A Barlow lens is a negative lens in a special mount. It fits in your focuser like an eyepiece does, and has an eyepiece holder of its own where you put an eyepiece.
The lens is an achromatic pair, or some more complex arrangement. By inserting it between the eyepiece and the tube, you increase the magnification - it is not obvious why a negative lens would increase magnification, but that's what happens. Most Barlow lenses are made for a fixed "Barlow facor" M of increase. M = 2 is very common, but values up to M = 3 are also available. The factor is usually printed on the barrel.
The Barlow factor depends on the focal length F of the negative lens (F is negative by definition!) and distance from the negative lens to the focal plane of the eyepiece. So, if you move the eyepiece outwards from the lens, you will increase the magnification! You can make an extension tube, much like a Barlow without the lens, and put between the Barlow and eyepiece to get just about any magnification you like.
If the extension tube adds a distance B, the new, larger factor M1 is given by:
M1 = M - B/F (note that F is negative!)
The focal length (by definition negative for a negative lens) is usually not given. However, if you know M and can measure the height of the outer barrel C - provided you do not have to re-focus much when changing - you can use the formula:
F = C / (2 - M - 1/M ) (note that F is negative!)
If you need to refocus much, measure the distance from the focuser base to any point on the eyepiece (focused), with and without Barlow. The difference is C. If you do not know M, you can measure the exit pupil with and without the Barlow, and calculate the ratio=M. Use a low-power eyepiece here, a magnifying glass (such as another low-power eyepiece held "backwards") and a fine ruler.
B = -F (M1 - M)
There is one thing you need to check before you try! If you add an extension tube, you need to move the focuser inwards by a distance P, given by the formula below. Find the focuser position with Barlow but no extension tube, and see that you have enough inward travel.
P = -F (1/M - 1/M1)
Strictly, the Barlow lens is (or should have been) designed for a particular M, and if you alter it, you will, in principle, have some spherical aberration. There is AFAIK no easy way to predict how much, but in my experience (shared by a few others) the result can be very good, nonetheless. What you want is small total aberration of the wavefront after it has passed through the telescope including Barlow, extension tube and eyepiece. Actually, a Barlow will decrease the spherical aberration (as well as astigmatism) of the eyepiece itself, by increasing the effective f/ ratio. Thus, the final criterion is in the star test at high magnifications - if the defocused image looks very different inside and outside focus, you have spherical aberration. If so, it may not be the fault of the extension tube - if you can, try the same test with a known good eyepiece giving a similar magnification.
On the other hand, the Barlow will keep (and even increase!) the original eye relief of the eyepiece even at high magnification. For this reason, and for the low astigmatism, some of the best eyepieces have an integral "Barlow" lens.
Suppose you have a 2x Barlow and a 10 mm eyepiece, getting the equivalent of 5 mm eyepiece focal length. Supose you want to try for 4 mm instead - you need M1 = 2.5. The outer barrel C is 50 mm, and you use the formula above for F=50/(2-2-1/2)=-100 mm. The next formula gives B=-100(2.5-2)=50 mm, and you need to make a tube that holds the eyepiece 50 mm from the Barlow - the PVC joint I use is bout 50 mm high. This done, calculate P=100(1/2-1/2.5)=10 mm - this is the extra focuser in-travel you need. Check that you have it.
Can you make extension tubes to get the full range of magnifications with only one Barlow and one eyepiece?
A "tele-extender" for SLR cameras is a negative lens combination that does for a camera what the Barlow does for a telescope. I had got one of those from a swap table, and decided to try making a Barlow lens out of it. As it happened, the lens assembly could be removed in one piece and put (with just a little packing tape) inside a piece of 1.25" aluminium tube that I had lying around (once it belonged to a vacuum cleaner). I cut it to a suitable length and added a PVC joint (designed to join PVC drain pipes) as an eyepiece holder. After a bit of cut-and-try, I could calculate, with moderate precision, a barlow factor M = 2.5 and a focal length F = 60 mm. I made one B=90 mm extension tube using similar material, increasing the Barlow factor by 1.5, and one B=180 mm giving a factor 3 increase - thus, I can get 2.5, 4, 5.5 and even 7! And, surprisingly, the performance is very good.
Using my 160 mm f/5.6 Dob and my trusty old no-name 25 mm Plössl, I get these magnifications (exit pupil in parenthesis, in mm):
36 (4.4), 90 (1.8), 144 (1.11), 200 (0.80), 250 (0.64)
This neatly covers just about any magnification I could wish, at a cost that is pretty hard to beat - I admit that the up to 300 mm (12") long combination looks a bit funny, but it works. On the picture, I pose with the 180 mm tube.
If you have a Barlow already, use it - just make sure your focuser in-travel is enough for the extension tubes you plan to make. If you plan to buy one to use with extension tubes, a "short" one means the tubes will be shorter and the in-travel less. For tubing, use what you can find that fits your Barlow and budget - PVC drainpipes, electrical conduit tubes, whatever (but don't forget to check the fit). For eyepiece holders, I suggest PVC joints for drain pipes. I haven't used locking screws on mine, but that is easy to add. One important thing to do is to cover the inside of the tube with non-reflecting material. Paint is of little use to stop the grazing reflections, but flocked self-adhesive plastic or paper is fine.
Nils Olof Carlin email@example.com
PS - You can find the focal lengths of TeleVue Barlows on their home page - no affiliation, but I don't know if any other manufacturer publishes them. From an "ask Al" message, I gather the equivalent focal length of their 5x "PowerMate" is about 37.5 mm - the design is a bit different from ordinary Barlows but it works the same and an extension tube can be used.