In this post we describe the ways to link a telescope to a spectrograph through optical fibres. We will see in which cases we need to reduce the telescope beam to decrease the FRD. We explain the ways to properly place the image of the star in front of the fibre input end. Finally, we describe the way to couple the fibre output beam into the collimator of the spectrograph.
1. Coupling lenses
As described in Characterization of optical fibres, a way to reduce the FRD is to launch the telescope beam into the fibre with the faster possible aperture (lower F/#). This can be achieved with appropriate lenses or a tapered fibre. As a general rule and following the FRD measurements taken in laboratory, a coupling lens should be used for telescope beams open between F/5 and F/8. For telescope beams slower than F/8 the use of a coupling lens is imperative! While for telescope beams faster than F/5 it is not really necessary. In terms of amateur telescopes, most Schmidt-Cassegrain or refractive models have F/10 apertures or slower. In such cases you must need to reduce the beam before the fibre in order to reduce the FRD. The Dobsonians and Newtons with apertures between F/5 and F/6 are at the limit but still can be coupled directly to the fibres. Some reflective telescopes allow using the prime focus opened to F/3 or F/4, in this case the fibres can be optimally coupled directly to these beams.
There are basically two ways to couple a telescope beam into a fibre, to project either, the image of the star directly on the fibre end or the pupil of the telescope. Both methods have their advantages and drawbacks as explained in the next two sections. Tapered fibres are discussed at the end of the post.
1.2. Image of the star on the fibre end
A positive lens of any focal distance may be used to image the star on the fibre input end. Figure 1 shows two optical layouts where the image of the star delivered by the telescope is projected on the fibre end. The distances of the telescope focal plane to the lens and from the lens to the fibre can be determined by using the well known Gauss equation. The reduction factor between the F/# of the telescope beam and the aperture of the beam transmitted along the fibre must be introduced in this equation.
In the upper layout of Figure 1 the star is a real object with respect to the lens. Its image is on the fibre and stays after the focal plane of the lens. Since the pupil is practically at infinity, its image is at the focal plane of the lens. In the lower layout the star is “injected” through the lens and therefore it is a virtual object to the lens. Its image is placed before the focal plane and therefore before the image of the telescope pupil!
If Ft is the aperture (F/#) of the telescope, Ff the wished aperture along the fibre, f the focal length of the lens, then the distances from the lens to the star (dt )and to the fibre (df ) are given by:
As a practical example, let’s assume that we have a single and thin lens with a focal distance of 10 mm and a telescope delivering a beam open to F/10. If we want to reduce the beam to for example F/4 into the fibre, we will have:
f = 10 mm, Ft = 10 and Ff = 4,
therefore the distances from the lens to the fibre and to the star (telescope focal plane) are
df = 14 mm and dt = 35 mm respectively
However, the reduction of the telescope aperture into the fibre applies only for a point on the optical axis. Some beams coming from any other point of the star (drawn in red) will enter into the fibre with higher apertures (F/#). Theoretically, only the beam from the axis would reach the spectrograph collimator without vignetting. Other beams along the field will be gradually vignetted. This vignetting depends basically of the focal length of the coupling lens. This issue is explained by the fact that the pupils of the telescope and the fibre are not properly matched. Indeed, the image of the telescope pupil is usually far away from the lens (focal distance of the telescope) and therefore its image will be close to the focal plane of the lens. On the other hand, the pupil of an optical fibre is at infinity (Characterization of optical fibres), so, smaller the focal length of the lens, closer the telescope pupil will be to the fibre and thus higher will be the vignetting for points out the centre.
The red beams in Figure 1 define an F/#v according to the following equation:
Where f is the focal length of the coupling lens, φf the diameter of the fibre, F/#T the telescope aperture and F/#f the wished beam aperture in the fibre.
