# Tapered optical fibres

### Introduction

Tapered optical fibres (tapers) can be used in astronomical instrumentation as an alternative to replace the conventional lenses to reduce the telescope aperture beam into the fibre. The purpose of the F/# reduction is to minimize the Focal Ratio Degradation (FRD).

Tapers are not “free of charge” artifacts to increase the resolving power of a spectrograph or to concentrate the flux of a star in a reduced spot! If a taper is not properly matched to the spectrograph, you may have significant flux losses. On the other hand, the price to pay to concentrate the flux in a spot is to increase the output beam aperture (F/#) in the same proportion (conservation of the geometrical etendue).

In this article we explain the advantages and drawbacks of these devices in the field of astronomical spectrographs.

### Description

Tapers are optical fibres where the core diameter changes gradually along the fibre (Figure 1). They are mainly used to couple high power laser beams.

Figure 1. Tapered optical fibre

In astronomy they can be used to couple the telescope beam to the spectrograph with the aim to reduce the FRD. Theoretically they are an attractive alternative to the standard coupling lenses:

1. No intermediate optics, therefore no alignments and no coupling losses
2. No especial opto-mechanical holders
3. High thermal and mechanical stability

However, practically there are a number of non negligible problems:

1. The taper has to be long and linear (perfect cone)
2. The taper itself generates focal ratio degradation! The measured optical efficiency is comparable to the solution with lenses for high diameter ratio tapers
3. They are very expensive, especially for custom fibre diameters

### Beam conversion

Depending of the direction of the cone, the taper can be used to increase or decrease the aperture (F/#) of an incoming beam.

If the taper is long enough (Length of the cone >> core diameter that in practical cases has to be longer than tens of cm), the output aperture (F/out) is given by:

$F{{/}_{out}}=\frac{{{\varnothing }_{in}}}{{{\varnothing }_{out}}}F{{/}_{in}}$

where F/in is the input beam aperture, φin the diameter of the taper receiving the flux and φout the output diameter. This equation is derived from the general principle of the conservation of the geometrical etendue in optical systems:

$A\cdot \Omega =A'\cdot \Omega '$

where A is the area of the object emitting a solid angle Ω and A’ the surface of the image receiving a solid angle Ω’. This is the principle of the “globo”, if you squeeze it, it will go out of your hand up and down. Similarly, if you want to squeeze the image of your star, the F/# of the image will be bigger!

For example, assume you have a taper where the input core diameter is 100 µm and the output fibre has a core of 25 µm. (taper with a 4:1 ratio) If your telescope provides an F/10 beam, the beam emerging from the output fibre end will have a theoretical aperture of F/2.5. To this result, one has to add the FRD along the fibre and the taper.

### Resolving power of a spectrograph

There are different ways to express the resolution of a spectrograph. A practical one to illustrate the relation with fibre diameter is the following:

$R=\frac{2\cdot f\cdot \sin (\theta )\cdot \cos (\varphi /2)}{d\cdot \cos (\theta +\varphi /2)}$

where f is the focal distance of the collimator, d the diameter of the fibre, θ the grating angle and φ the collimator-camera configuration angle. If this angle is constant, the resolution can be rewritten as

$R=k\cdot (F/\#)\cdot D/d$

where k is the constant related to the geometry of the spectrograph, F/# and D are the aperture and diameter of the collimator respectively.

Let’s assume that you have a fibre with a core diameter of 0.1 mm (100 µm) connected directly to your spectrograph where the aperture of the collimator and its diameter are F/10 and 20 mm respectively. You will have, therefore, a resolution of R = 2000k. Now, if you use a taper where the output fibre has a 25 µm core diameter you will quadruple the resolution (8000K)! BUT if you do not re-adapt your collimator to the new aperture (F/2.5) you will lose 84 % of the photons! Indeed, the new aperture (F/2.5) will require a collimator with a diameter of 80 mm. If you use the original collimator (20 mm diameter), the flux through it will be only (20/80)^2 = 16 %. If you decide to increase the diameter of the collimator by a factor of 4 to match the new aperture, you have to quadruple the size of the rest of the optical components: echelle, cross-disperser, objective and CCD camera. Everybody knows that the price of the optics and mechanics does not increase linearly with the size but exponentially.

The alternative is to re-adapt the fibre output beam to the collimator one. A positive lens can be used to convert the F/2.5 beam to an F/10. The problem is that the image of the output fibre end will be 100 µm again! You will come back to the original resolution of 2000k. Nothing was changed. As you can see all this exercise was done to minimize the FRD produced by the fibre.

In a previous post we have shown that a fibre linked to a telescope has to work with a beam as fast as possible (F/# < F/5). If you have a telescope providing a slow beam (F/10 as in our example) you have to reduce it. You can use a lens or a taper provided you match the output beam to the collimator of the spectrograph. Figure 2 shows 3 ways to match the beams between the telescope, fibre and spectrograph collimator.

Figure 2. Three ways to link a telescope to a spectrograph. a) The telescope beam is reduced by a taper. b) The input beam is reduced by a lens and returned to its original F/# with a similar lens. c) The coupling is the same as b) but with two symmetrical tapers

In Figure a) the F/10 telescope beam is injected on the input face of a 4:1 taper. The beam is reduced to F/2.5 along the  taper – fibre and the output beam is sent directly to the collimator. The  collimator diameter is asjusted to receive such aperture. This solution requires big optics (expensive). In Figure b) the telescope beam is reduced in the fibre  by means of  a common lens and the output beam re-adapted to a slow collimator (F/10). In this example with the same lens. Figure c) shows the same principle as in b) but with a double symmetrical taper.

