´
Optica y Electr´onica
TECHNICAL REPORT
No. 632
ASTROPHYSICS DEPARTMENT
CanHiS: Very High Resolution
Capability for the OAGH
Joannes Bosco Hern´andez- ´
Aguila
August 2016
Tonantzintla, Puebla
c
INAOE 2016
All rights reserved
The author hereby grants to INAOE permission to reproduce and to distribute publicly paper and electronic copies of this technical report
In this work we present the characterization and coupling process of the Cananea High-Resolution Spectrograph (CanHiS) of the Observatorio Astrofísico Guillermo Haro (OAGH), at Cananea, Sonora, from the acquisition of the instrument at the University of Arizona (UA), until its commissioning at the OAGH and the observation of the first spectra of stellar objects, aimed at determining the abundances of individual chemical species.
We first carried out a complete characterization of the opto-mechanical CanHiS properties and optical performance analysis, culminating with a comprehensive opto-mechanical simulation of the spectrograph, and we showed the advantages offered by CanHiS. Its principal feature is the adjustable quasi-Littrow mounting, that, combined with the use of medium-band interference filters, makes CanHiS an instrument with very high spectral resolution capabilities and efficiency (R ≈ 85 000−185 000, and ≈36%, respectively). Since the original opto-mechanical specifications were incomplete
and not precise, we carried out a complete and detailed characterization of the whole instrument. Finally, we conducted an analytical study of the expected performance and we compared it with the results of a commisioning observing run, showing an excellent match.
Contents
Abstract i
1 Overview 1
1.1 Test process description . . . 2
1.2 Opto-Mechanical Characterization Process . . . 7
1.2.1 Opto-Mechanical Simulation . . . 7
1.3 Mechanical coupling with the 2.1-m OAGH Telescope . . . 7
1.3.1 The First Light . . . 9
1.4 First Observing Runs . . . 10
1.5 Performance Specifications . . . 13
2 Optical and Mechanical Layout 17 2.0.1 The slit box . . . 18
2.1 The filter wheels . . . 20
2.1.1 The collimator, the camera off-axis system and the folding mirror system. . . 26
2.1.2 The Echelle grating and theM3 assembly . . . 29
2.1.3 A Spectrograph Total View . . . 33
3 Optical Performance Analysis 35 3.1 The Grating Equation . . . 35
3.1.1 Angular Dispersion . . . 35
3.1.2 Free Spectral Range (FSR) . . . 36
3.2 The blaze function . . . 36
3.3 The echelle grating . . . 38
3.4 The Quasi-Littrow Mounting (QLM) . . . 39
3.4.1 The echelle grating in QLM . . . 39
3.5 The blaze function in CanHiS . . . 41
3.6 Blaze Wavelength as Function of Crank Number . . . 42
3.6.1 FSR in CanHiS QLM . . . 44
3.7 Convolution of the blaze function and the filters response . . . 52
4 Expected Spectrograph Performance 55 4.1 Angular and linear dispersion. Plate factor . . . 55
4.2 Angular length of a FSR and linear length at the CCD . . . 56
4.3 Resolution and Spectral Resolving Power . . . 57
4.4 Pixel matching . . . 58
5 Optical simulation and Real Performance 59
A Design Drawings for the new CanHiS Mechanical Coupling 63
List of Figures
1.1 General view of CanHiS. . . 4
1.2 Green calibration lamp at 546 nm and UNe lamp. . . 5
1.3 First approach for supplying Sun-light to CanHiS. . . 6
1.4 Optical arrangement to supply a well-focusedf /12 pencil beam at Can-HiS entrance slit. . . 6
1.5 Spectral images of the Sun obtained with CanHiS during the test period. 7 1.6 Different stages of the CanHiS opto-mechanical characterization process. 8 1.7 New upper-side of the mechanical coupling manufactured at OAGH me-chanical shop. . . 10
1.8 CanHiS attached to the stage of the 2.1-m OAGH. . . 11
1.9 Spectral image of the UNe comparison lamp. . . 12
1.10 Procyon spectrum acquired as CanHiS first-light. . . 12
1.11 Normalized Eta Cas and HD 220334 spectra, acquired with CanHiS at the 2.1-m OAGH Telescope during the first observing run. . . 13
1.12 CanHiS as is view attached at the 2.1-m OAGH Telescope. . . 15
1.13 CanHiS optical sketch and mechanical layout. . . 16
2.1 Spectrograph entrance slit and slit box. . . 18
2.2 Different slit configurations. . . 19
2.3 Slit box control micrometers and slit rotation dial. . . 19
2.4 Solidworks modelling of the slit box. . . 21
2.5 CanHiS filter wheels. . . 22
2.6 Filter wheels modelling. . . 23
2.7 Normalized response for 14 intermediate band filters used by CanHiS. . 26
2.8 Parent blank collimator mirror optical sketch. . . 27
2.9 Parent blank camera mirror optical sketch. . . 28
2.10 Off-axis paraboloid collimator mirror M2 and tilted flat mirror M1. . . . 29
2.11 M1 mechanical support. . . 30
2.12 M1 modelling. . . 30
2.13 CanHiS Quasi-Littrow Mounting (QLM). . . 31
2.14 Mechanical linkage between the external crank, the echelle grating and the adjustable mirror M3. . . 32
2.15 Front face of the echelle grating . . . 33
2.16 Opto-mechanical CanHiS simulation. . . 34
3.1 Blazed grating. . . 37
3.2 Coordinates system for an echelle grating in a Littrow mounting. . . 38
3.3 Coordinates system for an echelle grating in a quasi-Littrow mounting. . 40
3.4 Geometry used to understand the anamorphic magnification. . . 42
3.5 Filter responses (blue curves) versus the wavelength intervals covered by different diffraction orders (red segments). . . 49
3.6 Filter responses (green curves) versus the wavelength intervals covered by different diffraction orders (blue segments). . . 50
3.7 Filter responses (red curves) versus the wavelength intervals covered by different diffraction orders (blue segments). . . 51
3.8 The efficiency of the observed spectral intervals (blue circles) is obtained by the convolution of the filters response (red curves) and the blaze func-tion (black curves). . . 52
3.9 Same as Fig. 3.8, but for different wavelength intervals. . . 53
5.1 Optical simulation of CanHiS on the blaze wavelength of 5 880 Å. . . 60
5.2 Spectral image of the Lithium interval of the UNe comparison lamp and its intensity spectrum. . . 61
List of Tables
1.1 CanHiS Optical-Mechanical Specifications . . . 3 1.2 CanHiS First-light set-up Parameters . . . 11 1.3 CanHiS Performance Specifications . . . 14
2.1 Numerical matching between micrometers scales and slit width, slit lenght and slit position. . . 20 2.2 List of filters mounted on the wheels, and its corresponding central
wave-length and measured FWHM. . . 23 2.3 Parent Paraboloid Mirrors Specifications . . . 27
3.1 Echelle spectrograph wavelengths as function of crank number and dis-persion order. Table shows the lowest orders disdis-persion (21—33) display-ing their corresponddisplay-ing wavelengths (the reddest, from 6 976 to 11 386 Å) and the crank value to select a specific lambda central valueλc. . . 43
3.2 Echelle spectrograph wavelength as function of crank number and dis-persion order. Table shows central orders disdis-persion (34—42) displaying their corresponding wavelengths (the central, from 7 033 to 5 482 Å) and the crank value to select a specific lambda central valueλc. . . 44
3.3 Echelle spectrograph wavelength as function of crank number and dis-persion order. The highest orders disdis-persion (43—60) displaying their corresponding wavelengths (the bluest, from 5 561 to 3 837 Å) and the crank value to select a specific lambda central valueλc. . . 45
3.4 Echelle spectrograph wavelength as function of crank number and dis-persion order. The highest orders disdis-persion (34—42) displaying their corresponding wavelengths (the bluest, from 5 482 to 7 033 Å) and the crank value to select a specific lambda central valueλc. . . 47
Chapter 1
Overview
CanHiS (Cananea High-Resolution Spectrograph) is af /13.5, “R3.2” very high spectral resolution echelle spectrograph (R ≈85 000−185 000), recently put into operation at the
Observatorio Astrofísico Guillermo Haro (OAGH), in Cananea, Sonora, managed by the Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE). The opto-mechanical configuration (adjustable quasi-Littrow mounting), combined with the use of medium-band interference filters, instead of a cross disperser system, for isolating individual dispersion orders, makes CanHiS the instrument with the highest spectral resolution and efficiency (≈36% according to Hunten et al., 1991), operating at a mexican facility.
