The échelle spectrograph SOFIN

Introduction

The high resolution échelle spectrograph SOFIN was designed and manufactured at the Crimean Astrophysical Observatory in collaboration with the Observatory of Helsinki University and installed at the Cassegrain focus of the 2.56m Nordic Optical Telescope, La Palma, Canary Islands. The observations with the instrument started in June 1991 (Tuominen 1992). The spectrograph is a unique instrument in its class and is one of the few very high resolution spectrographs in the northern hemisphere. The spectrograph was designed to allow stellar spectroscopy with three different spectral resolutions R = $\lambda$/$\Delta$$\lambda$ = 30 000, 80000, and 170000, depending on the brightness of the star and the scientific goals. The resolution is altered by changing one of the three different optical cameras whilst all other optical elements of the spectrograph remain unchanged. The spectrograph is equipped with a cross-dispersion prism to separate spectral orders so that many different wavelengths are recorded in a single CCD exposure. The higher the spectral resolution the smaller the part of the spectral range which can be covered by the CCD. The change of the spectral setting is done by turning the échelle grating and the cross-dispersion prism.

The spectrograph is completely remotely controlled from the host computer, except for a few minor functions. Such a design provides a high operational efficiency during the observations. The spectrograph control software is currently running under MS-DOS and is a menu-driven program with a range of services to assist the observations (Ilyin 1996). It includes a database of observations where the data records are stored, which can be retrieved and displayed later, a quick-look facility for an express analysis of the data obtained, the databases of observed objects and spectral regions to hold all information specific for the particular observation and the setup of the spectrograph. The data reduction facility is the second large part of the software developed for the spectrograph. The key points of the data reduction package are the simplicity and the high efficiency which in conjunction with the user-friendly interface provides an easy way to operate with the enormous amount and complex nature of the data.

Scientific applications

The long focus optical camera provides the highest spectral resolving power R = 180 000 and is used to resolve the fine structures of spectral lines and for accurate measurements of their positions. For instance, the interstellar NaID 5890Å is formed on the line of sight to some background stellar object and reflects the motions of interstellar clouds. The higher the resolution, the more components can be resolved, i.e. the smaller velocity differences between different clouds can be detected. Another astrophysical aspect is to use the high resolution camera to measure the isotopic shift and hyperfine structure of the spectral lines of chemically peculiar Ap stars and of cool stars (e.g. Wahlgren et al. 1999). In the case of the Ap stars, the isotopic ratio gives an estimate of the diffusion rate of the isotopes from the enriched regions of their origin to the upper atmosphere. The very important LiI6707Å line is a key element for studying the history of the universe, nucleosynthethis, stellar interiors, and stellar evolution. These measurements require the accurate observation of the position and intensity of fully resolved spectral lines with the subsequent accurate analysis and comparison with a synthetic spectrum. The high resolution camera allows to increase the accuracy of radial velocity measurements, since the error of the spectral line position determination decreases as the resolution grows. This also gives the possibility to investigate the instability and variations of the radial velocities of F, G, and K-type stars, which in the past were supposed to be very stable and were used as radial velocity standards. For some stars, these variations, seen as tiny Doppler shifts of the spectral line positions, are thought to be due to a low mass companion, like a brown dwarf or a large Jupiter-like planet, perturbing the position of the main star along the line of sight, although the variations may be caused as well by non-radial pulsations of a single star. To decide which explanation is correct for a particular star, accurate observations at the highest resolution are needed, allowing to study the line profile shapes in detail.

The second optical camera provides half of the resolving power of the previous one and corresponds to about 80000 near the échelle blaze angle. Reducing the resolution two times doubles the amount of light per resolution element, as well as allows for a doubling of the entrance slit width which reduces the light losses, and, therefore, fainter objects can be observed. The second optical camera is widely used for Doppler imaging of late type stars. These exhibit small bumps in a spectral line which drifts over the profile as the star rotates. The solution of the inverse problem, using models of stellar atmospheres, yields the geometrical distribution of the spots and a temperature map of the stellar surface (e.g. Bergyugina 1998). The medium resolution optical camera is used for the direct measurements of the stellar magnetic field with the Stokesmeter installed in front of the spectrograph entrance slit. The Stokesmeter divides the stellar light into two beams with opposite circular polarization. In the presence of a stellar magnetic field the spectral lines in these two beams are shifted with respect to each other and the stronger the field, the larger the offset. Measurements of the Zeeman splitting of a number of spectral lines gives the effective magnetic field of the star as a function of its rotational phase. Solving the inverse problem, using models of stellar atmospheres and taking into account the effect of Zeeman splitting, gives the distribution of the magnetic field strength over the stellar surface. Another aspect of the second camera applications is the observations of the non-radial pulsations of Ap stars on the short time scale of their variability with simultaneous recording of many spectral features in one exposure. Lines of certain chemical elements exhibit small variations of their radial velocities which can be attributed to the surface inhomogeneities. On the other hand, the amplitude of the oscillations derived from broad band photometry strongly decays from the blue to the red which is attributed to the wavelength dependence of limb-darkening and the steepness of the temperature gradient with respect to the optical thickness.

The low resolution camera provides a spectral resolution around 30000 which is typical for most other stellar spectrographs. Since there is virtually no light loss on the wide entrance slit and the spectral resolution is low, the camera is used for observations of objects as faint as 15^m with a 2.5m telescope. The camera allows to record in one exposure half of the whole optical spectrum ranging from the blue spectral orders at 3500Å to the red at 11000Å. Such curious objects as T Tauri and FU Orionis stars (e.g. Petrov et al. 1999), chromospherically active late type binary systems, red dwarfs, and cataclysmic variables are observable with the camera with most of the hydrogen lines, ultraviolet CaIIH&K, the infrared CaII triplet seen in the same échelle image which offers the possibility to analyze spectral lines originating in different layers of the star and the surrounding environment.

Design of the spectrograph

The optical design of the SOFIN spectrograph is similar to that of the échelle spectrograph for the CTIO 4m Blanco Telescope. The complete spectrograph design is described in four papers: the optics (Pronik 1995), the mechanics (Lagutin 1995), the electronics (Bukach & Zlotnikov 1995), and the software (Ilyin 1995, including results of the first observations).

The spectrograph is mounted on the Cassegrain rotating adapter of the alt-azimuth Nordic Optical Telescope (NOT). The diameters of the main and secondary mirror of the telescope are 2560 and 510mm. The effective focal length is 28160mm, which makes the scale in the focal plane 137$\mu$m/ 1$^{$\prime{^\prime}$}$.

The schematics of the optics is given in Fig.1.1 and the layout of the main components are shown in Figs.1.2 and 1.3. All components of the spectrograph are assembled in a rigid welded construction which has the dimension of 1800×800×800 mm³; the total weight without the CCD dewars is 240kg. The components of the spectrograph are described in the following.

Figure 1.1: The optical layout of the SOFIN spectrograph. The converging f /11 beam focussed onto the slit is redirected by the flat mirror to the collimator. The collimated beam illuminates the échelle grating which renders the light dispersed in wavelength to the cross-dispersion prism to separate the spectral orders in the direction perpendicular to the dispersion. The two flipped positions of the cross-dispersion prism are shown by the solid and dashed lines. Depending on the position, the beam is directed to one of the optical cameras mounted in the spectrograph. The échelle image is formed on the CCD attached to the optical camera.

Figure 1.2: The mechanical layout of the SOFIN spectrograph: the upper unit (1) is shown in details in Fig.1.3, flat mirror (2) with the adjustment mechanism, collimator (3), échelle grating (4), cross-dispersion prism (5) with the handle to flip the prism, two optical cameras (6 and 7) installed, CCD cameras (8), assembly frame (9), and power supply with the stepping motor amplifiers (10).

Figure 1.3: The mechanical layout of the upper unit: slit (1), filter turret and the shutter (2), pentaprism rotator (3), intensified guiding TV camera (4), flat field lamp (5), comparison spectrum ThAr lamp (6), an auxiliary viewing pupil (7), Zeeman analyzer (8), stray light absorber (9), and the assembly unit (10).

The entrance slit (1) in Fig.1.3 is situated at the distance 210mm below the attachment flange. The mirrored slit plane is tilted with respect to the optical axis by 13.5^o in order to be viewed by the intensified TV guiding camera. The slit width is remotely controlled and depends on the spectral resolution being used, whilst the width of the decker can be changed manually and defines the height of spectral orders and the interorder spacing.

