In this FAQ, we'll assume that you do not want to couple your camera to a microscope complete with its stage, condenser and so on, but rather you just want to stick a microscope objective in front of your camera sort of like a high power macro lens. In that case...How do I hook a microscope objective to my camera so I can shoot at 5X and above?
There are two fundamentally different approaches:
1. Get an old-style "finite" objective that has sufficiently small chromatic aberration, and stick it on tubes or bellows to get the appropriate extension.
2. Get a new-style "infinite" objective that has sufficiently small chromatic aberration, and stick it in front of an appropriately long "tube lens" such as a telephoto lens focused at infinity.
Notice that I've emphasized the issue of chromatic aberration. Most of the older "finite" objectives and some of the newer "infinite" objectives are specifically designed to have quite a bit of chromatic aberration that is expected to be canceled out by a "correcting eyepiece" or other optics that have just the right amount of opposite aberration. When an objective like that is used by itself, or with other optics that are not matched to it, the result can be some pretty extreme color fringes. Certain lines of objectives are known to be good by themselves. Using an unknown objective is like rolling the dice -- maybe it works, maybe it doesn't. When in doubt, post a request in the Equipment forum to see if anybody knows about the objective you're interested in.
OK, with that main caution out of the way, let's forge ahead.
Here are some pictures that illustrate the major components and strategies.
First, the classic approach of sticking a finite objective on extension.
Shown below is the Nikon CF N Plan Achromat 10X NA 0.30 objective, which is designed to be used on a microscope with 160 mm tube length, and with 0.17 mm cover glass. The "160" designation actually corresponds to 150 mm from the objective to the sensor plane, and despite its specification, this objective works very well with no cover glass.

Here are the components that go into this setup:

A: the objective itself
B: a mechanical adapter from the "RMS" thread of the objective to a standard M42x1 Pentax screw thread
C: M42 extension tubes as needed to make up the total extension
D: A "chipped" adapter that accepts M42x1 threaded optics on the lens side, and presents an interface to the camera that looks like a modern lens. This type of adapter "plays nicely" with the camera, so that all of its fancy metering modes including automatic flash will work properly. Using a purely mechanical adapter, without the chip, you may be limited to manual exposure with trial-and-error settings.
This particular objective (the CF N Plan Achromat 10X NA 0.30) delivers an excellent image over a 1.6 crop-factor sensor when used at 10X magnification with its rated 150 mm extension. It can also be extended beyond 150 mm to produce higher magnifications. However, it does not work well on shorter extensions because the corners develop severe astigmatism.
Some other lenses sacrifice a little resolution but are more tolerant of reduced magnification. One of those is the Nikon Finite Conjugate 10X objective listed by Edmund Optics. Here is that objective on 150 mm extension. Remove the extra tubes to reduce it to about 7X magnification.

Note that other objectives require different adapters and/or different extensions. Here is a Nikon CF M Plan 20X objective that is designed for use with a 210 mm tube and therefore works best with 200 mm extension.

The alternative approach, #2 in our initial list, is to use a new-style "infinite" objective and pair it with an appropriate "tube lens", such as a telephoto lens focused at infinity, to complete the image formation. (Historical note: the term "tube lens" comes from microscopy, where this lens sits at the bottom of what used to be an empty tube in the earlier finite designs.)
The Nikon CFI Plan Achromat 10X NA 0.25 objective has proved to be an excellent lens for this purpose. It delivers its rated magnification when used with a 200 mm telephoto or other tube lens, and it has been shown able to cover a full-frame sensor (36mm x 24mm) with high quality image when used with an appropriately wide telephoto. With an APS-sized sensor, this objective can also be pushed down to as low as 5X, by using rear lenses as short as 100 mm.
Here is an example configuration, used in conjunction with the Canon EF 55-200 mm f/4.5-5.6 lens set at 200 mm. In this case the adapter is 25mm to fit this objective and 52mm to fit the filter thread of this lens. One such adapter is currently HERE , or search ebay for m25 m52 adapter . (Other types of objectives and lenses may have different threads, so check before ordering.)
Note that there are no extension tubes in this setup. What may look like a tube is just the front section of the telephoto lens, which is zoomed out to its maximum length of 200 mm.

On smaller sensors, the objective can also be used with shorter tube lenses to give correspondingly lower magnifications. Here is an example using it in conjunction with an ancient 135 mm f/3.5 preset aperture telephoto, coupled to the camera with an M42x1 chipped adapter. This lens happens to have a 49 mm filter thread, so a 49 mm to 52 mm step-up ring is used to mate with the 52 mm adapter that fits the objective.

Note that in all these cases -- finite and infinite alike -- you have essentially no control over the aperture of the objective. The finite configurations simply do not have an adjustable aperture. In the infinity configurations, you can stop down the telephoto, but this will simply cause vignetting rather than increased DOF as you might expect.
Due to the high magnification and the wide aperture, microscope objectives give very shallow DOF -- typically around 0.01 mm for a 10X objective. As a result, most successful uses involve focus stacking, typically with a screw-driven rail or microscope focus block to provide small focus steps.
Traditionally focus stacking has been done using manual methods (turn dial, take picture, repeat many times), but automated systems have recently become available for affordable prices. This StackShot rail is one example:

It is always challenging to avoid motion blur due to camera vibration at high magnifications. Standard advice is to use flash, preferably with mirror lockup and second-curtain sync in a darkened room with a couple of seconds exposure time. It helps to use a powerful flash close to the subject, to take advantage of the ultra-short flash durations provided by modern consumer-grade flashes operating at much less than their rated output. If you have a Canon DSLR that supports LiveView with electronic first shutter curtain (EFSC), and you have a quiet environment, then you may also be able to use continuous illumination.
--Rik
Adding a couple of references for subsequent incorporation:
http://www.photomacrography.net/forum/v ... hp?t=12309
http://www.flickr.com/photos/johnhallmen/5379235010/
Further discussion: http://www.photomacrography.net/forum/v ... hp?t=23064
Edit: to fix typos
Edit CR: "Further Discussion" link
Edit: tweak wording and availability info
Edit: improve explanation of "tube lens".
Edit: add description of adapter for infinite objective to tube lens.
Edit: to clarify no extension tubes used with the telephoto.