As an example, if you want to reduce an F/10 telescope beam into an F/5 in a 50 µm fibre with a f = 5 mm coupling lens, the actual red beam in the fibre will degrade to F/4.5. However, if you use a lens with a longer focal length, for instance 100 mm, as in the bottom layout in Figure 16, the red beam will degrade only to F/4.8. In other words the vignetting will be significantly reduced.
Another way to overcome the incompatibility between the pupils is simply to enlarge the collimator aperture to accept all the beams. The price to pay is to enlarge all the optical components of the spectrograph accordingly! Still another way is to keep constant the collimator diameter but reducing the focal distance of the collimator in order to increase its F/#. The price to pay in this case is to reduce the resolving power of the spectrograph (next Post)
The ideal solution to this dilemma is to inject the telescope beam with an optical system which keeps the telescope pupil at infinity. In the optics jargon, this system is called telecentric. Figure 2 shows an example with 2 single lenses.
The image of the star (telescope focal plane) is placed at the focal plane of the first lens. The image of the star is sent to infinity and the image of the telescope pupil will fall on the focal plane image of the first lens. The second lens focuses the image of the star on the input fibre that corresponds to the focal plane image of the second lens. The second lens is placed in such a way that its object focal plane coincides with the image focal plane of the first lens. In this case, the second lens will send the image of the pupil to infinity. Under this configuration all rays issued from each point of the star will enter into the fibre with the same aperture (F/#). By the way, the ratio of telescope aperture and the beam aperture inside the fibre is given by the ratio of the focal distances between the lenses:
We notice that any couple of lenses with the appropriate focal distances ratio may be used. However, it is preferable to use lenses with small focal distances to have a compact design. Usually they are few millimetres only. The second lens may be designed in such a way that the rear surface of the lens is in contact with the fibre. In this case, two air-glass surfaces are eliminated and the optical efficiency is therefore increased.
The next step is to select the most appropriate coupling lenses. As amateurs, we look for the best performance-to-price ratio. By performance we meant both, the best mage quality and the highest optical transmission. Among the single lenses we can buy on the market (Linos, Edmund Optics, Thorlabs, etc.) there are the plano-convex, bi-convex, 6:1 radius ratio lens (convex lens where the radius of one surface is 6 times the radius of the other surface), gradient index rod lenses and doublets.
In the case to use a single lens and without going into much detail, the best choice (but the most expensive) are the doublets; they reduce spherical and chormatic aberrations. Next in the list are the bi-convex lenses. In both cases, the image quality increases with small focal lengths, but unfortunately the vignetting increases due to the pupil incompatibility. As for the aberrations only, there is a big difference between a doublet and a bi-convex lens. Optical design simulations show that when using a 5 mm focal length doublet to reduce an F/10 beam into a F/5 on a 50 µm fibre, only 4 % of the flux is lost for a star matching the fibre diameter. For a bi-convex lens the losses increases to 27 % ! (Zemax calculations)
As for the two lenses configuration, two doublets provide an excellent coupling efficiency. Figure 3 shows the layout of the two lenses for the fibre link for the HARPS spectrograph (Performances of HARPS and FEROS fibers in La Silla ESO Observatory, Proc. of SPIE Vol. 5492 pp 669-676, Glasgow, 2004). Note that the second doublet was extended in order to glue its surface to the fibre input end. Two air-glass surfaces are then eliminated. The F/8 telescope beam was reduced to F/4 in the fibre.
Still another compact, efficient coupling system is made with two gradient index 0.25 pitch rod lenses. For a good and detailed description of these GRIN lenses consult the Melles Griot Catalog at this site: (http://www.mellesgriot.com/pdf/0015.16-15.20.pdf). Figure 4 shows the optical layout with the two lenses glued together to the fibre input end. Each lens is cut to a 0.25 pitch but different focal distances. They can be used as collimators. When the image of the star is projected on the surface of the firs lens, its image will go to infinity. On the other hand the image of the telescope pupil will be located on the output surface of the lens. The second gradient index lens acting as a focusing lens brings the image of the star on the fibre and the pupil to infinity. As in the case of individual lenses, the ratio of focal distances equals the ratio of apertures between the telescope and the fibre (equation above).
