### Introduction

This post discusses how spectrograph features such as the resolving power and the “size” of its optical elements are affected when the spectrograph is linked to a telescope with an optical fibre.

The main aim of the post is to help in the design of a spectrograph taking into account the features of any given telescope together with the specifications of the required spectrograph. The basic parameters of a telescope are its size (aperture diameter), the focal ratio (F/#) and the average seeing at the observing location. For the spectrograph, on the other hand, the specifications are usually the required resolving power, the spectral range and the optical efficiency.

Most of the mentioned parameters above are related in a couple of equations and therefore they will tell you the kind of spectrograph you would like to attach to your telescope. These equations can also help you to know what you can do or what you can expect if you already have available a number of elements, such as an optical fibre, a diffraction grating, a CCD detector, a camera objective or a collimator.

We show first the equation for the case where the spectrograph is directly attached to the telescope. It will tell you the possibilities and limitations of your design. Second, we see how these parameters are affected when an optical fibre is used to link the telescope to the spectrograph.

Note that is not the intention of this post to deduce the equations appearing in the analysis; instead the equations will be used just as a tools to define the most appropriate spectrograph design. In our Spectrographs tab it is described in details the basic concepts on dispersers and spectrograph designs.

### Definitions

Before proceeding let´s clarify a few concepts like the definition of *resolution* and the *resolving power* of a spectrograph.

In the case of an ideal free aberration spectrograph where its slit is a point-like source and this source emits a monochromatic beam, the spectrum is given by the Airy pattern (1). If the slit is a “line-like” source, its image on the detector is a sinc function pattern [sinc(θ)=sin(θ)/θ]. If the slit is not a line-like but has a finite width, the spectrum is the convolution of the sinc function with the rectangular one. In all cases, the monochromatic image of the slit is the so-called point *spread function* (PSF) of the spectrograph. In a general case, the PSF includes the aberrations of the system.

The *resolution* of the spectrograph, denoted by *Δλ, *is the spectral coverage of the image of the slit on the detector. For a spectrograph working in the “visible” domain it is usually measured in nanometres (nm). On the other side, the *resolving power R* is a unit-less value defined by *λ/Δλ*, where *λ* is the wavelength of the monochromatic source.

The advantage of using *Δλ *is that its value is constant along the spectral order, while *λ/Δλ *is not. The latter, however, is more used as a generic performance of the spectrograph. Astronomers use *resolution *while engineers characterize the spectrograph by its *resolving power.*

A second clarification is about the resolving power difference between a diffraction limited spectrograph and a “geometrical” slit spectrograph. Most optical textbooks derive the spectral resolution of a diffraction grating working in Littrow configuration (Figure 1) by

*R = mN, *(eq. 1)

where *m *is the diffraction order at which the grating is working and *N * the total number of grooves in the grating. This equation works for a theoretical spectrograph where the slit is an ideal point-like source (zero diameter) and the optics is totally aberration free, i.e. diffraction case. Under these conditions and for a circular aperture of the collimator, the PSF is defined by the Airy pattern. The spectral range *Δλ * is defined by the Rayleigh condition where two spectral lines are resolved when the maximum of one falls on the first minimum of the second (see for example Optics, Hecht, Ch. 10 Diffraction).

Figure 1. Grating working in Littrow configuration. The incident and diffracted beams in red colour are equal with respect to the grating normal (only valid for λ = 500 nm)

Most conventional spectrographs don’t work with point-like sources but with rectangular slits having a finite width *w. *In Littrow mode the image of the slit on the detector has the same width *w. *If this width is much more bigger than the diffraction pattern and the optics of the spectrograph is aberrations free, the profile of the monochromatic slit is practically close to the rectangular function. The resolving power *R* is then given by

where *f *is the focal distance of the collimator and *θ* the Littrow angle (the ray incidence angle and diffracted angle with respect to the grating normal are the same as shown in Figure 1).

Note that this “geometrical” equation is quite different from the ideal case *R = mN. *It is independent of the number of grooves of the grating or of the working order. In other words, for a given optics configuration ( *f *and *w*), the resolving power depends solely of the grating angle with respect to the incident beam!