CanHiS, formerly the “LPL-Echelle Spectrograph”, was devised and constructed at the Lunar and Planetary Laboratory (LPL, at the University of Arizona –UA) by Pro-fessor D. Hunten, in the late 1970s. The original conception was that of R. A. Brown, who also took part in the detailed optical and mechanical design, in collaboration with R. L. Hilliard and, in the tuning and alignment, with N. M. Schneider. Operation, maintenance and upgrading were under the responsability of W. K. Wells and Prof. D. Hunten. The spectrograph operated from 1982 to 1985 at the now retired 0.61-m Telescope of the Whipple Observatory (today The Fred Lawrence Whipple Observa-tory), of the Harvard-Smithsonian Center for Astrophysics, and was later transferred to the 1.55-m Catalina Telescope of the University of Arizona Observatories (today The Steward Observatory 61′′Kuiper Telescope), until approximately the second half of the
1990s.
During its operation in those telescopes, the spectrograph produced numerous highly cited articles, mainly in planetary astronomy.1. LPL-Echelle made impact in planetary
sciences in such a way that Professor D. Hunten was awarded with the John Adam Fleming Medal at the AGU Spring Meeting Honors Ceremony, held on May 27, 1998, in Boston, Massachusetts, by the American Geophysical Union (AGU).
However, a failure in the spectrograph CCD, and the eventual impossibility for get-ting funds for a new detector, caused the LPL-Echelle to be decommissioned in the
1A list of investigations based on data obtained with the spectrograph until 1991 can be found in
Hunten et al. (1991)
late 1990s, stored in a tool room at the LPL and kept in good conditions to operate. Given these exceptional circumstances, in the first half of 2008, under a joint initiative between our research group at INAOE and from the LPL –lead again by Professor Don-ald Hunten, and Doctor Anne Sprague–, the LPL-Echelle was permanently donated to INAOE to operate at the OAGH. After the necessary legal formalities, the spectrograph was carried to the OAGH in July 2009 and re-baptized as CanHiS.
Because the spectrograph was designed to achieve different objectives than those of stellar atmospheres analysis, and was to be attached to a telescope with different op-tical and mechanical properties than the 2.1-m OAGH Telescope, the opto-mechanical coupling with its new host telescope was a crucial factor to preserve CanHiS top fea-tures. In order to design and manufacture the opto-mechanical elements for the best matching, full knowledge about spectrograph optical performance (and therefore inter-nal mechanical arrangement of its elements) was essential, requiring the detailed optical and mechanical designs. LPL researchers provided first-hand information about design and construction of the instrument and the know-how for its operation, and, addi-tionally, R. L. Hilliard shared the earliest spectrograph original blueprints. However, none of these informations contained external and internal definitive opto-mechanical spectrograph dimensions. Therefore, the starting tasks of this work were aimed at char-acterising the optical and mechanical performance of CanHiS and its components. This implied the undesirable but necessary opening of the instrument to measure the actual dimensions of its different parts required for its full opto-mechanical simulation.
In the next sections we will briefly describe the tests and characterization processes, and present the spectrograph opto-mechanical and performance specifications (Tables 1.1 and 1.3, respectively), obtained from the scarsely available original data, and mainly from the re-characterization process presented in this work. The later process takes into account its new 2.1-m OAGH host-telescope.
In Chapter 2, we show details of the CanHiS optical and mechanical layout. In Chap-ter 3 we present a complete examination of the CanHiS optical performance, focusing on all the variables under consideration for a spectrograph (such as angular dispersion, spectral resolution, free spectral range, etc.), and describe the advantages offered by CanHiS, with its quasi-Littrow mounting, compared with a classic echelle mounting. Finally, in Chapter 4 we present an analysis about the expected performance of Can-HiS, including an optical simulation, and compare this performance with a real data set. In this last section we demostrate the excellent match between the expected and observed data characteristics.
1.1
Test process description
Before the full spectrograph re-characterization, we conducted several tests with the main purpose of verifying the general state of the spectrograph. Using a Hitachi KP-D581 as test detector, we obtained spectral images with the spectrograph placed in the instrument room. These experiments were aimed at checking the optical alignment and mirror quality.
3
Table 1.1: CanHiS Optical-Mechanical Specifications
Overall focal ratio f /# f /13.5
ø= 57 mm deviation angle = 6.◦54
First diagonal flat M1
ø= 68 mm
f = 670.56 mm
angle off-axis = 6.◦54
Off-axis collimator M2
152×83 mm
variable tilt angle = 8◦
–18◦
Folding flat M3
165×105 mm
f = 670.56 mm angle off-axis = 4.◦87
Off-axis camera M4
ø= 57 mm deviation angle = 42◦.6
Last diagonal flat M5
The first test carried out was to project a HeNe laser at 633 nm on semi-transparent screens placed on both the principal optical path, before the spectrograph focal point, and on an external spectrograph entrance slit observation point (Fig.1.1a). The ex-pected outcome is the HeNe laser principal beam centered on the exit spectrograph optical path, when the laser beam is centered at the spectrograph entrance slit. Figure 1.1b shows the projected slit on the screen at the slit observation point by illuminating the entrance slit at the top, and Fig.1.1c the HeNe laser beam centered on the slit, as it is observed on the screen. Figure 1.1d presents the HeNe laser principal beam centered on the optical path, before the exit at the focal point, and after the beam has been subject to the spectrograph internal reflections and the Echelle dispersion, proving the correct spectrograph optical alignment.
In order to observe the entire slit (composed of a pair of adjustable polished jaws) at the spectrograph exit, the next test was made by illuminating the entrance slit with a calibrating monochromatic green source at 546 nm (Fig. 1.2a). Figure 1.2b shows the slit spectral image observed at the spectrograph exit. Because the incident light is monochromatic, the spectral image can only be obtained at the individual order in which the light is diffracted (order 43, see section 3.6). There is a slope at the slit lower end of the spectral image (Fig.1.2b), caused by the triangular shape of the frame in which the real slit is embedded (Fig.1.1b).
Another important test at this initial stage was to observe the Echelle dispersion using the UNe comparison lamp inside the spectrograph (Fig.1.2c). The spectral image was collected at the same spectral region of the green source, so it should be approxi-mately centered at 546 nm. There is some spectral overlap that is caused by defocusing,
(a) (b)
(c) (d)
Figure 1.1: a) General view of CanHiS, showing the site for places the CCD detector, at the principal optical path, and an external spectrograph entrance slit observation point; b) projected slit on a semi-transparent screen at the slit observation point on the spectrograph front face, by illuminating the entrance slit at the top. It can be seen the triangular shape of the frame in which the slit is embedded; c) HeNe laser beam centered on the slit (as it is observed on the screen placed at the slit observation point); d) HeNe laser principal beam centered on the optical path, before the exit focal point.
although it is not critical at this stage. In both the green source and UNe images, the slit images suffered from an undesirable tilt due to the quasi-Littrow configuration. This effect can be corrected adjusting appropriately the angle of the entrance slit with respect to the echelle grating.