Usually the gradient-index lenses are delivered with a single layer of MgF2 as anti-reflection coating. Since the lenses and fibre are glued together, the optical layout shows only one air-glass surface. This configuration is therefore highly efficient as for reflectivity losses. However they show some chromatic aberrations which reduce substantially the coupling efficiency to the fibre. For example, if a telescope delivers an F/10 beam with a star of 141 µm on the surface of a grin lens of 2.71 mm focal distance and the star is imaged on a 50 µm fibre by means of a second grin lens of 0.95 mm focal distance, the flux entering into the fibre will be 90% (Zemax calculation). Another drawback of this configuration is the difficulty to align and glue properly the two lenses and to the fibre, especially when the grin lenses have different diameters.
1.3. Image of the telescope pupil on the fibre end
An alternative to project the star on the input fibre end is to project the pupil of the telescope instead. The big advantage is that only a single lens can be used, and when using plan-convex or gradient index lenses they can be manufactured in such a way that its focal plane image lies on the back surface of the lens. The fibre can be directly glued eliminating two air-glass surfaces. The front surface of the lens can be coated minimizing the reflection losses. However, there are two big drawbacks: a) the focal length of the lens depends on the fibre diameter and usually is less than 1 mm, so they are very small lenses. b) this method is only useful for telescopes larger than 1 m. For smaller telescopes, the beam to be propagated along the fibre is not properly reduced (F/# < F/5, see Characterization of optical fibres)
Before going into details, let’s describe the locations of pupils in an optical system. There are four terms to be defined: the stop, the input, the intermediate and the exit pupils. Generally speaking, the light passing through an optical system is limited by a real aperture called the “stop”. The entrance pupil is the image of the stop for an observer located in the object space (Figure 5). The exit pupil is the image of the stop but as seen from the image space. The intermediate pupils in systems with several optical elements are the intermediate images of the stop by these elements.
When a single lens is used to image an object on a screen or detector, the stop is defined by the edge of the lens (Figure 5 a)). The lens itself limits the amount of light reaching the plane of the image, the input and exit pupils lie also on the stop plane. If a screen containing an aperture with a diameter smaller than the lens is placed in front of the lens, as in Figure 5 b), this aperture now limits the light beams. In this case the stop is defined by the aperture on the screen. For an observer placed in the object side, the image of the stop is the stop itself. The exit pupil is the image of the stop through the lens for an observer placed in the image space. In this case, the exit pupil is a virtual image.
In Figure c) the stop lies on the focal plane of the object. The entrance pupil is again defined by the stop but the exit pupil is at infinity. Finally when the stop is placed in the image space of the lens, the exit pupil is the stop itself but the entrance pupil is the image of the stop as seen from the object space. It is also a virtual image. Resuming: the entrance pupil is the image of the stop from the object space of the optical system and the exit pupil is image of the stop from the image space. Note that he stop may be defined by an intermediate lens. In elaborated optical systems, the images of the stop by the lenses or mirrors are intermediate pupils and they may be real or virtual
Most of refractive telescopes and binoculars are made by single doublets. The entrance lens defines the stop, the input and exit pupils. When you put an eyepiece at the focal plane of the telescope, the exit pupil lies very close to the focal plane of the eyepiece (the focal distance of the telescope is much bigger than the focal distance of the eyepiece). It is at this location where you must place your eye, or more exactly your pupil. It is interesting to note that the image of your eye will be indeed at the entrance lens of your telescope (Figure 6).
In Newtonian telescopes the stop is defined by the main parabolic mirror. In most Schmidt Cassegrain telescopes, the stop and also the entrance pupil is defined either by the support of the Schmidt plate or by the primary mirror. In any case the exit pupil is the image of these apertures and lie behind the M2 mirror. In all cases, the exit pupil is far away from the focal plane of the telescope. When you place an eye-piece to explore the sky images, the exit pupil is transported very close to the focal plane image of the eye-piece.