The filter and shutter unit (2) in Fig.1.3 is mounted below the entrance slit in one assembly unit. A hollow cylinder with two radial slots attached to the axis of a stepping motor constitutes the shutter. The turret with the filter wheel attached to a stepping motor contains eight holdings for filters which have 16mm in diameter and 6mm in thickness (the effective focal length of the collimator depends on the filter thickness). The filters reduce the amount of scattered light diffused on the spectrograph optical frames and surfaces, and are selected according to the spectral region being used. The bandpasses are given in the following table together with their numbers and positions on the turret. Some of the filters are composed of two pieces to keep the thickness constant: a metal ring spacer between the pieces eliminates possible optical interference on their surfaces.

0 0 Closed

1 7 Open

2 6 3000 - 25000 Neutral double

3 5 8500 - 25000 Infrared double

4 4 7000 - 25000 Red single

5 3 6000 - 25000 Orange single

6 2 3500 - 6000 Blue double

7 1 3500 - 7500 Transparent double

The beam switcher (3) in Fig.1.3 is used to render the light from the two calibration sources to the slit. It consists of a pentaprism attached to the stepping motor axis via a rotating arm. In the position shown with the solid line, the slit is unobscured for the light from the telescope, the position shown with the dashed line gives the light from the flat field lamp.

The calibration sources include a tungsten flat field lamp (5) in Fig.1.3, and a ThAr hollow cathode spectral lamp (6) manufactured by S. & J. Juniper & Co., UK. The pupils of the lamps are collimated at f /11.

The intensified CCD TV guiding camera (4) in Fig.1.3 is used for setting, focussing, and guiding the stellar image on the slit. The field of view on the TV screen is 70$^{$\prime{^\prime}$}$. With the maximum intensification a star as faint as 16^m can be seen. For bright objects a gray filter with an attenuation of 100 is put in front of the TV camera. The shutter, gray filter, and open diaphragm are installed on a remotely controlled linear shaft.

The analyzer of circular polarization (Stokesmeter) (8) in Fig.1.3 is mounted on a platform above the slit and can be positioned onto the optical axis when used for spectropolarimetry. The design of the Stokesmeter is described by Plachinda & Tarasova (1999) and is similar to that of Donati & Semel (1990). It consists of an achromatic (4000-6800Å) turnable quarter-wave plate, a beam splitter, made of a plate of Iceland spar, and a fixed achromatic quarter-wave plate on exit which converts the linearly polarized light into circularly polarized light to avoid linear polarized light attenuation on the échelle grating. The angle of the turnable quarter-wave plate on the entrance is controlled by a stepping motor; four exposures with the plate subsequently turned by 22.5^o allows to measure all four Stokes parameters. The image separation is 3$^{$\prime{^\prime}$}$ on the slit.

The diagonal flat mirror (2) in Fig.1.2 is located 300mm below the entrance slit and turns the optical axis by 98^oThe projected diameter is 32mm.

The parabolic collimator mirror (3) in Fig.1.2 has a diameter of 128mm and a focal length of 1396mm. The effective focal length of the collimator coupled with the filters is 1400mm. The mirror has been aligned once during assembly and any temperature changes of its focal length are compensated by the focussing of the optical cameras.

The photon counter (not shown but located between 2 and 3 in Fig.1.2) is used to estimate the amount of light passed through the slit of the spectrograph and consists of a small prism installed in the collimated beam and a photomultiplier. The photon count rate and accumulated sum during the object exposure are displayed in real-time in order to suggest the exposure time in case of observing in modest weather conditions.

The R2 échelle (4) in Fig.1.2 (Milton Roy Co., USA), has a grooved area of 128×256 mm² and is ruled with 79groovesmm^-1. The blaze angle is 63$.^o$435 (arctan 2). The incident and diffracted beams are separated by a fixed angle of 8^o and the angles are coplanar with the échelle normal. The échelle tilt mechanism changes the angle of incidence by turning the frame around ball-edged pivots within ±3^o by driving the tangent arm attached to the axis of the remotely controlled stepping motor. The motor step size is about one pixel on the CCD for the long camera. The working spectral orders are 20-65 (11300 - 3500Å).

The double-pass cross-dispersion prism (5) in Fig.1.2 is made of BK7 glass and mounted 800mm apart from the échelle. The prism apex angle is 17$.^o$0212 which makes the interorder spacing 75% of the order height at 5650Å in the central 40th order. The positioning mechanism of the prism is similar to that of the échelle and allows to change the spectral setting in the cross-dispersion direction. The motor step size is about one pixel on the CCD for the long camera. The prism assembly is mounted on a mechanism which allows to turn the prism around the optical axis to redirect the refracted beam to one of the two optical cameras installed simultaneously in the spectrograph.

Three optical cameras provide three different spectral resolutions; two of them (6 and 7 in Fig.1.2) can be installed in the optical ports at the same time and the resolution is altered by flipping the cross-dispersion prism. The short and long cameras are interchangeable, the medium camera is mounted permanently.

1. The long optical camera is a Cassegrain mirror system with an effective focal length of 2079mm and provides a resolving power ranging from 150000 to 185000 which depends on the échelle deflection angle. The entrance slit width projected on two CCD pixels is 38$\mu$m ( 0$.^{$\prime{^\prime}$}$28 on the sky) at the blaze angle. The light loss on the slit with a seeing of 1$^{$\prime{^\prime}$}$ is 75%. The length of the spectral orders is about 20Å around 5500Å which corresponds to a pixel size of about 1000ms^-1 in radial velocities. About 15 adjacent spectral orders can be covered in one CCD image. The vignetting in the image centre is about 7% and almost homogeneous along the image.

2. The medium optical camera is a Ritchey-Chrétien mirror system with an effective focal length of 1000mm with a spectral resolution ranging from 70000 to 86000. The entrance slit width projected on two CCD pixels is 82$\mu$m ( 0$.^{$\prime{^\prime}$}$6) at the blaze angle. The light loss on the slit is about 50% if the seeing is 1$^{$\prime{^\prime}$}$. The length of the spectral orders is about 40Å around 5500Å. The CCD format allows to record 12 such orders in one exposure. The pixel size corresponds to about 1900ms^-1 in radial velocities. The vignetting in the image centre is about 16% and increases towards the edges (68%).

3. The short optical camera is a meniscus system (two menisci and two mirrors) has an effective focal length of 348mm and provides a spectral resolution from 25000 to 30000 with the entrance slit width of 236$\mu$m ( 1$.^{$\prime{^\prime}$}$73) projected on two CCD pixels. There is virtually no light loss on the slit since it is wider than the average seeing. The light loss occurs mostly on the slit decker which is reduced for the camera as compared to the others for better order separation in cross-dispersion. The spectral format of the camera allows to record all spectral orders in one image from 20 to 67 with the length of one order being around 120Å at 5500Å. Two such exposures are necessary to get the complete overlap and full wavelength coverage of all spectral orders. The pixel size corresponds to 5700ms^-1 in radial velocities. The vignetting of the camera is 22% in the image centre and 92% at the edges is due to the large secondary mirror.

The parameters of the first two optical cameras are given in the following table (units are mm); the optical diagram of the third camera is given in the original paper of Pronik (1995):

$\renewedcommand{arraystretch}{1.2}\arraycolsep=1.5mm\begin{array}{lrrrrrrrrr} ... ... -937.50 & -635.30 & 1.247 & 12.2233 & 160 & 60 & -300 & 356.00 \\ \end{array}$

where f_cam is the focal length, R is the radius of the mirror curvature, e² is the eccentricity, D is the diameter of the mirror, d₁ is the distance between the mirrors, and d₂ is the distance between the focal plane and the secondary mirror.

Each camera is equipped with a focussing mechanism which is a turnable threaded ring to offset the image formed on the CCD along the optical axis whilst the positions of the optical elements of the camera remain unchanged. The CCD camera is attached to the optical camera via a bayonet adapter which preserves the CCD adjustment after re-installation of the optical and CCD cameras. The bayonet connectors are equipped with adjustment screws to align the CCD pixels with respect to the cross-dispersion direction.