The next test was to observe absorption lines from a stellar source, specifically the Sun. Because the spectrograph was not yet attached to the telescope, solar light was fed using a bundle of 33 optical fibers. The end exposed to solar light has linear shape (so spectral image presents straight lines along the dispersion axis, due to the individual fibers), and the end feeding the spectrograph has a circular shape. Sun light was supplied in two ways: first, placing the circular end shape ≈ 8′′.40 above the slit
(Fig. 1.3) —thus light projected into the spectrograph was not focused; and, second, through an optical arrangement to produce a f /12 beam light (consisting of a pair of
5
(a) (b)
(c)
Figure 1.2: a) Green calibration lamp at 546 nm; b) spectral image of the spectrograph entrance slit, illuminated at 546 nm, in the order 43. The shape of the slit, given by the adjustable jaws that make it up and a triangular frame, is visible; c) spectral image of the UNe comparison lamp attached to the spectrograph, at the same spectral region that the green source (approximately centered at 546 nm). Slit images show a tilt due to the quasi-Littrow mounting, that can be easily corrected by adjusting the angle of the entrance slit.
1.4a). Figure 1.4b shows the fiber circular end projected on the slit, as it is observed at the slit observation point. We used an Apogee U77 CCD of 512 times 512 pixels, of 24 × 24 µm. Figure 1.5a shows the solar spectral image obtained in the first case
(rotated 90◦ counter clockwise with respect the original image at the CCD). In the
image the dispersion axis is horizontal and the spatial axis vertical. The solar spectrum is centered approximately at 5 890 Å (the Sodium D), at the order 40. It can be seen the aforementioned straight lines due to the individual fibers, and a shaded zone that grows towards the lower right corner. This region is produced by the lack of illumination from the linear shape end at the fiber exposed to the sunlight. Fig. 1.5b presents the spectral image acquired under the second setup, centered at≈6 710 Å, at the order 35.
Because the spectrograph opto-mechanical elements sizes are well suited for accepting a f /12 pencil-beam instead of the original f /13.5, this setup yields a better spectral
(a) (b)
Figure 1.3: First approach for supplying Sun-light to CanHiS. a) Optical fiber mounting; b) circular end shape optical fiber,≈8′′.40 above the slit. Because of the lack of a focus system at the entry and the exit of the fiber, light projected into the spectrograph slit is a roughly inverted pencil-beam. Figure 1.5a shows the spectral image resulting.
(a) (b)
Figure 1.4: a) Optical arrangement to supply a well-focused f /12 pencil beam at the spectrograph entrance slit, coming from the optical fiber circular end shape. b) Fiber circular end projected on the spectrograph entrance slit, as is observed at the slit observation point, after becoming focused by the optical array for generating af /12 pencil beam. Figure 1.5b presents the spectral image at the exit.
In both instances, the curvature of the projected straight slit is visible (at the ab-sortion lines) as a consequence of the diffracted angle β dependence on the off-axis γ
angle at the echelle grating. Because of the short spectral coverage range, the charac-teristic echelle spectral curvature along the spectral axis is not visible.
Once the functionality of CanHiS was verified, the next steps were to carry out a whole opto-mechanical instrument characterization, and to attach it to its new host telescope, the 2.1-m OAGH Telescope. The detailed knowledge of CanHiS will allow to explore potential upgrades such as the inclusion of a cross disperser to increase the wavelength coverage, or to make CanHiS a fiber-fed spectrograph to enhance the mechanical stability.
7
(a) (b)
Figure 1.5: Spectral images of the Sun obtained with CanHiS during the test period before the spec-trograph re-characterization, using a fiber optic bundle with one end of linear shape, and the other of circular shape. a) Spectral image around the Sodium-D (≈5 890 Å), obtained by placing the circular end shape of the optical fiber≈8.′′40 above the spectrograph entrance slit, without any focusing system at the entrance and the exit of the fiber. b) Spectral image around the Lithium line (≈6 707 Å), through an optical arrangement to produce af /12 beam light, focused at the spectrograph slit. Although light going into the fiber is not focused, spectral image is better than in the prior situation. In both cases, there are horizontal straight lines due to the individual fibers at the linear shape end, and darker areas over the spectrum due to a not well illuminated portion of the fiber exposed to the sunlight.
1.2
Opto-Mechanical Characterization Process
1.2.1 Opto-Mechanical Simulation
One of the most important tests carried out for the CanHiS characterization was the opto-mechanical simulation of the spectrograph, which was developed independently us-ing the computer-aided optical and mechanics design software Zemax and Solidworks, respectively, and then joined together into a single simulation. Executing both mechan-ical and optmechan-ical simulations requires a precise and accurate knowledge of all dimensions of the spectrograph, which were not available, as previously mentioned. Consequently, it was necessary to open the spectrograph and measure every component involved in the optical and mechanical performance. Measurements were done at the instrument room and the mechanical shop of the OAGH (Figure 1.6). These measurements were the basis for developing both the necessary elements to attach the spectrograph to its new host telescope, and to carry out the opto-mechanical characterization.
1.3
Mechanical coupling with the 2.1-m OAGH Telescope
In order to attach CanHiS to the 2.1-m OAGH Telescope, preserving all its desirable top features, the first explored option was to design a focal expander to match the slightly fasterf /12 telescope beam with thef /13.5 spectrograph focal ratio. This would ensure
(a) (b) (c)
(d) (e)
Figure 1.6: Different stages of the CanHiS opto-mechanical characterization process. a) and b) Mea-suring of individual spectrograph elements; in a), the upper side of the original mechanical coupling between the host-telescope and the spectrograph is characterized; in b), an individual lense belonging to the guiding optics is measured with a depth gauge. c) and d) Characterization of the whole spectro-graph with the milling machine in the OAGH mechanical shop, and d) characterization with the milling machine and a depth gauge.
lead to a small loss in the instrument efficiency. However, a quick optical analysis showed that the principal optical elements had the suitable dimensions to accept the
f /12 telescope beam, without suffering any loss neither in resolution nor in efficiency. The only prerequisite was to ensure that the 2.1-m OAGH Telescope focal point was well focused at the spectrograph entrance slit, placed 7.′′20 below the spectrograph
upper face. Because the original mechanical coupling —consisting of a square box of 10′′.5 in height and 13
. ′′25
×13′′.25 by side, closed at both ends by circular plates of 22 . ′′0
9
Telescope properties, and also was too thick for placing the 2.1-m OAGH Telescope focal point at the CanHiS entrance slit (the total height of the coupling was 12′′.0), it
was necessary to manufacture a new coupling. Based on the original piece, the new coupling was designed with a new central structure of 1′′.93 in height and 13
. ′′25
×13′′.25
(a square box) in width and depth, and a new plate of 22′′.0 in diameter and 0 . ′′75 of
thickness, and a central hole of 4′′.
