Resolution Calculation
for a Slitless Spectrograph

By Dr. Doug West


Introduction

A slitless spectrograph is probably the simplest form of spectrograph that an amateur astronomer can build. The three main components of the system are the telescope, diffraction grating, and a detector. Virtually any form of telescope will work as long as it has quality optics and a sturdy mount. The diffraction grating is a transmission grating which is readily available. The detector can be either a photographic or CCD camera.

This article explains how to assemble a spectrograph from commercially available components and then calculate the spectral resolution of the instrument. A knowledge of the spectrograph’s resolution is important in astronomical object selection, for example, the spectra of late-type stars (red giants) are typically dominated by wide TiO absorption bands than only require a low resolution spectrograph to study. However, this type of spectrograph would not be suitable for measuring the radial velocity of a nearby star – this would require a much higher resolution.

System Overview

The spectrograph consists of four commercially available products – a Meade LX200 8" f/6.3 telescope, an SBIG ST-8 CCD camera, a Rainbow Optics transmission grating, and the MIRA 6.0 image processing software.


Figure 1 – Diagram of telescope with grating and CCD attached.

Spectrograph Basics

The term spectrometer refers to any of several types of spectroscopic instruments. A spectrograph is an instrument that records many spectral elements simultaneously with an area detector. The term "slitless" spectrograph can actually refer to two different optical configurations. The first is the spectrograph made by placing the dispersing element, prism or grating, in front of the telescope objective. The second type places the dispersing element between the objective and the focal plane. In this case, the light incident on the prism or grating is a converging beam. This type of spectrograph is also called a "nonobjective" grating spectrograph. Only the second type of "slitless" spectrograph will be discussed in this paper. This type of spectrograph is similar to the Monk-Gillieson spectrograph (Kaneko, et. al.). General spectrometer references are Schroeder (1987) and James.

The light ray in figure 2 emerges from the telescope and enters the transmission grating. The grating equation,

,

defines the relationship between the entrance angle (), deviation angle (), number of grooves per millimeter (m), wavelength (), and the integer number for the spectrum order (k). In the case of the slitless spectrograph, is 0, and the grating equation reduces to .


Figure 2 – Diffracted star image produced by a diffraction grating.

The spectrographs spectral resolution, R, determines the ability of the instrument to distinguish spectral features separated by . The spectral resolution is a dimensionless quantity that is a measure of spectral purity ().



Spectrographs are considered low resolution when they have R<1000.

The plate factor, P, is a measure of the change in wavelength along the surface of the CCD, or . The plate factor is usually expressed in angstrom per mm. It can be determined by observing a star’s spectrum that has known features, for example, the hydrogen lines in an A type star, and using the equation:



Here, n1 and n2 are the pixel values at and respectively and is the linear size of a pixel. Applying this equation with n1 = 691.9, n2 =864.4, =6562.8 A, = 4861.3 A, and = 9x10-3 mm yields P = 1094 A/mm.

For the case of the Meade LX200 8" f/6.3 telescope, a Rainbow Optics transmission grating, and an SBIG ST-8 CCD, the follow quantities are known:

  • d = distance from grating to CCD = 44.2 mm
  • k = order of spectrum = 1
  • m = number of grooves per mm = 200 g/mm
  • D = telescope diameter = 203.2 mm
  • F = telescope focal length = 1280.2 mm
  • N = F/D = focal ratio = 6.3
Since the entire surface of the grating is not illuminated the theoretical resolution becomes

R = mLk,

where L = gratings lit width (portion of grating illuminated by the star’s image from the telescope).



R = (200 g/mm)(7.02 mm)(1) = 1404   at = 6563 A (Hydrogen Alpha line)



As we shall soon see, the actual resolution of the spectrograph turns out to be much lower.