Two similar Astromed-3200 CCD cameras (8) in Fig.1.2 make use of two UV-coated EEV CCDs: P88100 ( 1152×298 pixels) and P88200 ( 1152×770 pixels) which are housed in liquid nitrogen cooled dewars and operated at a temperature of 150K. The pixel size is 22.5×22.5$\mu$m². The dewar window is made of Spectrosil B fused silica, 50mm in diameter and 2mm in thickness. The first one, the larger format CCD camera, is used with the long and short cameras, the second one, the smaller format CCD, is used with the medium resolution camera.

Figure 1.4: An échelle image of the RS CVn star IM Peg obtained with the medium camera; R = 80 000 in the 6173 & 6563Å spectral region. The spectral orders located during data reduction are shown. The wavelengths of the spectral orders increase from bottom to top. The O₂ atmospheric bands at 7600Å are clearly seen in the 5th order from the top.

Figure 1.5: A raw échelle image of $\alpha$ Tau obtained with the long optical camera; R = 160 000 in the 6427 & 7516Å spectral region. The wavelengths of the spectral orders increase from top to bottom.

Figure 1.6: A raw échelle image of the magnetic Ap star $\alpha^{2}_{}$ CVn recorded with the Zeeman analyzer with the medium optical camera R = 80 000 around 4500Å. The doubled spectral orders are the left and right polarized beams.

Figure 1.7: The reduced spectra of $\alpha$ Cas (V=2.20, K0IIIa) with resolution R = 160 000 in the 6427 & 7516Å spectral region.

Figure 1.8: Echelle images taken with the low resolution R = 30 000 optical camera. The horizontal curves indicate the position of the spectral orders found during data reduction. Top: preprocessed image with the scattered light surface subtracted prior to the weighted extraction of the spectral orders of the 20 min exposure of the FU Ori star V1057 Cyg (V=10) which gave a signal-to-noise ratio around 100 in the continuum at H$\alpha$. Bottom: The corresponding ThAr comparison spectrum image taken in the same spectral region. The bright Thorium lines in the red are overexposed to allow a large number of fainter lines to be seen and used for the calibration. The saturated lines produce vertical white ``tails'' areas, i.e. of depressed bias level.

Efficiency of the spectrograph

The overall efficiency of the system including telescope and spectrograph is defined as the fraction of the photons entering the main mirror which is detected by the CCD.

The direct measurement of the efficiency

Measurements of the spectrograph efficiency are usually done by observing stars with known energy distributions at different wavelengths, e.g. the set of bright secondary standard stars for flux calibration given in Taylor (1984) and Hamuy et al. (1992). Instead of the standard method, we used the programme observations of two G-type stars obtained during different observing runs at a range of zenith distances and meteorological conditions to estimate the efficiency in the red.

The central CCD order was used where the vignetting effect is minimal. The width of the cross-dispersion profile was used as the estimate of the seeing condition during the exposure. Assuming that the seeing profile is a Gaussian, the relative transmittance of the slit is given by

$$T={\rm erf}\left(\frac{w\,\sqrt{\ln 2}}{{\rm FWHM}}\right),$$ (1.1)

where the seeing profile width is given by FWHM, and w is the slit width. The transmittance is plotted as a function of the width ratio in Fig.1.9. The measured signal-to-noise ratio (SNR) at the selected order, corrected for the slit effect, and reduced to the same exposure time was plotted in logarithmic units with respect to the airmass of the observations. The upper envelope of the points was used for linear extrapolation to the zenith.

Figure 1.9: Throughput of the slit illuminated by a Gaussian seeing image. The horizontal axis is the ratio of the FWHM of the Gaussian seeing profile and the slit width.

58 observations of 31Aql (V=5.20, G7IV) were made with the long camera with the slit width 37$\mu$m ( 0$.^{$\prime{^\prime}$}$27) which gives the resolution R = 160 000 ( $\Delta$$\lambda$=0.020Å per pixel) in the central order 35 at 6440Å. Extrapolation to the zenith estimates SNR to be 470 in 10 min of exposure time without the light loss on the slit. The flux outside of the Earth atmosphere expected for the star is (interpolated from Allen 1976, p. 207). The monochromatic Earth atmosphere extinction at 6440Å is 0.057 per airmass (the extinction coefficients for the La Palma Observatory were provided by the Carlsberg Meridian Telescope). For the given diameters of the telescope mirrors (256 and 51 cm respectively), the expected SNR of the star at zenith is 2260. Hence, the efficiency for the long camera is estimated to be 4.3%.

92 observations of HD199178 (V=7.24 G2III) were made with the medium camera with the slit width 81$\mu$m ( 0$.^{$\prime{^\prime}$}$60) which provides the resolution R = 76 000 ( $\Delta$$\lambda$=0.042Å per pixel) in the central order 32 at 7000Å. Extrapolation to the zenith estimates SNR to be 300 in 10 min of exposure time without the light losses on the slit. The extra-atmosphere flux for the star is about . The atmospheric extinction at 7000Å is 0.033 per airmass, hence the expected SNR of the star at zenith is 1220 which gives the efficiency for the medium camera of about 6%.

The expected efficiency

The efficiency can also be estimated by calculating the transmissions of all optical surfaces of the spectrograph. The results of these detailed calculations are given in Tab.1.1. The telescope reflectance is given by direct measurements after the cleaning of the mirrors. The reflectance on a surface at normal incidence is calculated according to the Fresnel formula:

$$r=\left(\frac{n-1}{n+1}\right)^2,$$ (1.2)

where the refractive index n is 1.51 at 7000Å for BK7 glass. The internal transmittance of BK7 is 0.99988 per mm (Optics Guide, Melles Griot Co. 1988). The échelle reflectance was provided by its specification. The reflectance of all SOFIN mirrored surfaces aluminized in 1990 is estimated to be 0.8.

Table 1.1: The transmission table of the optical elements of the SOFIN spectrograph. The total transmission coefficients for each subsection are given in the rightmost column.

secondary mirror reflection 0.782 0.60

single pass, 3mm 0.99963

surface 0.96 0.92

single pass, 3mm 0.99963

surface 0.96 0.92

Collimator reflection 0.8

Echelle reflection 0.67

vignetting 0.87 0.37

first pass, 39mm 0.9953

reflection 0.8

second pass, 39mm 0.9953

surface 0.96 0.73

secondary mirror reflection 0.8 0.64

single pass, 3mm 0.99963

surface 0.96 0.92

The overall efficiency of the spectrograph and CCD is 6.8%. By including the telescope, the efficiency is reduced to 4%, which is in good agreement with the direct measurements.

For comparison, the coudé échelle spectrometer at the 2.7m telescope of the McDonald Observatory has an estimated efficiency of the spectrograph of 16%, mainly because of the silver coated surfaces (97.5% reflectance), and the higher CCD efficiency (77%).

A similar design échelle spectrograph at the 4m telescope of CTIO has an efficiency ranging from 4-8% depending on the optical system setup.

Model of the spectrograph

Present day CCDs which are used in the spectrograph are too small to cover the whole spectral range of the échelle image. Hence, the observations are carried out in selected settings of the échelle and prism angles with limited spectral coverage. It is a matter of importance to have an appropriate means to select the configuration of spectral lines (Fig.1.10) and the pointing model of the spectrograph which transforms the coordinates of the spectral setting into the instrumental units of échelle and prism.

Figure 1.10: The model of the focal plane of the spectrograph as seen on the computer screen: the red orders are at the bottom and the blue orders are at the top. The blaze axis is at the centre, the two white curves at 41% of the échelle efficiency indicate the interval of the full wavelength overlap. The CCD box and the wavelength coverage correspond to the medium optical camera in 6173 & 6563Å spectral region. The positions of some hydrogen and Ca lines are marked with crosses and boxes.

Spectral mosaic

The échelle grating equation is

$$\sin\alpha+\sin\beta=N\,k\,\lambda$$ (1.3)

with

$$\alpha=\theta+\gamma+\epsilon \quad\mbox{and}\quad \beta=\theta-\gamma+\epsilon+\delta,$$ (1.4)

where $\alpha$ and $\beta$ are the angles of incidence and diffraction, $\gamma$ is the fixed angle of 4^o (2$\gamma$ is the angular separation of collimator and prism), $\theta$ is the blaze angle, $\epsilon$ is the échelle deflection angle with respect to the blaze, $\delta$ is the field angle varying around the image centre in dispersion direction, N is the number of grooves per mm, k is the spectral order number, and $\lambda$ is the wavelength. The incidence and diffraction angles are in the same plane for this spectrograph (various configurations of échelle spectrographs were extensively discussed in Schroeder & Hilliard 1980).

