powerful tool for other science cases. Here we present the main characteristics of the instrument, its optical design and tolerance analysis. This spectrograph is a fiber fed, white pupil Echelle, which can yield R>50,000 within the spectral range of µm. We adopt echelle R6, grooves/mm and a VPH grating 333 lines/mm as cross disperser. The spectrograph will use a Teledyne H1RG IR detector in a Dewar at 80K. Keywords Infrared Spectroscopy; Ground based infrared instrumentation; Optical Design; Tolerance analysis I. INTRODUCTION After the first discovery of an extra solar planet in 1995 (Mayor et al., 1995), exoplanet science quickly became a main branch of astronomical research. The interest of the subject is high as it will shed light on the formation and evolution of planetary systems and ultimately, the likelihood of life in the universe. Several observational techniques have been applied to detect exoplanets: Radial velocity, Astrometry, Timing, Microlensing, Photometry and transits and Direct imaging (Perryman 2011). The Radial Velocity Technique (RV) measures Doppler shifts in stellar lines due to gravitational attraction by a nearby planet using a high resolution spectrograph. It is the oldest and the most successful detection method to date. In recent years, the exoplanet search has moved to cooler and lower mass stars. Due to their large numbers, M dwarfs are interesting candidates for planets in their habitable zone, a distance from a star where water stays liquid (Tarter et al. 2007). Also planet formation theories can be tested by M dwarf planet searches. The major challenge of detecting M dwarfs is that they are faint despite being majority of stars in our galaxy (Apps et al. 2010). Infrared spectroscopy is highly efficient, since the flux emitted by M5 dwarfs at 1 μm (Y-band) is several times higher than in the optical regime. To achieve RV precision of 2.2 m/s in Y-band spectroscopy, assuming S/N of 100 at 1 µm, requires spectral resolution of 60,000 for spectral-type M9 (Reiners et al. 2010). A number of near infrared instruments started operation in recent years, such as CRIRES (Moorwood et al. 2013), GIANO (Gennari et. al 2006), and IGRINS (Yuk et. al 2010); they report many interesting science results including Chemical abundances of Phosphorus, Sulfur and Strontium essential for galaxy evolution study (Caffau et al.2016), High precision near infrared radial velocity information (Bean et al. 2010, Terrien et al. 2010) and Detection of Neptune sized planet around M4.5 dwarf (Mann et al.2016). Further large instruments under construction are HPF (Mahadevan et. al 2014), IRD (Tamura M. et al. 2012), CARMENES (Quirrenbachetal A. et al. 2014) and SPIRou (Artigau E, et al. 2014) also optimized for an RV measurement. In this paper, we present the main characteristics of the optical design and tolerance analysis of TARdYS (Tao Aiuc high Resolution (d) Y band Spectrograph). This project is a collaboration between the center of astroengineering UC-AIUC, Chile and the University of Tokyo, Japan. TARdYS is an infrared high resolution echelle spectrograph of R = 66,000 covering Y-band near infrared region between µm around the Y band. The spectrograph will be installed on the Tokyo Atacama Observatory TAO 6.5 meter infraredoptimized telescope (Yoshii et al. 2010) at the Chajnator, at 5640 m altitude, in northern Chile, an