In the case of real spectrographs where optics are not usually diffraction limited but have small aberrations, the image profile of the slit is not any more rectangular. The edges are rounded and for narrow slits the profile may approach a Lorentzian or Gaussian curve. In any case, the *resolution Δλ* is now defined by the

*FWHM*(Full Width Half Maximum) of the spectral spread of the image of the slit on the detector. In astronomical spectroscopy

*Δλ*is measured in Angstroms for historical reasons. As for the resolving power, the slit width

*w*is replaced by the

*FWHM*and equation 2 becomes

As a numerical example: for a standard 50 mm grating length, 1200 lines/mm and blaze angle of 17.5 deg, the total number of grooves is 1200 x 50 = 60 000. Working at the first order, the theoretical resolving power (eq. 1) is 60 000.

However for a spectrograph with a collimator having a focal length of 100 mm and a 50 μm slit, the resolving power (eq. 2) is only 1 300 ! (~2 x 100 x tan 17.5 / 50). Using an Échelle grating working at 63 deg, the resolving power rises up to 8 000, still well below the theoretical one.

Actually the optics of the spectrographs are not perfect but introduce some aberrations. The image of the slit on the detector is blurred in some extent, making wider its FWHM and therefore the resolving power is further degraded (eq. 3).

### Coupling a spectrograph to a telescope

Figure 2 shows a layout of a slit spectrograph directly coupled to a telescope.

Figure 2. A spectrograph directly coupled to a telescope

The spectrograph is composed basically by 4 elements: the collimator, the disperser optics, the objective and the detector. In the astronomical jargon, the objective is called “camera”.

The telescope is represented by an ideal lens with a diameter *Φ*_{t} and a focal distance *f*_{t.} The slit subtends an aperture *δ* on the sky. The focal ratio of the telescope (*F/#*) matches the one of the collimator.

The circular cross section of the collimator beam with a diameter *Φ**o* defines the pupil of the spectrograph. This pupil is then deformed by the diffraction grating and takes an elliptical shape where its major axis is represented by *Φa*. The ratio between *Φa** * and *Φo *is defined by the *anamorphic magnification:*

This distortion produces a deformation of the image of the slit on the detector. For example, the image of a circular slit will be an ellipse. Its minor axis will be parallel to the dispersion direction on the detector if the angle of the incidence beam is bigger than the angle of the dispersed beam. This narrowing of the image reduces the FWHM of the PSF and therefore increases the resolving power. In the opposite case, when the angle of the diffracted beam is bigger than the angle of the incident beam, the resolving power is smaller. More about the anamorphic ratio can be found in this post (in preparation).

In the case of needing a spectrograph with a long slit, the instrument should be designed in such a way that the image of the telescope pupil should be placed close to the diffraction grating. This condition helps to reduce the size of the grating and camera. A discussion on pupils can be found in this post (in preparation).

Taking into account the telescope parameters, the resolving power of the spectrograph (Eq. 2) in Littrow configuration takes the form:

For a given telescope and a given seeing at the location of observation, this equation is very useful to have an idea of the pupil diameter in the spectrograph and therefore the “size” of the optical elements in your design. It is the war horse of astronomers and hence we will call it “astronomer’s equation”.

There are a number of interesting points: if you need to observe your stellar object with a given resolving power *R* and you want to catch as much as possible flux into the spectrograph, you have to wide the slit. But, to preserve the resolution you have to increase the collimator diameter in the same proportion. Note that the grating and camera grow with the collimator aperture.

Note also that the spectrograph size (collimator pupil, grating and camera) increase linearly with the telescope diameter. Unfortunately the price grows exponentially!

Example 1. You have a 30 cm telescope and you wish a spectrograph with a resolving power of R = 20 000. The seeing at your location is 2″ and you open the slit at this value. You want to record the entire spectrum in one shot, so you have in mind to build an échelle spectrograph. Assume you want to buy an échelle grating working at 63 deg blaze angle. With all these parameters in hand the collimator you need will have a diameter of 1.5 cm ( øo = 20000 x 2″ x 30 cm / 2 x tan 63 deg and 2″ = 9.7 x 10^ -6 radians)

This diameter gives you an idea of the “size” of the optical elements (collimator, grating and camera) you need to gather in order to reach this resolving power.

It is important to point out that this equation is valid for a spectrograph working in Littrow configuration. In the post *Spectrograph designs* we will discuss the equation for a general case.

### Coupling a spectrograph to a telescope with an optical fibre

Figure 3 shows a spectrograph linked to the telescope through an optical fibre. The image of the star is projected on the input fibre end and the output fibre end is also directly coupled to the collimator of the spectrograph.