The cross-dispersion prism equation describes the fact that the monochromatic ray crosses the surface two times at different angles and becomes internally reflected on the rear mirrored surface:

$$\arcsin\left(\frac{\sin\alpha_p}{n}\right)+\arcsin\left(\frac{\sin\beta_p}{n}\right)=2\,\theta_p,$$ (1.5)

where $\alpha_{p}^{}$ and $\beta_{p}^{}$ are the angles of incidence and refraction, $\theta_{p}^{}$ is the prism apex angle. The refraction angle is given by the following sum:

$$\beta_p=\alpha_p+\gamma_p+\delta_p,$$ (1.6)

where $\gamma_{p}^{}$ = 30^o is the fixed angular separation between the échelle and the optical camera, and $\delta_{p}^{}$ is the field angle varying around the image centre in cross-dispersion direction.

The refractive index n of BK7 is approximated by

$$n^2=A_1+A_2\lambda^2+A_3\lambda^{-2}+A_4\lambda^{-4}+A_5\lambda^{-6}+A_6\lambda^{-8}$$ (1.7)

with the wavelength $\lambda$ given in $\mu$m, and the polynomial coefficients (Pronik 1995)

$$\arraycolsep=5mm\begin{array}{ll} A_1=2.2699804 & A_2=-9.825... ...0\times 10^{-5} & A_6=5.81309000\times 10^{-7}. \\ \end{array}$$

The image formed in the focal plane is calculated in the coordinate system ($\epsilon$,$\beta_{p}^{}$) of the échelle deflection and the prism refraction angles as a function of $\epsilon$ and order number k:

$$\alpha_p(\lambda)=\alpha_p(\lambda(\epsilon,k))$$ (1.8)

in the image centre with the field angles $\delta$ = $\delta_{p}^{}$ = 0. A box in the focal plane which reflects the CCD format is calculated in the angular units according to the scaling factor of the selected optical camera. The CCD box can be positioned to the different angles of échelle $\epsilon$ and prism $\beta_{p}^{}$. The corresponding table of the wavelength coverage is calculated with the two angles fixed at the selected position and the field angles $\delta$ and $\delta_{p}^{}$ are varying within the CCD image:

$$\delta_p(\lambda)=\delta_p(\lambda(\delta,k)).$$ (1.9)

The échelle efficiency

The échelle grating has its maximum reflectance at the blaze angle which decreases as the angle changes. Also, as the incidence angle increases, the projected area of the grating is reduced which results in a beam area reduction. The total effect is calculated allowing to estimate the light collection efficiency at the given spectral setting.

The light reflectance with respect to the blaze intensity ( $\epsilon$ = 0) is

$$I_\lambda=\left(\frac{\sin u}{u}\right)^2, \quad\mbox{where}\quad u=\pi\left(k-\frac{\lambda_1}{\lambda}\right).$$ (1.10)

The quantity $\lambda_{1}^{}$ = 225 885.77Å is the wavelength in order k = 1 at the blaze angle ( N $\lambda_{1}^{}$ = 2 sin$\theta$ cos$\gamma$).

The transmittance of the light due to the change of the illumination area of the échelle by the collimated beam as a function of the grating angle is given by the ratio of the two areas S_ech/S_col:

$$\frac{1}{4}S_{\rm ech}=(\sin{2\rho}+2\rho)\,D^2_{\rm col}-\pi... ...quad \frac{1}{4}S_{\rm col}=\pi D^2_{\rm col}-\pi D^2_{\rm obs}$$ (1.11)

with

$$\rho$$ = $${\frac{{L_y}}{{D_{\rm col}}}}$$cos$$\alpha$$,

where L_y is the length of the longest échelle side (across the grooves), D_col is the diameter of collimator, and D_obs is the diameter of the diagonal flat mirror, which obscures the central part of the collimated beam. It is assumed that the collimated beam size equals the width of the échelle D_col = L_x. The transmittance decreases from 94% to 77% from the blue to the red side of the order; additionally, the resolution increases from blue to red.

The overall efficiency is shown in Fig.1.11 as obtained by the multiplication of the blaze function with the transmittance. The maximal efficiency at the blaze angle is about 86%.

The resolving power

The resolving power $R=\lambda/d\lambda$ is one of the basic parameters of the spectrograph and describes its ability to resolve narrow spectral lines. For the given focal length of the optical camera, the resolution is a function of the échelle angles and it changes along spectral orders. Qualitatively, it is obvious that with the increase of the angle of incidence, the projected density of the grooves becomes higher, which results in a larger dispersion.

The wavelength size d$\lambda$ of the resolution element formed by the slit projection onto the CCD is

$$N\,k\,d\lambda=\cos\beta\,d\beta, \quad\mbox{where}\quad d\beta=\frac{p}{f_{\rm cam}}$$ (1.12)

is the angular size of the resolution element given by the projected slit width p on the CCD and the focal length of the camera f_cam. Then the resolving power is

$$R=\frac{\lambda}{d\lambda}=\frac{\sin\alpha+\sin\beta}{\cos\beta}\cdot \frac{f_{\rm cam}}{p}$$ (1.13)

which is a function of the échelle deflection angle $\epsilon$, field angle $\delta$, and the projected slit width p (the entrance slit width also depends on the above two angles). For the CCD centre at $\delta$ = 0 it becomes

$$R=\frac{2\,\tan(\theta+\epsilon)}{1+\tan(\theta+\epsilon)\,\tan\gamma}\cdot \frac{f_{\rm cam}}{p}.$$ (1.14)

Figure 1.11: The échelle plus collimator efficiency is plotted with respect to the deflection angle in degrees. The uppermost curve corresponds to the red 20th order, and the innermost curve is for the blue 60th order. The resolving power is calculated for the three optical cameras for a resolution element of two pixels of the CCD.

The optimal slit width

The slit is called optimal when its image in the focal plane is optimally sampled by the CCD pixels. For the ordinary observing mode, the sampling interval is equal to one CCD pixel, which implies that the spatial cutoff (or Nyquist) frequency corresponds to two pixels on the CCD. The slit width which corresponds to an instrumental profile of two pixels FWHM is used for observations to provide the maximal resolving power for the given settings of the spectrograph.

The projected slit image is a convolution of three main profiles: the geometrical projection of the rectangular entrance slit, the aberration profile of the camera, and the diffractional profile of the monochromatic beam on the collimated pupil. To calculate the resulting width, it is assumed that the profiles are Gaussians, hence, their widths can be added in squares:

$$p^2=\left(\frac{f_{\rm cam}}{f_{\rm col}}\cdot \frac{\cos\al... ...ambda)+\left(\frac{\lambda}{D_{\rm col}}\,f_{\rm cam}\right)^2.$$ (1.15)

The last term describes the diffraction broadening of the slit image. The middle term is the camera aberration as a function of image position; for simplicity, the aberration spot size is set constant and equal to 8$\mu$m for all three optical cameras. The first term consists of the three factors: the magnification factor of the spectrograph, the angular magnification on the échelle, and the entrance slit width w. The magnification by the échelle follows from Eq.(1.3): $\cos\alpha\,d\alpha+\cos\beta\,d\beta=0$.

The essential setup parameters for the three optical cameras are given in the following table for the central 40th order at 5647Å at the blaze. The slit width w, given in $\mu$m, corresponds to two pixels on the CCD. The slit relative throughput T in percent is calculated for 1'' seeing. $\delta$$\lambda$ and $\delta$v are the pixel width in wavelength (mÅ) and radial velocities (ms^-1). $\Delta$$\lambda$ and $\Delta$k are the length in Å and the number of spectral orders in the CCD image.

$\renewedcommand{arraystretch}{1.3}\arraycolsep=2.2mm\begin{array}{\vert clrrrrr... ...} & 27\,000 & 236 & 1.73'' & 100 & 104 & 5550 & 120 & 40 \\ \hline\end{array}$

The pointing model of the spectrograph

The échelle and the cross-dispersion prism position angles are controlled by stepping motors. The pointing model establishes the relation between the step numbers and the position in wavelengths and order numbers for the centre of the CCD image and is used for the selection of spectral regions (Fig.1.10).

