The Barlow lens

An earlier tutorial discussed eyepieces and touched on the device known as a Barlow Lens. This article looks more closely at how the Barlow Lens works and explains how it often does not actually do what it says on the tin. For a refresher on the workings of telescope optics see this excellent tutorial by Paul Abel.

Basics
The Barlow Lens, invented in the nineteenth century by the British mathematician and physicist Peter Barlow (1776-1862), is a negative (concave) lens fitting inside the focuser of a telescope. Unlike the better known positive (convex) lens a negative one does not cause light passing through it to converge to a focus. On the contrary, a negative lens causes light entering it to diverge as if from a ‘virtual focus’ (Figure 1). For simplicity, in all the diagrams, the Barlow is shown as a single lens. In reality it will be comprised of two or more elements for better optical performance (particularly colour correction).

The Barlow Lens is placed a short distance inside the focus of the main telescope (Figure 2). Light entering has its path changed so that it converges less steeply. As a result the light leaving the Barlow appears to be coming from a much longer focal length telescope whilst actually moving the new focal plane back only a short distance.
The effective focal length obtained when using a Barlow with a telescope can be much greater than the actual physical distance between the objective lens/mirror and the focal plane. Since magnification is given by dividing the focal length of the telescope by the focal length of the eyepiece, increasing the effective focal length of the telescope in this way increases the magnification of any eyepiece placed after the Barlow. Equally the telescope’s focal ratio is also increased, so for example a 2x Barlow would convert an f/5 telescope to f/10.
Generally the Barlow lens is supplied mounted at one end of a short tube and an eyepiece fits into the other end. The whole assembly is then inserted into the focuser.
The most common Barlows double the magnification of any eyepiece and are known as 2x Barlows. Other amplification values are available, particularly for imaging where a long effective focal length is desirable.

• Since any eyepiece can now give two magnifications (with and without the Barlow) this potentially doubles the range of magnifications available.
• Because of the greater focal ratio provided by the Barlow the quality of the eyepiece needed in order to give a good image is reduced. An eyepiece that was quite indifferent at say f/5 could perform much better at f/10 when paired with a 2x Barlow.
• Short focal length eyepieces often have small eye relief which can be uncomfortable to use. When used with a Barlow, short to medium focal length eyepieces usually retain close to their original eye relief. Therefore, rather than use a short focal length eyepiece it can be more comfortable to use a longer focal length one combined with a Barlow. So for example a 20mm eyepiece with a 2x Barlow may well be easier to use than a 10mm on its own.
• Adding an extra optical element into the optical train has the potential to degrade the image because of any imperfections of its own. As long as you use a quality Barlow this should not be a significant problem.
• When using a Barlow with long focal length eyepieces, the eye relief can increase significantly beyond that for which the eyepiece was designed. This can result in ‘vignetting’, the falling off in light towards the edge of the field as the outer parts of the field of view are no longer able to “see” the full aperture of the objective. How important this may be to you depends on the construction of your eyepieces and on the type of observing you plan to do. If you are only interested in the centre of the field, for example when planet observing, then any vignetting at the edge is irrelevant. If however your interest is variable star observing then having a fully illuminated field is very important.
Barlow Maths

NB. In the interests of simplicity, in the following equations I have ignored sign conventions. As a result, Barlow focal lengths are positive when technically they should have negative values.

To make sense of any particular Barlow and determine the effect it has on magnification we need to know its amplification factor.

There are three things that determine this (Figure 3):
1. f – The focal length of the Barlow Lens.
2. d – How far inside the primary focal plane the Barlow is placed.
3. s – How far inside the new focal plane the Barlow is placed.
The amount of amplification (A) is given by either of these two formulae

i) A = (s/f) + 1
ii) A = f/(f-d)

These two equations tell us that that for any Barlow the amplification obtained can be varied by changing either the eyepiece/Barlow separation (s) or the Barlow’s distance inside the primary focal plane (d).

Because the Barlow lens is mounted in a short tube with the eyepiece fitted into the other end, the length of this short tube governs the eyepiece/Barlow separation so equation (i) is the most useful. The length of the short tube is generally such that the nominal magnification will then be achieved.

Consider the case where the Barlow lens is placed a distance inside the new focal plane equal to its focal length so s=f. In this case equation (i) becomes (f/f) + 1 = 2. Thus for any 2x Barlow the length of the tube will be roughly equal to the focal length of the Barlow.

Usually the Barlow will give its nominal amplification when the new focal point is at the top of the tube.

The real world
When the image is in focus the focal plane of the eyepiece and that of the objective/mirror, after travelling through the Barlow lens, are at the same position. To give the Barlow’s nominal magnification these focal planes should meet at the end of the Barlow tube.