LIMITS ON RESOLUTION
Due to optical aberrations and other reasons the actual resolution is much lower, hence, the spectral purity is reduced also. The main contributors to a lower resolution come from the chromatic coma, field curvature, the implications of the Sampling theorem, and the effect of the atmosphere on the effective full width at half maximum (FWHM) of the stars image. Spherical aberration and astigmatism are considered to be of less importance to the determination of resolution and are not discussed. Astigmatism is responsible for widening the spectrum in the direction perpendicular to the direction of the spectra on the CCD surface. This results in a lowering of the signal to noise ratio rather than reducing the resolution. The thickness of the grating contributes to the aberrations only minimally and is only considered if the grating face is not normal to the incident chief ray.

1. Chromatic Coma

The main aberration affecting this system is spectral or chromatic coma. The effect of chromatic coma on the spectral purity is given by (Schroeder 1987)



A characteristic of the coma is that 80% of the energy is concentrated in one half the image spot (Hoag 1970). Taking this into account, the formula for the spectral purity becomes



At = 6563 A and N = 6.3, the spectral purity is = 31 A. As the wavelength increases so does the chromatic coma. To reduce the effect of the coma on the spectral purity, a prism can be placed next to the grating, thus forming a "grism". The design of a grism spectrograph is covered in Bowen, Buil, Gavin, and Schroeder (1987).

2. Field Curvature

The aberration due to field curvature results from the cylindrical rather than planar nature of the spectrum’s focus. Referring to figure 2, as the distance x increases from the central star image the focus no longer is directly on the CCD surface. Since astigmatism has only a minor effect on spectral purity, the distance between the sagittal and tangential focal points is assumed to be zero.

To quantify the effects of field curvature on resolution, we assume that the focus of the image is a point. Using the property of similar triangles and the geometry in figure 3 the following relationship holds



Combining this with and solving for yields



The resolving power becomes



For this spectrograph at = 6563 A, , L = 7.02mm, and P = 1094 A/mm, the spectral purity = 67 A. The defocusing effect produced by field curvature significantly reduces the spectrograph’s resolution.


Figure 3 – Field curvature effect geometry

3. FWHM of Star Image

In a slitless spectrograph the full width at half maximum (FWHM) of the star’s image becomes the effective entrance slit width for the instrument. The larger the slit width the lower the resolution and spectral resolving power . The spectral resolving power is given by:

,

where r is the anamorphic factor,

, since = 0.

The spectral resolving power becomes



At a wavelength of 6563 A, = 7.54 degrees, and assuming a FWHM of 3 pixels, becomes

 = 29.8 A.

For this combination of telescope and camera, each pixel represents 1.45 arc seconds.

4. Sampling Theorem

When a continuous signal is sampled, such as a spectra, this process places a lower limit on the spectra purity. Simply put, the sampling theorem states that the highest frequency present in a sampled waveform is one half the sampling rate. The implication of this theorem on the spectrograph is that the smallest value of spectral purity possible is twice the wavelength interval covered by a pixel, or

= (2)(9x10-3)(1094 A/mm) = 19.7 A

Focusing Procedure to Minimize Field Curvature Effect

The field curvature aberration essentially produces a defocusing effect on the spectra. To partially correct the problem a simple focusing procedure can be used to minimize the effect. Figure 4 is a 0.5 second exposure of Sirius with the Hydrogen alpha (6563 A), Hydrogen beta (4861 A), and the Fraunhofer A atmospheric water absorption band (band-head at 7594 A) identified. Note that the throughs in the spectrum at approximately column numbers 450 and 620 are artifacts of the CCD spectral response. The Hydrogen beta line will be used to quantify the focus of the telescope/grating system.