00 of diameter for coupling to the mounting of the telescope. The plate to be coupled to the spectrograph was the already available piece (Figure 1.7b). The total height of the new coupling is 3′′.43, ensuring that the 2.1-m
OAGH Telescope focal point will be at the CanHiS entrance slit. Figures 1.7a and 1.7b present the new plate and parts of the new square box, respectively, and Figures 1.7c and 1.7d the new coupling completed and assembled, and the coupling attached to the 2.1-m OAGH telescope. The design of all the parts for the new coupling were made using Solidworks, and were fabricated at the mechanical shop of the OAGH. The design drawings of all the pieces designed for the new mechanical coupling are presented in Appendix A.
In Figure 1.8 we show CanHiS attached to the 2.1-m OAGH Telescope through its current mechanical coupling. In these panels the comissioning CCD VersArray camera is already attached to the spectrograph and ready to conduct the first light observations.
1.3.1 The First Light
After the fabrication of the mechanical coupling, we proceeded to attach CanHiS to the 2.1-m OAGH Telescope, and then to obtain CanHiS first-light in January 2011. The first astronomical target was Procyon, because of its brightness (mv = 0.37) and
spectral type (F5 IV-V). While we envisaged the use of CanHiS in abundace studies of much fainter solar-like stars (the faintest object observed until then with CanHiS was Io, with an apparent magnitude of 5.02), these properties would ensure an easy detection of numerous metallic spectral lines as expected for an intermediate F-type star. The region selected for the first observation was centered at the sodium-D line,
≈ 5 890 Å, using the filter named D, whose transmittance curve is centered at this
wavelength (see section 2.1). As indicated above, the CCD camera was a VersArray of 1 340× 1 300 pixels. Table 1.2 shows the general parameters of the observation, which will be explained below.
Figure 1.9 shows the spectral image of the UNe comparison lamp used to identify the spectral region selected. While we have not conducted a formal wavelength calibration, we made a carefully visual inspection to compare the spectral region with an Atlas of Uranium emission intensities2, identifyng the principal lines. It is important to note that
none of the spectrograph optical elements were calibrated for this first light observation (e.g. collimator, guiding camera and CCD). Fine focus mechanical coupling was still in the process of manufacturing.
Finally, Figure 1.10a presents the raw 1-D Procyon spectrum acquired as CanHiS
2An Atlas of Uranium Emission Intensities in A Hollow Cathode Discharge,LOS ALAMOS SCIENTIFIC
(a) (b)
(c) (d)
Figure 1.7: a) New upper-side manufactured at OAGH mechanical shop for the mechanical coupling between CanHiS and the 2.1-m OAGH Telescope. b) Poles manufactured at the OAGH mechanical shop, fixed at the circular plate from the original mechanical coupling, reused in the new coupling. c) Mechanical coupling completed. d) Mechanical coupling fixed at the 2.1-m OAGH telescope, ready to receive CanHiS.
first-light, with the instrumental set-up defined at Fig. 1.9. The overall shape of the spectrum corresponds to the convolution of the echelle grating blaze function and the filter used (see section 3.7). Figure 1.10b shows the same spectrum, roughly normalized to the continuum.
1.4
First Observing Runs
The first observing runs devoted to engineering time were carried out using the same VersArray CCD. These runs were aimed at characterizing the optical and mechanical CanHiS performance. Figure 1.11 shows the normalized uncalibrated spectra of the bright G-type starsEta Cas (mv = 3.45) and HD 220334 (mv = 6.62). In both cases,
the spectra have a high signal-to-noise (S/N≥150), and high-resolution (R ≈ 138 000
11
(a) (b)
Figure 1.8: CanHiS attached to the stage of the 2.1-m OAGH, its current host-telescope, through the upgraded mechanical coupling.
Table 1.2: CanHiS First-light set-up Parameters
Telescope
Primary mirror 2.1-m
Focus Cassegrain
Focal ratio f /12
Plate scale 8b′′.19 per mm
Spectrograph
Focal ratio f /13.5
Filter used D (5 890 Å)
≈90 Å
(limited at final image by the CCD size) Wavelength range
Order selected 35
4 750 (at order 35) Central wavelength number indicator
217 (not accounted for this test) Dial slit box indicator
≤25µm
(micrometer indicator ≥0′′.395)
Slit width
Detector
CCD VersArray thinned, back illuminated
Pixel size 20 µm, square format
Figure 1.9: Spectral image of the UNe comparison lamp, centered at ≈ 5 890 Å, based on a visual inspection to compare the spectral region (in pixels), with an Atlas of Uranium emission intensities. The broadest line (≈ 900 px) correspond to a Ne line, which is not taken into account in a formal wavelength calibration process.
(a) (b)
Figure 1.10: Procyon spectrum acquired as CanHiS first-light, with the same instrumental set-up defined at Fig. 1.9. a) Unnormalized Procyon spectrum, which shape obeys the convolution of the Echelle grating blaze function and the filter used. b) The same spectrum, but roughly normalized. The deepest absortion lines are the Na-D.
in the lithium feature at 6 707.76 Å. For comparison we include in both panels the very high resolution (R= 522 000) observed solar spectrum (Kurucz et al., 1984), degraded to the resolutions of each case. Dashed lines indicate some of the main absortion features of individual chemical species present in both stars.
13
(a)
(b)
Figure 1.11: a) NormalizedEta Cas spectrum, acquired with CanHiS at the 2.1-m OAGH Telescope,
during the first observing run. b) NormalizedHD 220334 spectrum, under the same characteristics that in a).
1.5
Performance Specifications
Based on the opto-mechanical characterization and simulation, and on the first and the follow-up observing runs (with CanHiS attached to the 2.1-m OAGH Telescope), and on the results obtained through a CCD camera e2v 42-40, of 2 048×2 048 pixel format,
as is shown in Table 1.3 (an example of the latter results —with the current CCD e2v 42-40, is presented in Section 4). It is very important to note that, as a result of the slightly faster f /12 incident beam towards the spectrograph, in addition to the robust optical design made by the team leaded by Prof. D. Hunten and the new baffle, CanHiS optical performance was improved significantly, increasing both the maximum resolving power, from 160 000 to≈185 000, and the efficiency.
Table 1.3: CanHiS Performance Specifications
Entrance aperture(at 2.1 m –f /12 Cananea Telescope)
Maximum (slit width ≈50 µm) 0b′′.
410
Minimum (slit width≈ 25µm) 0b′′.205
Spatial resolution(at 2.1 m –f /12 Cananea Telescope)
Maximum (slit length≈4.62 mm) 37b.′′
8
Minimum (slit length≈2.81mm) 23b.′′
0
Resolution δλ
Maximum (at minimum slit width) ≈ 0.03 Å
Minimum (at maximum slit width) ≈ 0.08 Å
Power resolution R=λc/δλ
Maximum ≈ 190 000
Minimum ≈85 000
≈50 Å (near to 4 500 Å) ≈90 Å (near to 6 300 Å)
≈140 Å (near to 7 800 Å)
Typical wavelength range∆λ
Limiting magnitudemv
≈10.2 mag
(atδλ≈0.200 Å, and λc≈ 6 707 Å)
Figure 1.12 shows CanHiS attached at the 2.1-m OAGH Telescope and Fig. 1.13 presents the whole opto-mechanical layout of CanHiS in an x, y, z coordinate system.
This plot was created on the basis of re-characterization and the simulation process carried out during this work. The depicted incident light is af /12 pencil-beam, similar
15
(a) (b)
Figure 1.13: CanHiS optical sketch and mechanical layout (in order to show the elements under the internal shelf, this is not visible in the upper-view)
Chapter 2
Optical and Mechanical Layout
According to Figure 1.13, the main-rectangular box of the spectrograph is divided in two chambers: the smallest upper cavity, enclosing guiding optics and comparison and flat-field sources, and the lower and principal compartment, containing the spectrograph optics, separated shelf. The total size of the spectrograph box is 8.50×47.9×25.0 inches, in thex, y, z axes, respectively.