Now, coming back to the injection of the telescope beam into the optical fibre, when you put a small lens just after the focal plane of the telescope, the image of the telescope stop or the exit pupil falls almost on the focal plane image of the lens, see Figure 7. It is at this location where you must place the input fibre end. In addition, the image of the star should be placed on the focal plane object of the lens. At this location, all rays emerging from any point of the star will enter parallel into the fibre. The aperture of the beam propagated along the fibre will be defined by the size of the star. If the star is placed on other locations along the optical axis, the beams will not enter parallel and will increase the beam aperture (F/#).
If s is the size of the star (seeing), F/#tel the beam aperture of the telescope and Φ the diameter of the fibre, the focal length of the lens (f) and the aperture of the beam in the fibre (F/#fibre) are given respectively by:
As a practical example, if you have a 1m telescope opened to F/10 and a 50 μm fibre, you will need a mini-lens with a focal distance of only 0.5 mm. If you want to have an aperture on the sky of say 2.5 arcsec, the size of this aperture at the focal plane of the telescope will have around 125 μm* . The beam to be propagated along the fibre will have therefore an aperture of F/4. As we discussed in Characterization of optical fibres, this aperture is at the limit in terms of FRD. In order to reduce the FRD, the beam has to be faster (<F/4), therefore the aperture on the sky or the telescope focal distance should be higher.
2. Tapered optical fibres
As mentioned in Section 3.1.4. tapers may be used to couple telescopes to spectrographs. They may replace the troublesome micro-lenses to reduce the FRD. The taper may be directly placed at the input fibre end to catch the flux of a large portion of the star and the output end to the spectrograph. It looks a very convenient device where most of the flux of the star enters into the fibre. It also increase the resolving power of the spectrograph because the equivalent “slit width” is smaller. However, there are a number of problems:
- The beam aperture of the output beam increases with the ratio of the diameters between the input and output fibre ends
- The taper must be linear and long, otherwise the FRD increases very fast
- Commercial tapers are very expensive (for amateur purposes)
- The measured efficiency is low than expected and therefore not interesting
If the taper is long enough (L >> core diameter, in practice longer than 10 cm), the output aperture (F/#out) is given by:
where F/#in is the input beam aperture, Φin the diameter of the taper receiving the flux and Φout the output diameter. For example, if you want to reduce your fibre from 100 µm to 25 µm and your telescope provides an F/10 beam, the output beam will be F/4. A taper with a 25 μm fibre is very interesting for amateur purposes but it is difficult to manufacture and therefore very expensive!
At ESO we have tested 4 tapers from different manufacturers. Two of them provided throughputs comparable to the ones of fibres coupled with lenses and the other two with much lower efficiencies. The Table below summarize the results.
|Taper||Input – output beams||Absolute efficiency (%)|
|600 to 200 μm||F/15 – F/5
F/11 – F/3.7
F/8 – F/2.7
|400 to 100 μm||F/15 – F/3.75
F/13 – F/3.25
F/11 – F/2.75
F/8 – F/2
|300 to 100 μm||F/15 – F/5
F/10 – F/3.3
|365 to 100 μm||F/15 – F/4.1
F/10 – F/2.7
As you can deduce from the results, the throughputs are not better than the solution with lenses. In conclusion, we do not advice to use tapers for coupling telescopes to spectrographs.
* If f is the focal distance of the telescope (f = Φ•F/#), one arcsec projected on the focal plane of the telescope, or plate scale will be s = 4.85•f, where the focal distance of the telescope is expressed in meters and s will be given in μm/arcsec. One arcsec = 4.85 E-6 radians. A very good approximation is to say that the plate scale (in μm/arcsec) is 5 times the focal distance of the telescope (in meters).
*European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Munich, Germany
Linking a telescope to a spectrograph through an optical fibre. Part I by CAOS group is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Germany License.
Based on a work at spectroscopy.wordpress.com.