The top end of the periscope is inserted into the Newtonian's focuser and the eyepiece is inserted into the bottom end a few feet lower. The optics consist of a pair of diagonal mirrors at the top and bottom, and a pair of positive lenses between the mirrors. The upper lens makes the light rays parallel, like an eyepiece, and the lower lens re-focuses them.
For the upper lens I used a 300 mm telephoto. The front of the lens faces downwards so the light goes through the lens "backwards". I attached a 1.25" prism diagonal to the back end of the telephoto. The other side of the diagonal is inserted into the Newtonian's focuser. The lower lens is a complete 80 mm refractor with a focal length of 500 mm.
To support the optics I used a 5" diameter aluminum tube with a 1" square aluminum tube fastened to its side. The telephoto and the upper end of the refractor are mounted in the 5" tube. The lower end of the refractor is attached to the 1" square tube. The bottom end of the square tube is attached to the mirror box of the Newtonian.
I've done quite a bit of "eyeball" testing of the periscope this summer. I don't have any images or other measurements. As far as I can tell, the periscope does not introduce any additional coma or astigmatism, except that if the optics are badly misaligned the star images become astigmatic. The field appears to be perfectly flat. There is a small amount of false color on bright planets and the brightest stars. I believe this is caused by the prism diagonal at the top. I plan to replace this diagonal with a dielectric mirror diagonal. There is a noticeable loss of light, maybe 20-30 percent, which I was expecting because the telephoto is a little smaller than it should be and is over 30 years old so the coatings are probably not up to modern standards. To avoid loss of light the upper lens should be at least as "fast" (low focal ratio) as the main mirror. My mirror is f/4.5 and the telephoto is labeled as f/4.5 but the front lens is too small for this ratio so the actual speed appears to be about 4.7. With multicoated lenses of the proper size and dielectric mirrors you should be able to limit the light loss to 2 or 3 percent.
The optics seem to be very forgiving of misalignment. If not aligned correctly, you will lose some light and may not see the entire field, but other distortions do not appear until the misalignment is extreme. The telephoto is held in the tube with six screws, much like a finder scope, and the front end of the refractor is positioned with three screws, so it is easy to make precise adjustments. It is possible to get pretty good alignment simply by "eyeballing" through the refractor, and a laser collimator can be used to improve on this.
Since the lenses that I used were of different focal lengths (300 mm at the top and 500 mm at the bottom), there is increased magnification by that ratio and consequently a smaller field of view. If you wanted to maintain the full field of view you could use lenses of equal focal length, but large fast lenses tend to be expensive. Several manufacturers make 300 mm f/4 prime (non-zoom) camera lenses and these cost around $1100 new but I've seen a lot of used ones advertised for around $400. I think these would work well with most telescopes. Possibly an objective from a binocular or spotting scope or a finder scope could be used, but I'm not sure whether the optical quality would be adequate.
You should be able to fit a periscope onto almost any large Newtonian with minor modifications. My periscope weighs about 15 lbs. so the main scope needs to be re-balanced. Another consideration is that the focal plane has to be far enough out to meet the periscope's focus. You will need to provide an attachment point on the mirror box for the bottom of the periscope.
One interesting benefit is that images are erect and correct, not mirror reversed. I mounted a 1x finder and a RACI (right-angle correct image) finder on the periscope, so the views through all three are in the same direction, a great benefit for star hopping. Since you use a diagonal at the eyepiece, you can rotate it to provide a comfortable viewing position. You don't need an adjustable chair. If the length of the periscope is set to provide standing-height viewing at zenith, all you need is a 24" stool for the lower elevations.
It is easy to tailor the length of the periscope to provide exactly the eyepiece height you want. Moving the refractor up and down doesn't change the view until you reach a limit where the field of view starts to narrow. The maximum length of the periscope depends on the diameter of the optics. As an example, my periscope has a 64 mm diameter upper lens and 80 mm lower lens. I want to maintain a 46 mm diameter field at the eyepiece. Picture a light ray traveling from the top edge of the upper lens, through the center of the lower lens, and hitting the bottom edge of the focal plane. In my case the maximum separation of the lenses works out to 500 / 23 x 32 = 696 mm or about 28". At this separation there will be a "soft" edge of the image at the point 23 mm from the optical axis. Add to this about 10" for the length of the telephoto plus diagonal and 17" for the refractor, and the total length of the periscope can be up to 55", measured from the upper to the lower eyepiece locations. With larger optics it could be much longer.
Given the excellent performance of my periscope using off-the-shelf optics, I am convinced that it is possible to build periscopes that would satisfy the most demanding observers. When periscopes become commonplace and eliminate ladders, giant Newtonians should become a lot more popular.
The photo below shows the upper part of the periscope attached to my 20" Newtonian. If anyone is interested, I will post more details in the Telescope Making forum.