Barlow lenses are common "tools" in the amateur kit, used to increase the magnification of regular eyepieces. The theory is relatively simple but the practical uses are sometimes not what one would expect. To clear this up in my own mind and, possibly, help others I've scribbled these notes.
Let's consider a Newtonian optical path in which a concave mirror forms an image of a distant target field at a distance F from the surface of the mirror. The optical path is folded to the side to make this image field accessible using a flat diagonal mirror. Various design considerations are involved in placing this diagonal mirror but, for this discussion, let's assume the reflected real image plane is located at the tube wall of the telescope. There is, of course, a hole cut in the tube wall so the light can get out. It is over this hole that the focusing mechanism is placed.
Now, consider modern (ie, non-Galilean) eyepieces which are essentially positive lenses that magnify the real image created by the objective optics in a telescope (whether mirror or lens). These eyepieces are characterized by a focal length and a focal plane. Images (or objects) located at the focal plane can be viewed by the eye at magnified values which depend on the focal length of the eyepiece. Eyepiece designers and manufacturers usually place an aperture in the focal plane of the eyepiece called a field stop. In older eyepiece designs (eg, orthoscopics, Plossls, etc) this field stop in in "front" of the first element of the eyepiece (ie, in the direction from which the light is coming). In later designs (eg, Naglers) the field stop is inside the eyepiece and not physically accessible. What is important is that the real image to be examined must be located in the field stop and this is accomplished by moving the eyepiece back and forth along the optical path (ie, focusing). The optical elements of an eyepiece are not floating in space but are placed in a tube of some sort and this is where differences can occur between eyepieces. I'm not aware of any standards in the field and I'm quite sure some of my eyepieces wouldn't adhere to such a standard if it exists. In particular, all eyepieces I've ever seen (except for some home-made ones long, long ago) are in a tube that has two diameters with a shoulder that prevents the eyepiece from sinking too far into the eyepiece tube (which is a good thing - don't want an eyepiece sliding into the telescope tube). That shoulder, and the overall placement of the optical elements within the tube, is what determines where the field stop is located relative to the focusing tube and physical location of the focus plane of the telescope. In most cases there is enough latitude in the in/out motion of the focus tube to bring the eyepiece into focus but it is often the case that switching eyepieces requires re-focusing because the field stop in the new eyepiece is not in the same geometrical location as the previous one. Parfocalization is the process of aligning the physical locations of the field stops in a set of eyepieces to solve this problem. Most manufacturers create sets (or series) of eyepieces meant to be used together that have this feature. Televue, for example, locates the field stop some 6 mm in front of the shoulder of (most) their latest offerings so all their eyepieces are essentially parfocalized.
On to Barlows. Barlow lenses are "negative" lenses which cause incoming light to diverge. When placed in the path of the converging beam from the objective of a telescope, that beam is caused to converge more slowly and the focal plane is moved out farther than it was. This produces an apparent increase in the focal length of the telescope. Because the apparent magnification of an eyepiece is given by the simple mathematical ratio (telescope focal length/eyepiece focal length) the same eyepiece can have two different magnifications using the Barlow. Neat. But there are design considerations and constraints. In effect, adding a Barlow creates a new eyepiece (and that seems to be the idea behind the Naglers and other modern eyepieces where the Barlow is built-in). The relationship between the negative element and the "regular" eyepiece determines the overall performance and the use of the Barlow. But the Barlow concept is a "modular" concept, ie, the user is free to mix and match the elements but the lack of standardization causes mis-understandings.
We need a little mathematics to proceed (sorry!). There is a deceptively simple looking relationship between the location of the telescope's focal plane, the focal length of the negative element and the new location of the focal plane given by
1/f = 1/q - 1/p where
f is the focal length of the Barlow (a negative number)
p is the distance beyond the Barlow where the focal plane of the telescope would be without the presence of the Barlow, and
q is the distance beyond the Barlow where the re-imaged focal plane is located.
(I'll try to insert a little drawing here some day)
The apparent magnification of the telescope's focal length caused by the Barlow is the ration q/p. So, if q is twice as far as p, the telescope will appear to have had it's focal length doubled.
There are some constraints. If p = f, then q is at infinity and there is no real image created for the "regular" eyepiece to work on (plus it would have to be located at infinity which is a long, long focus tube). On the other hand, no "regular" eyepiece is needed. Just put your eyeball in the path and you have a telescope! This is the way Galilean telescopes (cheap "opera" glasses) work. So, the lesson is the closer p is to f, the farther away q is and the greater the apparent magnification (q/p). So, p needs to be less than f to keep things practical. p is, of course, controlled by where the focus tube is.
On the other hand, Barlow lenses are placed in tubes also and the length of the tube (relative to the Barlow focal length) comes into play when using a "regular" eyepiece and this is where things get interesting. Let's consider an illustrative example, the ubiquitous 2X Barlow. To get 2X we need q/p to equal 2. This is achieved by making p = f/2. In this case, q comes out to be equal to f. We could chose lots of values for f but let's just take 50 mm (negative, of course). Thus, when the Barlow (in its tube) is placed inside our Newtonian tube wall 25 mm, the new focal plane will be 50 mm from the Barlow element or 25 mm outside the telescope tube wall. If we want to achieve the rated value of 2X with the aforementioned Naglers, we need to make our Barlow tube to be 56 mm long (plus any extra in front of the negative element). If our "regular" eyepiece has a different relationship between its field stop and the shoulder, we will get a different relationship for the overall Barlow assembly.
Suppose the field stop is farther from the negative element thereby increasing q. If we have the room to do so, we could move the whole assembly deeper into the telescope tube thereby increasing p as well. When we find a good focus (which we should be able to do) we will be ok, but at a higher magnification than we expected. The reverse is true if the field stop is closer to the negative element, we should be able to achieve focus but at a lower magnification than 2X. We could get a variable magnification if we have room to slide the eyepiece into or out of the Barlow tube.