To obtain the reference points, a series of comparison spectrum images were made in a grid of 5×7 different positions in dispersion direction and across the orders for the three optical cameras. The images were processed to obtain the central order number and its wavelength. Bivariate polynomial linear least-squares fits were used to approximate the échelle deflection angle $\epsilon$ in Eq.(1.3) and the central order number k as functions of the step motor numbers x and y as follows

$$\epsilon(x,y)={\bf q'}(y){\bf E}\,{\bf p}(x) \quad\mbox{and}\quad k(x,y)={\bf q'}(y){\bf C}\,{\bf p}(x),$$ (1.16)

where $\bf E_{{{1}\times {2}}}^{}$ and $\bf C_{{{4}\times {1}}}^{}$ are the matrices of polynomial coefficients associated with the vectors of the Chebyshev polynomials $\bf q$(y) and $\bf p$(x) of the corresponding degrees. The polynomial coefficients were found sufficient for the SOFIN spectrograph to describe the surface with an appropriate accuracy, although, a more dense grid in the cross-dispersion direction would provide an accuracy similar to that achieved in the dispersion direction.

Stability of the spectrograph

As the alt-azimuth mounted telescope tracks the position of a star, the spectrograph mounted on the rotating adapter is changing its orientation which is described by the parallactic angle. The parallactic angle between the hour circle and the vertical circle (Woolard & Clemence 1966, p.55) is calculated from

$$\arraycolsep=1mm\renewcommand {\arraystretch}{1.2}\begin{arra... ...}=\sin\phi\,\cos\delta-\cos\phi\,\sin\delta\cos{h}, \end{array}$$ (1.17)

where q is the parallactic angle, z is the zenith distance, $\phi$ is the latitude of the telescope in the northern hemisphere, $\delta$ is the declination of the object, and h is its hour angle. Fig.1.12 shows how the rotator parallactic angle and the zenith distance of an object are changing in time. For a star with $\delta$ $\approx$ $\phi$ the rotator angle is changing very fast but the zenith distance is not. Apart from this declination, the slower change of the rotator angle is compensated by the near constancy of the zenith distance at the meridian. The most critical parts are on the east and west where both angles are changing very rapidly.

Figure 1.12: Change of the parallactic angle of the telescope rotator as function of hour angle (left) and zenith distance (right) at the latitude $\phi$ = 28.76^o of the NOT for stars with different declinations. The length of the curves is defined by the visibility of a star above the horizon. The time interval between two dots in the right panel is 10 min of hour angle.

Figure 1.13: The bottom view of the spectrograph mounted on the telescope rotator adapter. Clockwise rotation corresponds to negative angles, and anticlockwise rotation corresponds to positive angles of the rotator. The angle between the optical cameras is 60^o. When the rotator angle is -30^o the medium 2nd camera is oriented in the plane of the vertical circle of the telescope.

A change of the spectrograph orientation unavoidably results in a drift of spectral line positions, an effect known as the flexure of Cassegrain mounted spectrographs. Different optical components may change their positions in a different way as the spectrograph changes its orientation which makes the overall effect very complicated.

Figure 1.14: The line drift in two optical cameras as a function of the spectrograph spatial orientation. The shift in rows is along the dispersion direction and the shift in columns is in the cross-dispersion direction. The telescope rotator angle (R) is changing from -90^o to +90^o and the zenith distance (Z) is changing from 0^o to 70^o. The scale size for the medium camera plot is half that for the long camera plot.

To evaluate the amplitude and the behaviour of the line drifts, an experiment was made in December 1997 for the long 1st and medium 2nd optical cameras as follows. To ensure that the ambient temperature changes are minimal, the experiment was carried out during a cloudy night with the telescope dome closed and the air conditioning system switched on. The altitude of the telescope was gradually decreased from 90^o to 20^o; at each altitude the rotator was turned from -90^o to +90^o, and at each rotator angle an exposure of the comparison spectrum was made. Then the whole sequence was repeated for the other camera. The line displacements were measured with respect to a group of lines at the image centre of the very first exposure with the cross-correlation technique. The results of the measurements are shown in Fig.1.14, and the corresponding surfaces are shown in Fig.1.15 and Fig.1.16 which were approximated by a bivariate smoothing spline.

Figure 1.15: The surface fit shows the drift of a spectral line in the dispersion direction (upper panel), in the cross-dispersion direction (middle panel), and the change of the line FWHM as a function of zenith distance and rotator angle for the long 1st optical camera.

Figure 1.16: The surface fit shows the drift of a spectral line in the dispersion direction (upper panel), in the cross-dispersion direction (middle panel), and the change of the line FWHM as a function of zenith distance and rotator angle for the medium 2nd optical camera.

As was expected, the 1st camera traces show a twice larger amplitude due to the doubled focal length as compared to the other camera. At each trace when the altitude is fixed there is a turning point at which the drift in the direction of rows (the dispersion direction) is minimal and is at +30^o of the rotator angle for the 1st camera and -30^o for the 2nd. This is simply explained by the fact that the angle between the échelle-prism axis and the optical camera is 30^o: at the corresponding rotator angle the camera is exactly in the plane of the vertical circle, therefore its bending is minimal (Fig.1.13). As the zenith distance is changing, the lines are moving back and forth in the dispersion direction. The minimal change of the drift occurs at 30^o zenith distance. There is no feasible explanation why it happens at this angle: probably there are some changes in the resulting vector of forces in the échelle turnable frame drawn out by the braced springs, but at 30^o of zenith distance the angle between the échelle plane and the vertical is only 30^o. The échelle stands vertical at a zenith distance of 60^o, hence, the échelle unit counteracts the springs.

The change of the spectral line width from one exposure to the other is shown in the surface fits. These surfaces are less conclusive but show that the focussing of the optical camera is degrading at certain orientations of the spectrograph and it is definitely degrading with increase of the zenith distance. One conclusion that can be made is that at the rotator angles ( ±30^o), where the drift in the dispersion direction is minimal, the change of the focus is maximal.

The results of this experiment have merely practical consequences: to minimize the drift in the dispersion direction during observations the rotator angle for the long 1st camera should be around +30^o (the same is true for the short 3rd camera where the measurements were not made but the mechanical configuration is the same), and -30^o for the medium 2nd camera.

In order to obtain the optimal quality (stable lines and maximal resolution) long exposures should be subdivided into a series of short ones depending on the object position, with subsequent correction of the shifts.

Line broadening due to flexure

The drift of a spectral line across the CCD pixels during the integration results in an apparent shift of the line centre from where it is expected and leads to the increase of the line width. If the line shape can be described by a rectangular profile, then apparently, a shift of such a line by the amount $\Delta$ increases the line width by the same amount. Most of the real spectral lines can be described as Gaussian profiles; in the latter case the increase of the width due to the shift is less significant.

The whole process can be described as a convolution of the true line profile p(x) with the shift function s(x) which results in the smeared profile f (x). Assume that the shift function is a rectangular profile of the width $\Delta$, i.e. the shift of the line is constant in time:

$$s(u)=\left\{\arraycolsep=3mm\renewcommand {\arraystretch}{1.5... ... < u < +\Delta/2\\ 0, & {\rm otherwise}.\\ \end{array}\right.$$ (1.18)

Let the line p(u) be a Gaussian profile of the width $\sigma$, then the resulting profile is

$$\arraycolsep=1mm\renewcommand {\arraystretch}{2}\begin{array}... ...u= \frac{\sqrt{\pi}}{2}\sigma\,\left(y_2-y_1\right) \end{array}$$ (1.19)

where

$$y_1={\rm erf}\left(\frac{x-\Delta/2}{\sigma}\right) \quad\mbox{and}\quad y_2={\rm erf}\left(\frac{x+\Delta/2}{\sigma}\right).$$ (1.20)

The resulting profile is not a Gaussian anymore but the difference of two error function profiles. The resulting full width w_f is calculated at half amplitude of the profile and plotted with respect to the width w_p and the shift $\Delta$ in the following form:

$$\frac{w_f}{w_p}=F\left(\frac{\Delta}{w_p}\right).$$ (1.21)

A more simple but less realistic case is when the shift function can be described by a Gaussian profile, i.e. the line position is a normally distributed random number with a mean equal to the expected line position and the width $\sigma_{2}^{}$ (similar to the width $\Delta$ of the rectangular shift function). Then, the resulting profile is a convolution of the two Gaussians:

$$\arraycolsep=1mm\renewcommand {\arraystretch}{2}\begin{array}... ...ma_1\sigma_2\cdot e^{-x^2/(\sigma^2_1+\sigma^2_2)}. \end{array}$$ (1.22)

The resulting curves for the two shift functions are shown in Fig.1.17. A rectangular shift of a Gaussian profile of 2 pixels FWHM by two pixels would increase its width by 25%; a shift by four pixels doubles the line width.