This procedure will require that the telescope have a focusing aid to obtain the best possible focus of the central star image and a digital readout on the focusing knob. The procedure is as follows:

  1. Select a bright spectral type A star. It is best to use a type A star for this procedure because they have definite hydrogen lines.
  2. Obtain the best focus on the central star image using the focusing aid. Record the focus on the digital focus knob.
  3. Sequentially increment the focus knob and take spectra at each interval. Record the focus knob setting with each spectra. Re-focus the spectrograph back on the central star image and then sequentially decrement the focus knob and take spectra at each interval. Record the focus knob setting with each spectra. The total change in focus introduced by this procedure is only a fraction of a turn on the focusing knob.
  4. For each spectra at the different focus positions measure the pixel height of the Hydrogen beta line. This step requires image processing software that will allow the spectra to be "stripped" out of the CCD image. The author uses MIRA 6.0 software for this purpose. From this data a graph similar to figure 5 will be created. When the height (delta in figure 4) of the Hydrogen beta line is maximized, then the telescope is focused at this point.

Figure 4 – Spectrum of Sirius focused on central star image


Figure 5 – Variation in the depth of the Hydrogen beta line as the focus is changed.

After this procedure has been completed you will now have the focusing offset required to change the focal point of the spectrograph from the central star to the Hydrogen beta wavelength (4861 A). Determination of the focusing offset only needs to be accomplished once and then this offset can be used with all subsequent spectra.

Figure 6 illustrates the effects of focusing has on the Hydrogen lines. Note that the hydrogen lines are deeper and more distinct in figure 6 as compared to the case where the spectrograph is focused on the central star as in figure 4.


Figure 6 – Spectrum of Sirius focused on the H beta line

Combining the Resolution Limitations

The resulting spectral purity due to the aberrations of chromatic coma, star’s FWHM, and field curvature are plotted in figure 7. The limitation placed on the spectral purity resulting from the Sampling Theorem where small compared to the aberrations and was not plotted. As illustrated in figure 7, spectral purity is limited by the star’s FWHM to approximately 30 angstrom until 6300 angstroms. After this point the chromatic coma aberration dominates the spectral purity. The focusing procedure previously discussed was employed to reduce the effects of the curvature of field. Note that the curvature of field line in figure 7, labeled "Field", has a minimum at the wavelength of the Hydrogen beta line.

Using the spectral purity and wavelength values from figure 7, the resolution varies from R = 138 at 4000 angstroms to R = 213 at 10,000 angstrom. Resolution within this range qualifies this spectrograph as low-resolution.


Figure 7 – Spectral purity as a function of wavelength with the effects of each aberration shown separately.

CONCLUSIONS
A spectrograph can be constructed from commercially available components that is capable of recording spectra useful for astronomical purposes. The spectral purity is limited by the effects of the curvature of field and chromatic coma. A simple focusing procedure allows the curvature of field aberration to be minimized. The resulting resolution of the spectrograph varies from R =138 at 4000 angstrom to R=213 at 10,000 angstrom.

ACKNOWLEDGEMENTS
The author would like to thank Dr. David Alexander for his help, encouragement, and guidance through the development of the spectrograph. A debt of gratitude is also owed to Maurice Gavin, Joe Sivo, Steve Dearden, and Christian Buil for their helpful advice.

REFERENCES
[1] D. Schroeder, "Astronomical Optics", Academic Press Inc., 1987.

[2] Christian Buil’s web site - www.astrosurf.org/buil.

[3] Maurice Gavin’s web site - www.astroman.fsnet.co.uk.

[4] L.S. Bowen and A.H. Vaughan, Jr., "Nonobjective Gratings", Publications Astronomical Society of the Pacific, vol 85, April 1973, pages 174-176.

[5] J.F. James and R.S. Sternberg, "The Design of Optical Spectrometers", Chapman and Hall Ltd., 1969.

[6] T. Kaneko, T. Namioka, and M. Seya, "Monk-Gillieson Monochromator", Applied Optics, vol. 10, no. 2, Feb 1971, pages 367-381.

[7] A. Hoag and D. Schroeder, "Nonobjective Grating spectroscopy", Publ. Astron. Soc. Pac. 82, pages 1141-1145, 1970.

- Ed. BDM