Inside the upper bay, the central region of the incoming beam from the host telescope passes through the center hole of the 45◦ tilted guide mirror G
1, focusing the light to
the slightly tilted spectrograph entrance-slit. G1 reflects the total field observed by
the telescope towards the guide optics, meanwhile the slit slices the central part of the focused light inwards the spectrograph, and reflects the rest into a lateral microscope (the slit observation point, Fig. 1.1a).
At the principal compartment, a band-pass filter first cuts a narrow spectral band from the sliced beam of approximately 50, 90 or 140 Å (and centered at 4 500, 6 300 and 7 800 Å, respectively, depending on the filter used –see section 2.1 and Chapter 3). The tilted flat mirror M1 sends the spectral band selected to the off-axis paraboloid
mirror M2, collimating the pencil-beam towards the echelle grating. Diffracted light is
then reflected in the adjustable flat mirror M3, and sent towards the off-axis paraboloid
mirror camera M4. Finally, the spectrum reaches the CCD detector from the tilted-flat
mirror M5.
In order to also understand the enhanced CanHiS capabilities, we conducted a de-tailed analysis of each of its opto-mechanical elements. As previously mentioned, this analysis had been carried out using the software for mechanical design Solidworks, and for optical design Zemax. When possible, individual opto-mechanical elements had been measured by us, then optically and mechanically analyzed separately, and finally matched into one single opto-mechanical element.
2.0.1 The slit box
The spectrograph entrance slit consists of a pair of adjustable polished jaws (Fig. 2.1a), placed into a box tilted 6.◦
54 with respect the x-y spectrograph plane (Figs. 2.1b and 2.1c).
(a) (b)
(c)
Figure 2.1: a) spectrograph entrance slit; b) and c) general view of the slit box, and a side view of the slit box placed into the spectrograph, respectively. At the side view, the slope of 6.◦54 of the slit (along
with the slit box), with respect thex-yspectrograph plane, is observed.
The slit box provides four independent degrees of freedom to the slit: a) slit width1,
from less than 25 to≈ 50 µm; b) slit length, from ≈ 2.81 to 4.62 mm (Figs. 2.2a and
2.2b); c) slit position over the observation plane, into a square spatial range of ≈1.81
mm by side (Figs. 2.2c and 2.2d); and d) slit rotation. Three independent micrometers control the width, the length and the position of the slit, and a dial and a set screw at the basis of the box control the slit rotation (Figure 2.3). Table 2.1 presents the relationship between the micrometers scale and the real slit width and slit length, along with its arc-second equivalent at the OAGH telescope.
1Slit width possible is greater than 600µm; however, slit becomes unuseful above 50µm, presumably
19
(a) (b)
(c) (d)
Figure 2.2: a) and b) two different slit lengths; c) and d) two different positions for the slit over the observation plane. Note that a brass mask states the length and slit position; slit box (and the slit are fixed.
Table 2.1: Numerical matching between micrometers scales and slit width, slit lenght and slit position. Slit length and slit position measurements are experimental, whereas the slit width had been calculated based on Uranium-Neon comparison lamp exposures and the Eq. 3.13a. For the slit position, micrometer value of 0′′.300 is established as the zero-point value (0.00 mm) over the observation plane (the center of the CCD at the spatial direction), keeping the dispersion axis of the spectrum parallel to one side of the CCD and the spatial axis perpendicular. If dispersion axis is inverted and horizontal in the CCD, the minus sign represents slit upwards displacements respect the dispersion axis; if vertical, represents displacements to the left.
micrometer slit width slit length slit position
scale [inches] [µm] [arc-seconds] [mm] [arc-seconds] [mm] [arc-seconds]
0.075 — — 4.62 37.79 — —
0.100 — — 4.20 34.37 −1.20 −9.83
0.125 — — 3.78 30.94 — —
0.150 — — 3.38 27.62 — —
0.175 — — 2.94 24.09 — —
0.200 — — 2.54 20.77 −0.59 −4.86
0.225 — — 2.11 17.24 — —
0.250 — — 1.63 13.37 — —
0.275 — — 1.22 09.94 — —
0.300 — — 1.05 08.62 0.00 0.00
0.325 — — — — — —
0.350 — — 0.73 05.97 — —
0.375 — — — — — —
0.380 50.76 0.42 — — — —
0.385 — — — — — —
0.390 — — — — — —
>0.395 21.70 0.18 — — — —
0.400 — — — + 0.61 + 4.97
The light reaching the slit goes through the entrance hole of the slit box, with 0′′.
634 of diameter, and 0′′.420 over the slit. The total inclination of the slit box results in a real
diameter of 0′′.630, sufficient for the 0 .
′′374 incident beam diameter from the telescope.
Likewise, the sliced beam leaving the slit box, of 0.′′
374 in length, passes through the exit hole, of 0′′.630 of diameter and placed 0
.
′′420 below the slit. Figure 2.4 shows the
simplified slit box mechanical diagrams.
2.1
The filter wheels
A set of fifteen circular transmission filters mounted on a pair of independent hand-movable concentric wheels form the intermediate-band filters system (Figure 2.5). Wheels are joined through the axis by a pole fixed to a basis attached to the internal spectro-graph front face (parallel to y-z plane), and screwed on the lower face of the splitting
21
(a) (b)
(c)
Figure 2.4: a) Isometric view, b) bottom view, and c), side view of a hidden lines visibles model, for a modelling of the slit box. Real dimensions, which were made individually for each part of the element and used to modelling it at Solidworks, are presented.
faced-on with respect the beam from the slit box (thex-y plane) by turning the corre-sponding wheel. Filters central bandwidth is identified by letters at the upper wheel and by numbers at the lower. Selection of the filter is made by choosing the corresponding label attached to the edge of the wheels by means of a small section protruding at the spectrograph rear side (Fig. 2.5d). The filter wheels general mechanical diagrams are shown in Figure 2.6.
In order to know the actual filters response, the filters named with the letters from
BtoK, and with the numbers from4 to8, had been characterized at the Colorimetry
Lab of INAOE. Filter labeled 9 was not analyzed, because of the lack of a suitable
detector for the wavelength in which this is centered (10 970 Å). Table 2.2 lists the central wavelengths for the filters and the order in which they are set, along with its identifier and the FWHM measured. To give clearance to the light at the filter wheel not in use, a transparent window made of BK7 glass (named as CLEAR and labeled
A and 0) is mounted in both wheels. Also, the lower wheel hosts three independent
dimmers, named ND 1, ND 2 and ND 3, which attenuate the light in 10 %, 1 % and 0.1%, respectively. Note that the filters central wavelength had been selected in order
(a) (b)
(c) (d)
Figure 2.5: a) Upper filter wheel, letter labeled; b) Bottom filter wheel, number labeled; c) and d) lateral view of the filter-wheels, showing the identification labels and the section that protrudes at the spectrograph rear side.
to get the highest efficiency at the characteristic wavelengths of chemical species of astrophysical interest: for example, filter F (5 007 Å), is centered at the [O III] line;
filterD(5 890 Å), at the sodium D1 and D2 lines (5 890 and 5 896 Å, respectively); and
filter7, at Hα (6 563 Å).