Figure 1.17: Relative increase of a Gaussian profiles FWHM as a function of the line shift with respect to its width. The bottom curve corresponds to a rectangular shift function, and the upper curve is for a Gaussian shift function.

To verify this in practice, we made an observation of $\alpha$ Cyg (A2 I) whose spectrum has very narrow interstellar absorption features in the NaI doublet. The observation was made with the 1st camera when the telescope was normally tracking the star and then when the telescope rotator was turned by 50^o during the exposure. The comparison lines shifted by 0.5 pixels (425ms^-1) during the exposure. The width of narrow absorption lines is about 4-5 pixels. The two spectra are shown in Fig.1.18. No increase of the line width due to the shift was found in this experiment, because the expected increase of the line width is less than 1%.

Figure 1.18: The reduced spectrum of $\alpha$ Cyg (V=1.25 A2Iae) taken with the long 1st camera (R=176000 at this spectral region) showing the profile of the interstellar NaI D₂ at 5889.9512Å. The two spectra were taken under different conditions: when the telescope was only tracking, and when the rotator was turned from 0^o to +50^o step by step during its 5 min exposure at the zenith distance of 20^o. The rotation caused a shift of the comparison spectrum lines by 0.5 pixels (one pixel is 850ms^-1). No apparent change of the width in the profile components is seen. The decomposition of the line profile onto a number of Gaussians also gives the same widths for the two spectra.

Statical stability of the spectrograph

In this section we give an investigation of the statical stability of the spectrograph when the external long-term factors of the flexure due to telescope tracking are excluded. The importance of such a test is dictated by the need of understanding the nature of the positional instability of spectral lines due to instrumental effects. This is especially important when we study short term variations of spectral line profiles (e.g. non-radial pulsations or other monitoring programmes) where the physical effect could be compatible in amplitude with the effects of instrumental nature. On the other hand, slow variations (tens of minutes) of the line positions due to environmental effects give us additional suggestions of the proper operational modes of the SOFIN spectrograph which can be used to minimize the instrumental instability during observations and taken into account during data reduction.

Figure 1.19: The stability test of the spectrograph in stand-by position. The horizontal axis is time in minutes. The panels from top to bottom are the image drift in CCD rows and columns, line FWHM, the ambient temperature in C, and the CCD temperature in K.

The instrumental profile

The instrumental profile of a spectrograph defines its resolving power, a key parameter which determines the ability of the spectrograph to resolve narrow spectral lines. The resolving power is measured from the core of the profile, and its extended wings define the amount of scattered light of the spectrograph. This is another important parameter which can tell us how much the apparent line intensities differ from the true values and, therefore, defines the precision with which e.g. the equivalent widths of the lines can be measured.

Obtaining the instrumental profile

Following the prescriptions of Tull et al. (1995), Diego et al. (1995) and Barlow et al. (1995), the instrumental profile of SOFIN was measured in November 1998 for the 1st and the 2nd optical cameras. A HeNe laser was used as a source installed in front of the entrance slit. A small milk glass diffuser of the appropriate size was installed between the slit and the laser to ensure that the convergence of the entrance beam is f /11 and, therefore, the camera is uniformly illuminated. The spectral settings of the two cameras were adjusted so that the 6328Å laser line appears close to the centre of the CCD image. The best focus was found prior to the series of exposures. The exposure times for the two cameras were set differently (10 s and 1 s, respectively) to achieve 25000-30000 ADUs per pixel at the maximum of the HeNe line profile (the dynamical range of the CCD is 65535 ADUs). A series of 200 exposures was carried out for the 1st and 2nd cameras giving a total signal-to-noise ratio at the line maximum of 10000 and 5500, respectively.

A similar measurement of the instrumental profile was carried out in April 2000 for the low resolution 3rd camera. A series of 150 exposures was obtained which yields a total signal-to-noise ratio of 3500.

A CCD bias column, averaged from the overscan in columns, was obtained and subtracted from every image separately to ensure that bias variations would not introduce any systematics. The photon noise is estimated from the Poisson statistics for the known CCD gain factor. A standard unweighted integration across the order was done to obtain the spectra of the line. The series of exposures took from 1 to 2 hours, therefore, a change of the ambient temperature and the CCD dewar weight due to liquid nitrogen evaporation could shift the line position. To eliminate the effect, the spectra were cross-correlated with respect to the first profile and the apparent shift in pixels was removed from each spectrum. The image in Fig.1.20 shows a sum of all debiased individual images of the 1st camera, co-aligned in the dispersion direction.

Figure 1.20: An image of the HeNe 6328Å line obtained with the 1st camera. The image is a sum of 200 individual exposures co-aligned with each other.

We found that the immediate summation of all spectra results in an apparent increase of the core width of the resulting profile. The effect is well understood taking into account that due to the line drift and only a few pixels present in the line core, the undersampling of the core is severe, i.e. cannot be well described with the large pixel size. The increase of the line width will otherwise lead to the underestimation of the resolving power of the spectrograph. Therefore, we used the change of the line position in order to improve the sampling of the profile as follows.

Figure 1.21: The instrumental profiles for the three optical cameras. The dots show the overlaid pixels of individual spectra coaligned to each other due to the line drift during the run of exposures. A spline fit is shown as a thin line for the camera 1 and 2. A jagged thin line in the camera 3 plot is the sum of individual spectra. The widths at the three intensity levels are indicated with arrows.

In order to exclude the effect of changes of the line intensity during the run due to variations of the exposure time and laser instability, the spectra were rescaled to the first spectrum. The linear scaling coefficients are determined by the cross-correlation method at the maximum of the cross-correlation function. The wavelength calibration was established from a ThAr comparison spectrum, the image of which was obtained before the series. The wavelengths of the spectra were transformed into radial velocities with respect to the line centre at 6328.160Å. The spectra overlaid (interleaved) with each other in radial velocities show the structure of the profile at a number of pixels which were drifted over the time span of the run. The resulting profile was obtained by fitting a smoothing spline to the pixels obtained as a weighted average over a small interval in velocity scale (about 25ms^-1).

The result of such a procedure is shown in Fig.1.21 for the 1st and the 2nd cameras. In the case of camera 1, the overall line drift during the run is almost one pixel which resulted in the complete overlapping of the pixels. In the case of camera 2, the overall line drift was essentially smaller but well enough for the spline approximation. Unfortunately (for this application), the relative stability of the short camera is a few times higher than that of the other two. The corresponding pixels of the line profile were fixed to almost the same position during the run, giving no possibility to improve the sampling interval. Therefore, a sum of the cross-correlated spectra with small drifts corrected was calculated with the subsequent wavelength and radial velocity transformations.

The measurement of the resolving power

The resolving power definition involves the measurement of the line width. The line width for the 1st and 2nd camera was measured directly from the well-sampled profile. For the undersampled 3rd camera spectrum, the FWHM was measured from the weighted fit of a Gaussian to the core of the profile. Since only a few pixels are present in the core, the narrowest Gaussian was selected among all possible combinations of the pixels involved in the fit. The minimal number of pixels used for the fit is 4. This procedure is justified, because the very central part of the profile can be described by a Gaussian shape, as discussed later. In this section we assume, in the first approximation, that the internal line width of the HeNe laser line is negligible.

For the high resolution 1st camera, according to the model, the expected resolving power at this wavelength is R = 168 000 ( R = $\lambda$/$\Delta$$\lambda$ = c/v) per resolution element of 2 pixels FWHM on the CCD. The measured width of the HeNe line is 1.80kms^-1 FWHM (2.04 pixels), which gives R = 167 000. The 2nd camera profile yields a line width of 3.48kms^-1 FWHM (1.88 pixels) which results in R = 86 200. The resolving power according to the model is R = 81 000 for the 2 pixels FWHM resolution element. The width obtained with the low resolution 3rd camera is 10.7kms^-1 FWHM (2.02 pixels) and corresponds to R = 28 000, which is close to the expected value R = 28 000 for 2 pixels.