2 excellent site condition on earth for infrared astronomy with a fraction of 82% photometric nights, a median seeing of 0.69 in V-band and 0.85 mm water vapor. II. SPECTROGRAPH DESIGN ELEMENTS The spectrograph design is driven by science requirements, and made to fit into a constrained budget. We employ a 1024x1024 pixel size Teledyne H1RG infrared detector. This leads to the main disperser choice of an echelle R6 grating, which disperses all the Y-band within the limited detector space. The focal length of the spectrograph camera is 200 mm. We derive the focal length of the collimator from the relation between the image width of the fiber and the detector pixel size to meet the Nyquist sampling criterion. A. Fiber-feed and image slicer The spectrograph entrance pupil concept is adopted from a FIbre Dual Echelle Optical Spectrograph (FIDEOS) at AIUC (Tala et. al 2014). The light from dual fibers (50 μm diameter, f/4), which correspond to 0.4 arc second diameter on the sky, are fed through a simplified version of the Bowen-Walraven image slicer. Given a suitable distance and angle of its two mirrors, the image slicer creates a two-slice image of a circular fiber containing an astronomical object image. The image slicer works to increase the resolution of the spectrograph to R= 66,000 (Tala M. et al., 2016). Table 1. The main characteristics of TARdYS Spectral Resolution 66,000 Spectral Coverage µm Order 133 rd 175 th Detector Teledyne H1RG; 1kx1k Spectral Sampling 2 pixels Figure 1: The optical layout of TARdYS B. White Pupil Configuration In the concept design of TARdYS, we started with a classical spectrograph design (Berdja et al. 2012); the light beam reflects on a collimating mirror once and then reflects onto an echelle grating, a VPH grating and the camera optical elements respectively. This preliminary design shows some aberration especially tilt of the slit image. Applying Baranne s classic white pupil configuration (Baranne 1972) and a parabolic mirror (K = -1) as a collimator, we have cancelled spherical aberration, coma, astigmatism, distortion and chromatic aberrations according to Gratton s analytic calculations of Third-Order aberration (Gratton et. al 2000). This configuration (Fig. 1) has 3 extra reflections; the losses can be minimized by using a highly reflective coating, and the design delivers better image quality than the classic design for the same resolution. After the image slicer, the light beam of 28 mm diameter goes directly to the 140 mm diameter sized parabolic collimator with focal length 550 mm. We use an off axis angle of 3.8 degrees to use a subaperture of the main mirror as off-axis parabola. We apply a gold coating to a ZERODUR mirror to achieve the highest efficiency within the spectrograph s working wavelength ( μm) as shown in Table 1.

3 The beam reflects once at a fold mirror in the middle of the configuration (Fig.1), then gets reflected by the main mirror again before passing through the dispersing elements. C. Gratings We apply two dispersive element in TARdYS. First, a Richardson s R6 echelle grating corresponds to blaze angle of 80.6, lines/mm. An echellogram shown in Figure 2 is calculated based on a diffraction equation (Schroeder 1967). mλ = d(sini + sinθ) where m is the order, d is spacing between grooves, i is the angle of incidence and θ is the angle of diffraction. The orders are set to be 133 rd th. The angle of incident equals to angle of reflection = 9.4 in Littrow configuration. Second, a Volume Phase Holographic (VPH) Transmission Grating from Kaiser Optical System, Inc. is used as a cross disperser to obtain a sufficient interorder spacing in the final echelle spectrum. Zemax allows us to calculate the inter-order separation value. The separation value increases as the wavelength increases to the redder region in Figure 3. We finally decided to use a VPH grating 333 lines/mm. The two gratings will disperse the incoming light over the wavelength range μm into a total of 42 orders (133 to 175); the spectral format on the detector is shown in Figure 2. Figure 2: The echellogram shows the free spectral range of 42 orders on the detector of 1024x1024 pixels Figure 3: Inter-order separation using VPH grating 333 lines/mm as cross-disperser. The spacing in the plot is generated from the merit-function in Zemax D. Camera Figure 4: Camera part of TARdYs spectrograph consists of one doublet and two fused silica lenses, only the last lens (on the left) will be cooled in cryogenic. The ray traces all selected 7 configurations of different orders shown in different colors. The camera part of TARdYS was designed for room-temperature assembly. Nevertheless, to achieve low noise, the detector and the last lens are placed in a cryogenic Dewar cooled to 80K. The design of the camera part started from 4 lenses: a doublet and 3 singlets. The doublet combination of a crown glass and a flint glass was selected by the hammer optimization tool in Zemax considering the transmission wavelength and relative cost. We flattened one middle lens until it could be removed. Our final design is depicted in Figure 4. We select one doublet of SK2 and SF6 and two fusedsilica lenses to yield a focal length of 200 mm in the spectrograph camera. Only the last lens is in the cryogenic Dewar. E. Wavelength Calibration Source In optical astronomy, thorium-argon lamps are widely used as a calibration source. However, uranium lamps exhibit more lines in Y-band in the infrared (λ = µm) as shown in figure 5.