In the real world, however, the eyepiece focal plane is often not at the top of the barrel where the end of the Barlow tube will be when the eyepiece is inserted. What’s more, unless you have a set of ‘parfocal’ eyepieces which have their focal planes all in the same place you will find that different eyepieces will have different positions and hence different amplifications.

How big a problem is this? Let’s consider in detail the Barlows in Figure 4.

The table below gives specific details on all three. The nominal focal distance is the distance of the new focus beyond/inside the top of the Barlow tube.

 Barlow 1 Barlow 2 Barlow 3 Nominal Amplification 2x 2x 4x Focal length 80mm 70mm 25mm Nominal focal distance 0mm 0mm 4mm beyond
Within my set of eyepieces the focal planes range from 18mm above to 13mm below the top of the Barlow tube. The table below shows the actual amplification given by each Barlow for these two extremes.
 Barlow 1 Barlow 2 Barlow 3 Nominal Amplification 2x 2x 4x Eyepiece 1 (18mm beyond) 2.2x 2.3x 4.4x Eyepiece 2 (13mm inside) 1.8x 1.8x 3.2x
It is worth noting that the 4x Barlow was designed for imaging where the camera’s focal plane is some distance beyond the top of the Barlow.
Looking at the first Barlow in the table, if an unamplified eyepiece gave a magnification of 100x then with this Barlow it potentially would be either 220x or 180x rather than the predicted 200x, quite a difference. Obviously we are looking at the extremes here to make a point, but when considering magnifications it would make sense to round to the nearest ten or so since the accuracy of amplification does not justify more than this.

More serious is when a Barlow is used with a camera such as a DSLR, for example when planetary imaging. Here the focal plane is well behind the nominal focal point, in this case Barlow 1 would give an amplification of 2.7x rather than 2.0x.

Measuring the focal length
From the above it is clear that knowing a Barlow’s focal length can be useful. However, since it is a negative lens light is not brought to a focus. An internet search will turn up a number of methods, usually involving pairing the negative lens with one that is positive. This is fine if you have a suitable positive lens to use in the first place.
What follows is a method based on one described here which requires no additional lens or telescope and works by simply allowing the Sun’s rays to shine through the Barlow onto a sheet of paper (Figure 5). Even though this is a diverging lens, exercise caution and ensure that you never look directly at the Sun either with or without optical aid.
1. Make sure that the light path is not obstructed. In practice this will mean that the Sun’s rays will need to enter from the eyepiece end and exit from the Barlow end. Many Barlows have the lens itself mounted in a separate cell which can be detached from the main tube (Figure 4). This can be helpful, but never remove the actual lens from its cell.
2. Measure the diameter of the Barlow lens. You may find there is an internal stop and the functioning diameter is less than that of the lens itself. For example the lens of the 4x Barlow in Figure 4 is 17mm across but looking into it there is a stop of only 13.5mm. This latter measurement is the one to be used.
3. Allow the Sun to shine through the Barlow onto a piece of paper. You will see a circular patch of light projected onto the paper. Adjust the distance between paper and lens until the circle is as large as possible without being so faint that it is difficult to see.
4. Draw a circle on the paper roughly that size.
5. With the Sun shining through the Barlow adjust its distance from the paper so that the circle of light just fills the circle drawn on the paper.
6. Measure the distance between the centre of the lens and the paper.
7. Repeat the measurements a few times and average the result.
8. The focal length of the Barlow is given by:
f = D*L/(C-D)

Where
D = Barlow diameter
L = separation of lens and paper
C = diameter of light circle on paper

Varying the amplification
It is possible to obtain different amplifications by fitting extension tube(s) between the Barlow and the eyepiece. However there are a couple of points to consider.
Firstly, by changing the separation the Barlow is no longer working the way it was designed and there is a risk of it introducing aberrations with the resulting image being impaired.
Secondly, as you increase the separation you also have to move the Barlow further inside the original focal point by racking in the focuser (equation ii). This places a physical limit on how far the Barlow can be moved inside focus. You also need to consider that if you move the Barlow too far inside focus it will start to truncate the light cone entering it resulting in vignetting.

In conclusion
The Barlow lens is a really useful accessory to have. As long as you do not need to know the precise magnification or image scale that you are using just take the amplification stamped on the tube. If you need to be more precise, are using it to form an image for a camera or are just plain interested then the above will go a long way to allowing you to work out the true amplification.

Either way a Barlow is definitely worth having!

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The author thanks Martin Lewis and Es Reid of the BAA Equipment and Techniques Section for their comments on the technical aspects of this tutorial which have greatly improved its clarity.