Figure 2.7 presents the normalized response for the fourteen filters characterized, and for the transparent windows and dimmers. Filter responses had been separated in three intervals, considering the filter central wavelengths: a blue region, from 3 725 to 5 007 Å (Fig. 2.7a); an intermediate region, from 5 890 to 6 723 Å (Fig. 2.7b); and a red region, from 7 325 to 8 273 Å (Fig. 2.7c). In Figure 2.7a, filterG(3 725 Å) is practically
useless because of the very small filter efficiency, and filtersBand4(4 085 and 4 227 Å)
have less than the 50 % of efficiency. Finally, filterF (5 007 Å), has approximately an
efficiency of 65 %. It is of primary importance to mention, however, that filter low efficiency is highly compensated with the total spectrograph efficiency, as described in section 3.5 and Chapter 4.
23
(a) (b)
(c)
Figure 2.6: Filter wheels modelling.
Table 2.2: List of filters mounted on the wheels, and its corresponding central wavelength and measured FWHM.
Upper Central FWHM Lower Central FWHM
Wheel lambda Wheel lambda
A CLEAR — 0 CLEAR —
B 4086 Å Å 1 ND 1 —
C 6723 Å Å 2 ND 2 —
D 5890 Å Å 3 ND 3 —
E 6306 Å Å 4 4227 Å Å
F 5007 Å Å 5 4589 Å Å
G 3725 Å Å 6 7325 Å Å
H 7682 Å Å 7 6563 Å Å
J 8151 Å Å 8 8047 Å Å
but there is a small overlapping between the filters 7 and C (centered at 6 563 and
6 723 Å, respectively); also, as it will be explained in section 3.6.1, this overlap occurs at wavelengths where the blaze function turns it into an unimportant problem. However, in the red region (Fig. 2.7c), there is significant overlap in the transmission of the three filters with the longest wavelengths: 8,JandK, (8 047, 8 151 and 8 273 Å, respectively),
25
(a)
(c)
Figure 2.7: Normalized response for 14 intermediate band filters used by CanHiS. Dashed lines depicts the response for the BK7 windows, whereas dotted lines shows the response for the dimmers. In a), filter G(3 725 Å) is practically useless because of the very small filter efficiency, and filtersBand 4
(4 085 and 4 227 Å, respectively) have less than the 50 % of efficiency, whereas filter F(5 007 Å) has
approximately an efficiency of 65 %. In b), all the filters have an efficiency up to 90%, but filters7and C (centered at 6 563 and 6 723 Å, respectively) suffer of a small overlapping between them, although
this occurs at wavelengths where the blaze function makes it unimportant. At c), the red region, filters
8,JandK, show serious overlapping, particularly the filters centered at 8 151 and 8 273 Å.
2.1.1 The collimator, the camera off-axis system and the folding
mir-ror system.
The spectrograph collimator and camera mirrors (M2 and M4, respectively) consist
of paraboloid sections with an off-axis configuration, providing a compact design and preventing first and second order aberrations. Such configuration makes use of tilted flat mirrors (M1 and M5), folding the incoming beam light into the desired entry off-axis
angle towards M2, and folding the outgoing beam light into the necessary exit off-axis
angle, towards the detector. Each off-axis angle is calculated according to the size of each off-axis mirror, as described below.
27
2.1.1.1 The off-axis paraboloid collimator and camera mirrors M2 and M4
The paraboloid collimator mirror, as well as the paraboloid camera mirror, are both off-axis sections of an on-off-axis parent paraboloid. Notwithstanding precedent manufacturer information (private communication), shape, size and performance of both mirrors show that each one was cut from two different parent mirrors of different diameters but the same optical characteristics, and not from a single parent blank. The optical features of parent mirrors are listed in Table 2.3.
Table 2.3: Parent Paraboloid Mirrors Specifications
Parent blank collimator diameter 9′′
Parent blank camera mirror 10′′.50
Focal length 26′′.
40
Central radius of curvature 52′′.80
Center thickness >1′′
Shape on-axis paraboloid
Tolerance λ/8 at 5890 Å
Material fused silica
The collimator mirror is a circular cut of 2′′.67 in diameter, off-axis angle of 6 . ◦57 and
off-axis lineal distance of 3.′′
04, whereas camera mirror is a semi-rectangular cut of 6′′.
50×
3′′.50, off-axis angle of 4 .
◦87 and off-axis lineal distance of 2 .
′′25. Optical performance
analysis and Zemax simulations show that both off-axis angles are slightly different than the value of 6◦.
50 indicated in Hunten et al. (1991); their actual values are 6◦.
57 and 4◦.87, respectively, because of the different cut diameter size and, therefore, off-axis
lineal distances of 3′′.04 and 2 .
′′25 for M
2 and M4, respectively. Figures 2.8 and 2.9 show
the sketches of the cuttings over their respective parent blank, and the off-axis angles and lineal distances.
Figure 2.9: Parent blank camera mirror optical sketch.
2.1.1.2 The folding mirror system.
Figure 2.10 shows the off-axis collimator system. The illustrated configuration ensures that the fraction of light sliced by the entrance slit has the correct entry angle towards the echelle grating.
The M1 flat mirror, with an angle of 48◦.3 with respect to the x-z plane, folds the
incoming beam-light from the entrance slit by an angle of 96◦.6, with respect to the
same plane. This angle provides the off-axis angle of 6.◦
57, which is needed to match the paraboloid off-axis collimator mirror M2. For the principal ray, the distance from
the entrance slit towards the center of M1 is 5′′.00, and from this point to the center of
M2 is 21′′.4, resulting in a total of 26′′.4 (or 671 mm), equivalent to the focal length of
the paraboloid collimating mirror M2.
A similar configuration to that of the collimator, but parallel to the x-y spectro-graph plane, is used to achieve that the diffracted beam, emerging from the echelle and reflected by the plane mirror M3, reaches the detector, once it has been focused by the
camera mirror M4. The flat mirror M5 has a tilt angle of 47◦.4 (with respect to the
plane y-z, i.e. the spectrograph rear side), in order to fold the outcoming
diffracted-beam (from the camera mirror) to an angle of 4◦.87, with respect to the same plane.
2.1.1.3 The tilted flat mirror M1
The tilted flat mirror M1, of 2′′.25 in diameter, is a folding mirror with no optical
power. However, the mirror tilt and the diameter size of the input/output baffles in the mechanical support are decisive factors to ensure the correct operation of the system. Due to the off-axis configuration of the collimator mirror M2, the M1 tilt angle has to
achieve the desired entry off-axis collimator angle of 6◦.57. Therefore, the M
1mechanical
support (Figure 2.11) consists of an aluminium holder and a cell with three position bolts to set a tilt angle of 48◦.3 with respect to the
x-zplane. Figure 2.12 shows a sketch
29
Figure 2.10: Off-axis paraboloid collimator mirror M2 and tilted flat mirror M1. The fraction of light
sliced by the entrance slit is folded by the flat mirror M1, which has a tilt angle of 48◦.3 respect the
planex-z, aimed to divert the incoming beam-light a total of 96◦.6. This angle provides to the light the necessary off-axis angle of 6◦.57 for the paraboloid off-axis collimator mirror M
2. For the principal ray,
distance from the entrance slit towards the center of M1 is 5′′.00, and from this point to the center of
M2 of 21′′.4, resulting in a total of 26′′.4 (or 671 mm), equivalent to the focal length of the paraboloid
collimator mirror M2.
2.1.2 The Echelle grating and the M3 assembly
The spectroscopic CanHiS configuration is based on thequasi-Littrow mounting(QLM,
see section 3.4), one of the three possible echelle spectrograph designs referred to the orientation of the incident light beam with respect to normal of the echelle grating, according to Schroeder & Hilliard (1980). In QLM, the so-called off-axis angle γ (see
also section 3.4.1), which depends on the orientation of the Echelle grooves faces with respect to the incident beam, is not zero.