4 Our first generation calibration unit is based on FIDEOS Fiber link scheme. We plan to use a uranium lamp instead of the thorium-argon lamp. The fiber connector receives the image of the calibration source formed by an achromatic doublet lens (f = 100 mm) and feeds the calibration spectrum to the spectrograph. 2008) and argon (Whaling et al. 2002). Note that the distributions are collected from various source; they are not necessarily proportional to the intrinsic distribution density. III. PERFORMANCE A. Spot diagram A spot diagram shows how a point source of light appears on the image plane after passing through all optical elements in the spectrograph. In Figure 6, we show the optimized spot diagram. The black circle illustrates Airy Disk - the best possible image that an optical system could achieve or so called Diffraction Limited. The spot diagram in our design exhibits good fit within the airy disk. The feature in the central region shows small defocus but that should not be a problem for exoplanet detection. Figure 5: Histograms of the number of lines of standard uranium (Redman et al. 2011), thorium (Kerber et al. Figure 6: Spot diagrams of rays traced across the focal plane on the image surface. The top panel shows configurations in the infrared region. The central panel on is spot matrix in central orders. And the bottom panel is the bluest order. The black circle is the airy disk indicating the diffraction limit of each diagram. B. Tolerance analysis The tolerance analysis focuses on the camera part of the spectrometer. It is a critical part of the system because the camera has to be cooled down to 80K in cryogenic Dewar. We performed a testplate fit to all lenses to reduce fabrication cost and delivery time. The testplate fit did not change the optical performance significantly. For the tolerance analysis, we forced ray aiming in Zemax for reliable results. For the alignment tolerances, the doublet was treated as a group; additionally, we allowed the lenses to pivot about the center of curvature of the mating

5 surface, to account for tolerances during cementing. The summary of the general tolerance variation is shown in Table. 2. Table. 2 Camera Tolerance Criteria Surface Criteria Value Doublet Surface radius Center thickness Wedge Irregularity Tilt Decenter 1 fringe ± 0.05 mm ± 0.01 mm ± λ/8 Group Spacing ± 0.05 mm Spacing L3 & L4 Surface radius Center thickness Wedge Irregularity Surface Tilt Surface Decenter Element Tilt Element Decenter 1 fringe ± 0.05 mm ± 0.01 mm ± λ/8 ± 0.1 mm ± 0.05 mm We used a test wavelength of 1 μm to run sensitivity analysis. We use detector tilt and focus as compensators, and optimize the design using Damped Least Squares. In a Monte Carlo procedure, Zemax generated lens tolerances randomly from within the assigned range. The distribution of merit function values for 200 trials is shown in Fig 7. Figure 7: Histogram shows the merit value of 200 Monte Carlo draws of design meeting the tolerance criteria. C. Thermal analysis TARdYS system environment of the original design was set at a temperature of 20 C and pressure 1 atm. As the last lens is cooled to cryogenic temperatures, we analysed the optical performance for L4 at 73K (-200 C). In order to obtain a reliable refractive index at cryogenic temperatures, we used the actual measurements of Cryogenic, High-Accuracy Refraction Measuring System (CHARMS) at NASA s Goddard Space Flight Center by Douglas B. Leviton and Bradley J. Frey. We implemented the material at cold temperature by performing a glass fitting using the Sellmeier 1 formula in the Zemax glass fitting tool: where n is the refractive index, λ is the wavelength [μm], and K 1,2,3 and L 1,2,3 are determined Sellmeier coefficients. We then obtain a new glass property of Fused Silica and assign it for thermal analysis at -200 C. Initially the spot diagram has an airy disk size of 8.5 μm. The original setup shows a defocused spot diagram in the cold environment. The focusing problem can be solved by increasing the distance between the last Fused Silica lens and the detector. We optimized the thickness of the surface by using a merit function that considers all orders. For an optimally focused setup, the optics are diffraction limited over the whole field. The required detector tilt is 0.26 in cross dispersion direction. IV. SUMMARY TARdYS is a fiber-fed infrared echelle spectrograph planned to be installed at the TAO 6.5 meter telescope in Chile. The concept design of TARdYS, which based on the white pupil configuration, was presented. The tolerance and thermal analysis of the spectrograph s camera predicts diffraction limited performance assuming realistic manufacturing and alignment tolerances. The spectrograph is currently under construction in the opto-mechanic phase and expected to be commissioned in TARdYS will open up opportunities for high-resolution Y-band infrared spectroscopy including studies of M-dwarfs and searches for their planets.

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