Figure 3. Spectrograph linked to the telescope through an optical fibre

Unlike the direct telescope-to-spectrograph coupling, the slit now has a circular shape. This difference affects substantially both, the throughput and resolving power of the spectrograph. A fibre with diameter matching the seeing transmits 50% of the flux, whereas a slit with the same width passes 75%. More on flux calculation for a slit and circular slits can be found in our Flux Calculator here.

Following the data reduction where the illuminated pixels by the image of the fibre are binned perpendicular to the dispersion, it is found that circular slits provide between 15 to 20% more resolving power that the one generated by a square slit (its width equals the diameter of the fibre).

This gain in resolution is compensated by a reduction of the flux entering into the collimator. Indeed, when superposed, the circle reduces the area of a square by around 20% (more precisely by *π/4*). In our post *Resolving power for rectangular and circular slits *(to be completed) we propose an explanation of this effect. The resolving power increase by a factor 2/√3 = 1.16 :

Another substantial difference with respect to the direct telescope-to-spectrograph coupling is the non conservation of the focal ratio by the fibre or *focal ratio degradation* (FDR), this is the F/# of the output beam is always bigger than the F/# of the input beam. More about FDR can be found in this post. The posts *Linking a Telescope to a spectrograph through an optical fibre Part I and Part II *describe different ways to inject the telescope beam into the fibre taking into account the focal ratio degradation.

The adaptation of focal ratios between the fibre and the spectrograph collimator has a significant impact on the computation of the resolving power. The astronomer equation (eq. 3) would be valid if:

- The fibre had a square or rectangular section
- The focal ratio of the input and output beams in the fibre were the same and
- The core size of the fibre would change according to the sky aperture (
*δ*)

When designing a fibre linked spectrograph, usually the fibre diameter (“slit width”) and the focal ratio of the fibre beams are fixed. These conditions make sometimes confusion among astronomers who use the astronomer’s equation for a “dynamical” spectrograph where the slit aperture and focal ratio of the collimator follow the parameters of the telescope. This equation however does not take into account the focal ratio degradation (FRD). For a fibre linked spectrograph is better to use the equation 2 and consider the spectrograph as an independent entity.

Example 2. Let’s take the same telescope and results of example 1: you have a 30 cm telescope, you wish a spectrograph with R = 20 000 and you find that the pupil is 1.5 cm. Given these parameters, you want to know the fibre diameter you need to link the telescope to the spectrograph. Let’s assume that the telescope has a focal ratio of F/10. The plate scale (the projection of the sky on the telescope focal plane in microns per arc seconds) is therefore 29 μm/” (= 1″ x F/10 x 30 cm). As you need R = 20 000 for a sky aperture of 2″, the slit width will have a width of 58 μm. You will need thus a fibre of this size working at F/10. Unfortunately it is not advisable to work with fibres working at F/10 because of the focal ratio degradation. If you look at Figure 9 in this post, the efficiency at F/10 for a 60 or 50 μm fibres is less than 25%. However, If you work at F/5, the efficiency increases up to 50%. In order to reduce the focal ratio at F/5 you have to put a lens in front of the fibre input end, but then the scale plate will be reduced to the half. In other words the 2″ sky aperture becomes 29 μm. The fibre diameter must therefore be 29 μm.

On the market, the fibre standard diameters are 25 and 50 μm. Since the latter is easer to purchase and to work with, you decide to inject at F/10 directly on the fibre end and open your collimator at the fibre output end at F/5. The flux gathering increase significantly but the diameter of your collimator (1.5 cm) must be now 3 cm to cover the F/5 beam. So your optics must be twice bigger if you want to reach 20 000 of resolving power! If you keep your optics to 1.5 cm diameter, you have to reduce the focal distance of your collimator to 7.5 cm to accept the F/5. In this case and following equation 2, you reduce the resolution by half. The alternative (working with a 25 μm fibre and using lenses to convert the F/10 telescope beam in F/5 into the fibre) is certainly not easier to carry out but surely cheaper than buying bigger optics (collimator, grating and camera)!

Notes:

(1) It is valid only for a Littrow configuration, otherwise the Airy pattern is deformed by the anamorphic magnification

### Author

Gerardo Avila

European Southern Observatory

**Linking a telescope to a spectrograph through an optical fibre** by CAOS group is licensed under a Creative Commons Attribution-Non-Commercial-No Derivative Works 3.0 Germany License.

Based on a work at spectroscopy.wordpress.com.