Being a rare feature used in spectrographs (e.g. Schroeder 1967), CanHiS makes use of a variableγ angle controlled by a high-precision mechanical arrangement (Figs. 2.13 and 2.14). The system rotates the echelle grating with respect to the incident light beam so that theγ angle varies from 8◦
to 18◦
, at the same time, as the M3mirror turns
to follow the diffracted beam, maintaining the angular separation between the echelle grating and M3 always equal to 2γ (Fig. 2.13).
A pair of turntables linked through a bar join the echelle grating and the fold mirror M3 (Fig. 2.14b). A lead screw (Fig. 2.14a), coupled to the echelle turntable (Fig.
2.14b) on one end and to a high-precision external crank (Fig. 2.14c) on the other end, controls the positions of M3 and the grating turntables.
When the lead screw is moved, the echelle grating rotates with respect to its normal, remaining constant at its blaze angleδ, while the faces of its grooves turn perpendicular
(a) (b)
(c) (d)
Figure 2.11: M1 mechanical support. Circular cut at the upper side is for give clearance to the filter
wheels.
(a) (b)
31
(a) (b)
(c)
(d)
Figure 2.13: Quasi-Littrow Mounting for CanHiS. Schemes a), b) and c) show the three “extreme” positions for the echelle grating and the M3 mirror, corresponding to γ angle values of 18, 13 and 8
degrees, respectively. At the images, only the echelle grating (which is face-on) is visible; mirrors M1,
M2, M3 and M4 are side-view. M5 is visible, because of are tilted 47◦.4 respect the spectrograph rear
side. Diffracted wavelengths increases asγ decreases into the same order. d) shows the real QLM at CanHiS; M2 and M4 mirrors are visible, while M3 are not at the image. The echelle grating, above the
bar that join togheter the grating cell and M4, is covered by a cap.
to the normal. M3 moves parallel to the line A-A (see Figure 1.13), rotating with respect
to the echelle grooves faces. These features ensure that the central wavelength diffracted by the echelle will always be at the peak of the blaze function and sent towards the center of the CCD detector. Along with the silver-coated mirrors, this configuration is the major player for achieving the high spectrograph efficiency (see 3.4).
The γ angle values depend on the resolution supplied by the crank, which provides a continuous range of reference values between 0000 and 7298; hence,γ resolution is as
good as 1.37×10−3
degrees.
(a)
(b) (c)
Figure 2.14: Mechanical linkage between the external crank, the echelle grating and the adjustable mirror M3. a) QLM joined to the control external crank, through a lead screw. b) Detail of the support
for the Echelle mounting and M3. It is observed the turntables for both optical elements and the joining bar that moves them at the same time. c) Detail of the crank showing a numerical value correlated with theγ value angle. Number 5260 is equivalent to
Bausch & Lomb (now Richardson Gratings, Newport Corporation) supplied the echelle grating, ruled at 79.01 l mm−1
and intended to be blazed at 76◦
(“R4”; tan δ = 4). Hunten et al. (1991) explains that the measurements made by Brown et al. (1982) show that the real blaze angleδ is 72◦.5 (“R3.2”), which implies a loss of 20 % in the angular
33
dispersion and the resolution-throughput product, because such blaze angle does not match the incident and diffracted angles under QLM (the above cited 76◦ value, see
??); however, our measurements, performed directly on the spectrograph, show that
the grating has been tilted with respect to they-z plane of the spectrograph by 72◦.5,
offsetting the error in the grating blaze angle.
The grating is trimmed at two corners to allow clearance to the light into the optical path between the flat mirror M1 and the collimator M2. Because of the quasi-Littrow
configuration, the grating is mounted onto a cell that leaves a nearly ellipsoidal entrance for the light with major and minor axes of 8′′.50 and 3
.
′′00, respectively (Fig. 2.15a).
The complete set of the echelle grating and the cell can be removed from its support (Figs. 2.14) for its eventual replacement by another dispersing element (e.g. anechelette
dispersion grating or a back-reflecting prism), in order to get lower dispersions. The configuration shown in Fig. 2.13c, where γ = 8◦ (and M
3 is as far as possible from
the grating mounting), closely approaches the Littrow condition. In an echelette the
grooves must be perpendicular to they-zspectrograph plane, or to the diffraction edge
of a prism, if this is the dispersor.
Finally, M3 is a rectangular flat mirror of 6′′.00 × 2′′.80, enclosed by a rectangular
cell with a frontal frame of 0′′.
53 by side (Fig. 2.15b). Note that the M3 major axis is
rotated 90◦ with respect to the Echelle grating, as an outcome of the conical diffraction,
result of the γ angle different from zero. The M4 camera mirror follows this rotated
configuration.
(a) (b)
Figure 2.15: a) Front face of the echelle grating, showing its trimmed corners and the off-angleγ. At the image,γvalue is equal to 18◦. This angle can be appreciated observing M
3, which is placed at the
left edge of the lead screw, corresponding to 18◦, in accord the scheme shown in Fig. 2.13a. b) Detail
of the M3 mirror. Notice that M3 major axis is 90◦rotated respect the echelle grating.
(a)
(b)
Figure 2.16: Opto-mechanical CanHiS simulation, in which we have made independent analysis, using Zemax for the optical, and Solidworks for the mechanical. As a last step, we have joined the results and generate the simulation shown.
Chapter 3
Optical Performance Analysis
The configuration spectrograph was originally dictated by the necessity to face planetary astronomy challenges, which require very high efficiency, high spatial resolution along the spectrograph entrance slit and very high spectral resolution. Such requirements were all satisfied adopting the QLM and medium-band filters to select individual orders, instead of a cross-disperser system. The QLM offers several advantages over an Littrow-only arrangement, as will be described in the optical performance analysis in what follows. Some of the material presented in this section can be found in textbooks (e.g. Schroeder, 2000) and is included here for easy reference.
3.1
The Grating Equation
The analytical expression used in the modelling of a reflection grating, is expressed as the product of the dispersion order m and the diffracted wavelength λ, as given by equation 3.1:
mλ=σ(sinα+ sinβ), (3.1)
whereσ is the step of the grid, being the distance between the centers of equally spaced
grating rectangular grooves, or the width of equally spaced gratingblazedgrooves (also
called the grating constant). α and β are the incident and diffracted angles of the principal light ray, respectively, measured with respect to the normalN.
3.1.1 Angular Dispersion
One of the most important figures of merit in a dispersing element is its angular dis-persion, A =δβ/δλ, defined as the angular difference δβ between two diffracted rays
emerging from the disperser over the diffracted rays wavelength differenceδλ. Therefore, for a reflecting dispersion grating, the angular dispersion is obtained by differentiating
β with respect toλin Equation 3.1, maintaining α constant (Eq. 3.2):
A= dβ dλ =
m
σcosβ; (3.2a)
A= sinα+ sinβ
λcosβ . (3.2b)
3.1.2 Free Spectral Range (FSR)
For a specific diffraction grating, and according to Eq. 3.1, a particular wavelength
λ with incidence angle α can be diffracted in more than one angle β, whenever the
condition for constructive interference is to be satisfied (integer values ofm). Therefore,
in succesive orders, there are two wavelengthsλandλ′
, satisfyingmλ= (m+1)λ′
, where the spectral interval ∆λbetweenλ′ and
λis called theFree Spectral RangeFSR. It can
be shown that FSR can be expressed as (Eq. 3.3):
∆λ=λ′−λ= λ
m. (3.3)
3.2
The blaze function
By definition (Schroeder, 2000), the absolute efficiency of the normalized intensity of a diffracted wave by a grating of N equally spaced grooves, and with center to center
spacingσ, is the fraction of energy at a given wavelength incident on the grating, into
a certain diffracted order, fixed by the so-called blaze function BF. The normalized
intensity is provided by the product of a factor representing a multi-wave interference system —the interference function IF, times the blaze function BF— which represents a single-slit diffraction enveloping function (Eq. 3.4; Schroeder, 2000):
i(α, β) = IF·BF =
sinN ν′ Nsinν′
2sinν
ν 2
, (3.4)
where both factors have a maximum value of unity, and 2ν′
is the phase difference between the centers of adjacent grooves, and ν is the phase difference between the
center and edge of one groove, that depends on the groove physical shape.