The natural width of the HeNe laser 6328Å line

The natural width of the HeNe 6328Å is assumed to be negligible for medium resolutions. According to Bloom (1966), the Doppler width of the line is at most 1.7GHz, Melles Griot (1988) uses a typical value of 1.4GHz (18.7mÅ or 886ms^-1)^1.1. This value agrees with the direct measurements of Tull (1972) obtained with an échelle spectrograph and scanner (they used a R = 60 000 spectrograph). On the other hand, Barlow et al. (1995) refer to a width of 0.03mÅ or 1.4ms^-1. Their measurements of the instrumental profile with the Ultra High Resolution Facility (UHRF) at AAT (R = 10⁶) yields a core width of 300ms^-1 which implies that the natural width of their laser is indeed very small.

Figure 1.22: A schematic diagram of the fine structure of the HeNe laser 6328.16Å line which was used for the instrumental profile calibration. The Doppler profile width is 1500MHz (T = 400 K), the resonance mode spacing is 300MHz (50cm cavity length), and the single mode width is 1MHz (this width is exaggerated 10 times for clarity). The width of the picture corresponds to two pixels ( 2×882 m s$^-1$) of the high resolution 1st camera.

To clarify the question of the true shape of the HeNe laser line, we refer to excellent introductions by Siegman (1971) and Svelto & Hanna (1989), some details are also given in Gray (1992, Ch.12). Fig.1.22 shows the fine structure of the HeNe 6328.16Å line, which consists of a number of single modes superimposed on a Doppler broadened profile. The width of a single mode is defined by the internal width of an atomic transition and is typically 1MHz FWHM (13.3 $\mu$Å) for a HeNe laser. The mode spacing, or cavity resonance function, is defined by $\Delta$$\nu$ = c/2L and is caused by the multiple passing of the light packet between the two cavity mirrors of the laser, the distance between them is denoted by L. The shorter the cavity, the less modes are present (a laser with a very short cavity, L < 15 cm, operates in a single mode). In our case, L $\approx$ 50 cm which implies that the spacing is about 300MHz or 4mÅ. The intensities of the resonance modes are distributed with a thermal Doppler broadening function (a Gaussian profile), which depends on the gas temperature and its atomic number. For Ne (m=20) at 6328Å and for the gas temperature T = 400 K, a width of 1500MHz or 20mÅ FWHM for the thermal profile is obtained. All unstabilized HeNe lasers are subject to mode sweeping, an effect caused by thermal changes of the cavity length. The most important thing is, that it is a long term effect as the cavity length changes with the ambient temperature which may result in variations of the mode spacing and their frequencies.

Figure 1.23: Shown in the middle is the comb of the HeNe laser line fine structure, its convolutions with a Gaussian profile (thin curves), where the narrow profile is for camera 1, the wider is for camera 2, the measured profiles approximated by a spline (thick curves), and the step functions which are the undersampled profiles calculated as the sum of the individual spectra (given for comparison).

The described structure of the HeNe line would now explain the fact that the natural width is laser specific. For such a measurement when the natural width is comparable with the spectrograph resolution, a laser with a short cavity working in a single mode would be required.

Now, when we established the model of the line structure, we obtained the possibility to exclude it from the measurements by deconvolution of the laser comb. Instead, we convolve the laser profile with Gaussians of different widths and compare it with the measurements. Here, we have to assume that the convolution function has a Gaussian shape, resulting from the geometrical slit width, diffraction, and aberrations. The convolutions of the laser comb with Gaussians of 1.7 pixels FWHM for the two cameras are shown in Fig.1.23. The corresponding widths in velocities are 1.5 and 3.2 kms^-1 which yield a higher resolving power of R = 200 000 and 93000 for the pure Gaussian shape assumption for the 1st and 2nd cameras, respectively.

The features

The core of the instrumental profile of SOFIN can be approximated by a Gaussian profile. Fig.1.24 shows the degree of deviation of the core from a Gaussian. Comparison with the profile of the 2nd and 3rd cameras shows a similar behaviour. Down to the level of 10% of the central intensity the Gaussian is a good approximation to the instrumental profile, but at the level, where the Gaussian reaches 1% of the central intensity, the instrumental profile still has about 3-4% intensity.

The extended wings of the profile constitute the amount of scattered light added to the spectrum in the dispersion direction. For the 2nd camera profile, for instance, the contribution is about 0.1% of the central intensity at 30kms^-1 (0.6Å) from the line centre. Examination of the original image shows that there is no difference in the amount of scattered light along and across the dispersion around the central peak. The scattered light across the spectral orders can be approximated within the interorder gaps and removed from the image. This is not the case for the scattering in the dispersion direction, as was discussed in Gray (1992, Ch.12).

The scattered light intensity differs by a factor of 2 for the two cameras (the ratio of their focal lengths) over the whole range of ±100 km s$^-1$ as could be seen from the ratio of the two profiles. Griffin (1969) pointed out by using a log-log plot that the wings exhibit an inverse-square decline. The fit of a Lorentzian profile to the extended wings shows that the shape of the whole profile (except the satellites) can be well approximated by the sum of the two (but not with a Voigt profile which involves the convolution of the two). The FWHM of the Lorentzian profile is two times less than the FWHM of the Gaussian which fits the core. For the accurate convolution or deconvolution of the spectral lines, the instrumental profile should be used with the scattered light (far wings) subtracted from it. In practice, however, it is nearly impossible to distinguish between the scattered light and true features of the instrumental profile. Fortunately, for SOFIN, the scattered light contribution is so low, that it can be neglected.

Figure 1.24: Comparison of the instrumental profile for the 1st camera with the best fit of a Gaussian profile (1.90kms^-1 FWHM) to the line core and a Lorentzian profile (0.95kms^-1 FWHM) to the extended wings.

Griffin (1969) distinguishes the following components of the instrumental profile:

1. The main peak (has been discussed above).
2. Rowland ghosts. This feature should be symmetrical with respect to the main peak. Most likely, they are absent in the observed profiles.
3. Diffracted wings. The diffraction maxima are unresolved in our case: the position of the first minimum is at 0.4kms^-1, i.e. within one resolution element. The maximum N = 30 at 12kms^-1 has only 0.01% of the central intensity.
4. Small-angle scattering in the optical system.
5. Halation and scattering in the photographic emulsion (not present here).
6. True scattered light. Has been attributed above to the broad scattering wings.

Tull et al. (1995) add a number of new items:

1. Internal reflections and interference in the CCD. The two instrumental profiles were obtained by using two different CCDs (but of the same type). The structure of the wings obtained is much the same, so that this component can be neglected.
2. Widening due to CCD transparency; the same argument as in the previous item.
3. Charge transfer inefficiency, which makes a characteristic tail in the direction of parallel (along the dispersion) or serial transfers (depending on which of these two registers is not fully optimized). The CCDs being used are optimized and specially tested for the effect.
4. External reflections (stray light images). This is the most plausible explanation for the satellites in the scattering wings. The equal positions of the satellites for the two cameras imply that the effect is taking place before entering the elements of the optical camera. Filling the collimator pupil would make external reflections on the collimator and échelle frames. Small-angle back-reflections on the optical elements are also possible. One of the features of SOFIN is a small flat mirror situated in the collimated beam. The mirror is a key element of the spectrograph design: it turns the light beam emerging from the slit by a right angle and renders it to the collimator; the use of the mirror reduces the size of the spectrograph and, therefore, improves its mechanical stability. However, the design has its drawback: the mirror is situated in the collimated beam which allows small-angle reflections back to the rear of the slit which results in the appearance of additional off-axis beams. The reflections which occur in the direction of the slit width will be seen as the displaced features in the dispersion direction, i.e. the satellites in the instrumental profile. The rear of the slit is protected from the back-illumination effect by a black metal shield with a hole, although, the size of the outlet leaves room for the small-angle back reflections. One other possibility for back-reflections in SOFIN is a filter installed between the slit and the flat mirror. The filter is composed of two pieces separated by a spacer: an ``ideal'' optical element to generate off-axis beams.