### Efficiency

Usually the tapers are made from the same fibre at which they are attached. Therefore, the internal transmission should be the same as the original fibre. However, a taper is not a perfect dielectric straight cone and therefore it shows FRD as all multi-mode fibres. Remember that the main cause of FRD is bends and micro-bends along the fibre.

We have measured the throughput including the FRD of 2 tapers. One of them was provided by C-Technologies (not existing anymore) and one from Fiberguide Industries. The length of the pigtail fibre was 35m long for both fibres. We choose this length since at that time we ordered (end of 80’s) we wanted to link the 3.6m telescope in La SIlla Observatory to the CES, a high resolution spectrograph situated in the Coude Room one floor below. Figure 3 a) plots the total optical efficiency of the C-Technologies taper including the FRD. Click on the Figure to expand the graph. The spectral range of the light source was limited to the visible range with a broad-band filter (400 – 700 nm).

Figure 3. a) Total efficiency of a 600 to 200 μm taper. Fibre: 35 m. Taper length: few centimetres. b) Total efficiency of a 400 to 100 μm taper. Fibre: 35 m. Taper length: 50 cm

The taper had a useful input diameter of 600 µm and a fibre pigtail with 200 µm core diameter. If an F/15 beam is launched into the taper, the optimal output beam should have an aperture of F/5 (= 200/600xF/15). At this output aperture we measured only 35 % of the incoming flux! For an input F/11 beam, 53 % of the flux is collected at an output beam of F/3.7. For F/8 the efficiency is slightly better: 65 % at F/2.7. Table 1 summarizes the results. It is important to point out that this taper was one of the first in the market (1987) and that the taper length was only few centimetres long.

Table 1. Throughput for a 600 to 200 µm taper and few centimetres long. Fibre length: 35 m

 F/input – F/output Total efficiency (%) F/15 – F/5 35 F/11 – F/3.7 53 F/8 – F/2.7 65

For amateur applications the length of the fibre is indeed very long while 10 or 20 m would be enough. However, the extrapolation to a fibre of 20 m increases the flux by few percent only.

Finally if one wants to increase the flux into the spectrograph you can work at lower F/# provided you increase the optics of the spectrograph.

Figure 3 b) shows the resultats for the Fiberguide taper. The input diameter is 400 µm, the output fibre has 100 µm diameter and the taper length was 50 cm (1989). The throughput is apparently better. Table 2 shows the results

Table 2. Throughput for a 400 to 100 µm taper with a length of 50 cm. Fibre length: 35 m

 F/input – F/output Total efficiency (%) F/15 – F/3.75 53 F/13 – F/3.25 57 F/11 – F/2.75 61 F/8 – F/2 70 (extrapolated)

The efficiency is better since the ratio of taper diameters is higher, the optimal output beam aperture is smaller and therefore the FRD is lower.

### Tapers for amateur spectroscopy

Tapered optical fibres may be a useful device to link small telescopes to spectrographs provided they comply the following requirements:

1. A taper is useful for telescopes with F/# > 4. For faster telescopes, a single fibre may be connected directly. The FRD decreases with the F/#!
2. Small telescopes provide small plate scales, therefore the taper should be as small as possible. If for instance, your telescope provides an F/10 beam, the taper should have at least a ratio 3:1 (the input fibre diameter is 3 times bigger than the fibre) to minimize the FRD. In order to increase the resolution of your spectrograph the fibre must have the smallest core diameter. On the market few manufacturers offer fibres with 25 μm, so a taper with this fibre and an input of at least 75 μm would be very attractive! However most taper manufacturers they can not reach such diameters. Fiberguide Industries for example, offers tapers with fibres down to 50 μm.
3. The total optical efficiency of the taper should be bigger than the alternative with relay lenses. This is a trade-off between efficiency against cost. The tapers are expensive and are worth only if they are efficient!

For those courageous amateurs trying to make tapers, we would advice to convey your efforts for the lens relay solution instead. Tapers require a heavy technical process.

### Conclusions

1. A tapered optical fibre may be a good alternative to the telescope beam reduction with classical lenses for the optimization of the FRD. Amateur spectroscopy requires pigtail fibres with core diameter of the order of 25 μm only. To our knowledge only Fibertech is able to manufacture tapers in this range
2. Because of the conservation of the geometrical etendue (AΩ = A’Ω’), tapers do not increase the resolving power of spectrographs without a price to pay: light looses if the aperture of the collimator is not re-adapted
3. For amateur spectroscopy, tapers are worth use if they are more efficient than the alternative with relay lenses

### Authors:

G. Avila* , C. Guirao*

*European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Munich, Germany

Tapered Optical Fibres 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.

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1 Comment

Filed under Fibres

### One response to “Tapered optical fibres”

1. I used a 3mm plastic fibere for my funnel to pipette setup kàin my spectro

that is and other range also my scope is off axis and no obtruction 26 inch x3 scope

better result

jacques savard@videotron.ca jack 47’N 71’w