If the grating is composed by rectangular, equally spaced grooves of width b, the
phase difference in the interference function takes the form 2ν′
= 2πσ
λ (sinβ+ sinα);
under this configuration, there is a maximum whenν′ =
mπ, and the phase difference
leads to the grating equation. From the blaze function, ν= πb
λ(sinβ+ sinα) and has a
maximum whenν= 0; therefore,α=−βand the enveloping blaze function achieved will be simmetrical, with the maximum of energy at the center of the interference function, precisely at the specular reflection (m = 0).
37
For a grating with grooves of widthb, tilted by an angleδwith respect to the surface,
and having right-angle corners (Fig. 3.1), the phase difference from the center of the groove to the edge is (Eq. 3.5; Schroeder, 2000):
Figure 3.1: Modified image from Schroeder (2000), where can be observed a blazed grating at an angleδ. The clockwise angle between a facet normal and the incident rayαis calledθ, and the counterclockwise angle with the diffracted ray β is called θ′. Incident and diffracted rays can be expressed then as
α=δ+θ andβ=δ−θ′, respectively.
ν = πσcosδ
λ (sinθ−sinθ ′
);
= πσcosδ
λ [sin(β−δ) + sin(α−δ)],
(3.5)
where the angular positions of the nonzero orders are determined (in addition toσ and
δ), by the incident angleα, which, along withβ, is measured with respect to the grating normal, not with respect to the faces of the grooves. Nevertheless, since the peak of the blaze is obtained when α = β, which corresponds to specular reflection for each
groove face (as is the case for purely rectangular grooves, whenν = 0), the enveloping blaze function moves until its peak matches an interference maximum from the IF, in a dispersed order different fromm = 0. From Eq. 3.5,α+β= 2δ at the maximum of
the BF (ν = 0). The clockwise angle between a facet normal and the incident rayα is calledθ, and the counterclockwise angle with the diffracted ray β is calledθ′ (Fig. 3.1);
hence,α =δ+θ and β =δ−θ′. With fixed values of
α and β, the peak of the blaze
can only be accomplished for a single λvalue, called the blaze wavelength λb. At this blaze wavelength, the grating equation 3.1 can be expressed as (Eq. 3.6):
3.3
The echelle grating
An echelle is a diffraction grating with some fundamental differences with respect to the common gratings —called echelette gratings, in general—, providing the possibility to
get very high angular dispersions (and therefore very high resolving power,R ≥50 000),
using high dispersion orders at large angles of incidence and diffraction, as can be inferred from Equation 3.2. The critical parameter to achieve this performance is the high blaze angle δ value, tipically of 63◦.5 or 76
.
◦0. Since high dispersion depends on
large angles of incidence and diffraction (and high dispersion orders), and not on the density grating of grooves per mm, echelle gratings have low groove densities (between 31 and 300 l mm−1). Also, echelles must be used at high orders, typically among 10 and
100. The blaze angle values are conventionally used for describing the echelle gratings, expressed as aRvalue, whereR= tanδ. For the aforementioned usual values,δ = 63◦.5
describes anR-2 grating echelle, andδ = 76◦.0, an
R-4 echelle.
It is evident that, for ablazed grating—a grating with a blaze angleδin its grooves,
the peak of the blaze function occurs whenθ=θ′, which can be achieved with normal
incidence on the grating (α = 0◦
), or at the so-called Littrow configuration, when α=β =δ, and θ=θ′ = 0◦ (so
θ=θ′ is simply expressed as θ).
Echelle grating designs are particularly useful for Littrow mounting. Figure 3.2 shows ax-y-z coordinate system defining the orientation of an echelle at this
configu-ration, whereαand β are, respectively, the incidence and the diffraction angles for the principal ray, and δ the echelle blaze angle. Grating normal is labeled as N in Figure
3.2, whereas the normal to the facet grooves is aligned with zaxis. Note that the
inci-dent light is diffracted by the facets, unlike as an echelette (as in Figure 3.1), allowing the use of large angles of incidence and diffraction. In practice, the angular difference between the incident and the diffracted anglesθ cannot be equal to zero in a reflection
grating because the incident light would return on itself.
Figure 3.2: Coordinates system for an echelle grating in a Littrow mounting, whereα=β=δ, andθ≈ 0. (Modified image from Chaffee & Schroeder, 1976).
Another advantage of the echelle gratings is its blaze function shape, which takes a symmetrical profile whenm≫1, unlike the BF shape in echelettes, which is
39
from the blaze peak (at λb), a modified grating equation (Eq. 3.1) can be used for
modelling all the wavelengths diffracted by a reflection grating. Therefore, the equation grating for a pair of wavelengthsλ+ and λ−, separated by an angular distance of ε±,
respectively, fromλb, is (Eq. 3.7; Schroeder, 2000):
mλ±=σ[sinα+ sin(βb±ε±)], (3.7)
whereβbdenotes the peak of the BF operating on the diffracted beam, andε±the angles
at which the BF reachs a value of 0.405 (at phase difference ν =±π/2). The angular
width between these values is λ/b, and for grooves with shapes as those in Figure 3.1, ε±will beλ±/2σcosδ. Assuming thatε±is small, and cosβb ≈cosδ(Schroeder, 2000),
then we can derive from Equation 3.7 the wavelength limits for which the BF drops to a value of ≈ 40 %. For large m, as in an Echelle grating, this values are (Eq. 3.8;
Schroeder, 2000):
λ+−λ− = λb
m. (3.8)
Equation 3.8 is similar to that of the FSR (Eq. 3.3); since the BF, at the wavelength limits, has a value of 0.405, it means that all the wavelengths into a BF curve cover a FSR section that is above of the 40 % of the blaze peak value, for each orderm.
3.4
The Quasi-Littrow Mounting (QLM)
3.4.1 The echelle grating in QLM
QLM can be considered a slightly modified Littrow mounting case, where the light reaching the grating has an incident angle with respect to the plane y-z —an off-axis
angle—, calledγ, considering the configuration presented in Figure 3.3. Due to thisγ
angle, the collimated light arriving at the grating sees a shortened groove by a factor equal to cosγ, so that the grating equation 3.1 is multiplied by this factor, because the
groovet side, at Figure 3.2, is nowtcosγ (Fig. 3.3b, Eq. 3.9):
mλ=σcosγ(sinα+ sinβ). (3.9)
Moreover, because of in QLM θ = 0, and therefore α = β = δ, Equation 3.9 is simplified to:
mλ= 2σsinδcosγ. (3.10)
In Figure 3.3a,α and β angles are measured with respect to thex-zplane, showing the off-axisγ angle, whereas the blaze angleδis still lie on they-zplane. Because of the γ angle, in QLM the incident and the diffracted principal rays follow different paths,