It has been a long discussion in the literature (e.g. Griffin 1969), that the instrumental profile obtained in the laboratory may differ from that obtained on the telescope. This important issue depends on the design of the spectrograph. Typically, the way how the collimator is illuminated is different for the calibration source and for the stellar image. In the former case, the collimator is uniformly illuminated, in the latter case, a central shadow area is present due to the secondary mirror of the telescope, hence, a difference in the shape of the profile may be expected, as well as a possible systematic shift in the line positions from the two sources. In the case of SOFIN, the diagonal flat mirror reduces the effect: the secondary mirror of the NOT is 20% of the primary in diameter, the flat mirror is 25% of the collimator. Therefore, the shadow from the telescope mirror is completely inside the shadow due to the flat mirror and one should not expect any geometrical difference in the collimator illumination from the calibration and stellar sources. A non-uniform illumination of the collimator by the stellar source, which is also subject to temporal variations as opposed to the uniform illumination by the calibration source, causes the difference between the instrumental profiles obtained in the two cases.

Comparison with the solar spectrum

In this section we apply the measured instrumental profile to real observations. The solar spectrum is used and compared with the McMath FTS Solar Atlas (Kurucz et al. 1984), which has a resolving power of about 400000 and a noise level below 0.1%. Another advantage is that the solar spectrum obtained with FTS is free of most instrumental effects, like the scattered light, and it is more accurate unlike a synthetic spectrum which may contain uncertainties in the line parameters.

The solar spectra of day light were obtained with the 1st and 2nd optical cameras in July 1996 in the spectral region around 5620Å. The telescope was pointed without tracking to the sky with the dome and mirror opened. Pointing to the sky ensures that the collimator illumination is similar to the calibration or laser source. The exposure times were 15 and 8 min, yielding the signal-to-noise ratios 400 and 500 for the 1st and 2nd camera, respectively. The spectrum with the 3rd camera was obtained in similar conditions in June 1997 with the signal-to-noise ratio 260 in 8 min of exposure time.

A standard data reduction procedure was applied to the spectra including scattered light removal and wavelength calibration. No corrections to the wavelength scale other than the heliocentric correction were applied. The solar FTS spectrum was convolved with the measured instrumental profiles of the 1st, 2nd, and 3rd cameras in radial velocity scale. The continuum was derived by fitting a spline to the ratio of the observed spectra and the convolved FTS spectra.

Figure 1.25: Comparison between the solar spectrum obtained with the 1st camera (dots) and the FTS spectrum convolved with the HeNe instrumental profile (thin line). The difference spectrum enlarged 10 times is plotted at the bottom. The mean difference is 0.4% rms, the maximal deviation is 2.8%.

The two spectra for the 1st camera are shown in Fig.1.25: an excellent agreement with the two spectrographs at different line depths and consistency with the instrumental profile derived two years later. The difference between two spectra shows a scatter of 0.4% rms (depends on how many lines are present in the spectrum) which is higher than noise level of 0.25% due to photon statistics. The largest systematic deviations in the blue part of the spectrum occur possibly because of some small uncertainties in the wavelength scale. The comparison of the instrumental profile with a Gaussian in the previous sections shows that there is a systematic deviation at levels below 10% of the peak intensity. To check how significant the deviation is, we convolved the FTS spectrum with a number of Gaussians of different widths. For a convolution with 2 pixels FWHM all line profiles are 5% deeper than the observed spectrum. The best fit can be achieved with 4 pixels FWHM as far as the line depth is concerned, although the convolved line widths are then 10% larger than in the observed spectrum. Hence, it can be inferred that the profile wings below 0.1 are important and one should expect systematic deviations when using a pure Gaussian profile for the convolution with the synthetic data.

Figure 1.26: The solar FTS spectrum is convolved with the instrumental profile of the 2nd camera (thin line) and overplotted with the observed spectrum (dots). The difference spectrum enlarged 10 times is plotted at the bottom. The mean difference is 0.3% rms, the maximal deviation is 1.3%.

In the case of the 2nd camera, the observed spectrum is in best agreement with the convolved FTS spectrum (Fig.1.26). The noise level in the difference spectrum 0.3% rms is almost completely due to the statistical noise level of 0.2%. However, a contribution to the excess could possibly be due to the fine structure of HeNe laser line present in the instrumental profile.

Figure 1.27: The solar spectrum observed with the 3rd camera (dots) and the FTS spectrum convolved with the HeNe laser instrumental profile. The difference spectrum enlarged 10 times is plotted at the bottom. The mean difference is 0.7% rms, the maximal deviation is 4%.

The comparison between the convolved FTS spectrum and the observed one with the camera 3 is given in Fig.1.27. The convolved spectrum shows small, but systematical deviations of the line widths and depths from the observations: the lines are more narrow and shallow than in the observed spectrum, especially at the edges of the image.

The real resolving power

The above comparison with the observed spectra proves that the measured instrumental profile almost corresponds to the real profile. The core of the profile is broader due to the fine structure of the HeNe line but the near and far scattering wings are introducing a real effect when convolved with the FTS spectrum. The resolving power was measured from the deconvolved core of the profile and is in a good agreement with the expected values. A matter of practical considerations, is whether the real profile with its features is able to resolve narrow spectral lines and what is the real resolving power.

Figure 1.28: The convolution of two delta-functions of different separations with a Gaussian and with the instrumental profile of the 1st camera. The width of the convolved Gaussian profile is 2 pixels FWHM. The 1st camera instrumental profile is able to resolve two sharp features separated by less than 2 pixels (1.8kms^-1) which corresponds to the resolving power better than R = 167 000. The asymmetry and shift of the convolved instrumental profile with respect to the Gaussian is apparent; this could lead to a systematical error in radial velocities.

Two delta-functions were created in velocity scale with different separations to mimic narrow spectral features (Fig.1.28). A convolution with a Gaussian profile of 2 pixels FWHM is given for comparison. The resolution can be defined as the minimal distance between two narrow features at which the convolved profile can be still numerically separated. The numerical separation involves either decomposition of the two profiles or deconvolution with the known instrumental function. The visual separation tells us that the convolved profile is still resolved at 2 pixels separation of the delta-functions.

Figure 1.29: The spectrum of $\alpha$ Cyg (V=1.25 A2Iae) taken with the 1st camera (R=176000 at this spectral region) showing the profile of interstellar NaI D₂ at 5889.9512Å. The profile is decomposed into 10 Gaussians. The line at -31kms^-1 with FWHM=4.6kms^-1 is a telluric line.

One other test for the resolving power involves observation of interstellar clouds towards $\alpha$ Cyg (this spectrum was also used in Sec.1.6.1 to demonstrate the effect of line broadening due to mechanical flexure). The spectrum of interstellar NaI D₂ at 5889.9512Å was taken with the 1st camera ( R = 176 000 at this spectral region) and is shown in Fig.1.29. The profile was decomposed into 10 Gaussians ( $\chi_{\nu}^{2}$ = 7.7) and compared with observations of Wayte et al. (1978) and Blades et al. (1980), made with a Michelson interferometer ( R = 500 000). They have resolved a hyperfine structure of the interstellar profile on the line of sight in a number of narrow $\approx$ 1kms^-1 components. The component at +1kms^-1 is a blend of two narrow profiles separated by 1kms^-1 with the total width 2kms^-1. Convolution with a two pixels Gaussian instrumental profile (the width is determined from the ThAr comparison spectrum lines) gives FWHM=2.6kms^-1 which is exactly the width of the decomposed component.

Discussion

The above analysis has shown that the measured resolving power is consistent with that which is expected according to the design parameters. However, the estimated resolving power is higher than expected especially after the fine structure of the laser line was eliminated. The reduction of the profile width can be interpreted in different ways as follows.

Firstly, the observed line width could be narrower if the collimated beam does not fill the pupil completely, which results in only partial illumination of the optics of the spectrograph, i.e. the aberration spot becomes narrower. This problem was overcome by using a light diffuser situated at a certain distance above the slit. Secondly, there might be some uncertainty in the estimate of the aberration spot for the optical cameras which is involved into the calculation of the slit width. And the last uncertainty may come from the slit calibration which is possibly resulting in a slit which is narrower than expected in linear scale. Nevertheless, the obtained instrumental profile is legitimate for the CCD image centre and for the optimal focus of the spectrograph.

The measured instrumental profiles show the typical shape common to all grating spectrographs (Dravins 1993). The core of the profiles can be approximated by a Gaussian down to 10% of its maximal intensity. Different kinds of estimations of the convolution width based on the sum of squares of widths remain valid up to the above accuracy. For higher accuracies, the real measured instrumental profile should be used for the convolution with the template (synthetic) spectrum to achieve the maximal correspondence to the observed spectrum. One should not use a Gaussian approximation of the PSF for the convolution.