Prisms:
Prisms are, as you might say, a necessary evil in optical design. It if weren’t for our unreasonable prejudice for seeing the world right side up and right way around we wouldn’t need prisms in binoculars and spotting scopes at all.
You might remember the experiments with chickens. Chickens were fitted with spectacles that inverted and reversed their view of the world. They stumbled around for a few days, but most of them quickly adapted to the new view, and were soon functioning normally (as normally as a chicken can. Evidently the chicken brains “remapped” the visual space for the new conditions. If chickens can get used to an upside down, reversed view of the world, certainly we ought to be able to. Amateur astronomers (in a not often documented example of convergent evolution) demonstrate this same ability, though with the popularity of cat scopes, increasingly they only have to deal with an image that is reversed right for left.
The rest of us, though, continue to show a strong preference for “erect” images…which is another way of saying right side up, right way around…and all modern binoculars and almost all spotting scopes today employ a prism erector system.

Prisms accomplish this feat by “bending” or, more properly, reflecting light around a series of corners. When light strikes a glass-air boundary at an angle, some of it is always reflected back, while the rest passes through. Think of standing outside trying to look through a window into the room beyond. The amount that is reflected back is determined by the angle at which the light strikes the glass-air boundary (angle of incidence), and by the index of refraction of the glass. Index of refraction is a measure of how sharply glass bends light that enters and leaves its substance, and is closely related to the density of the glass. What is of interest in prisms is what happens when the light that is already inside hits a boundary. There is an angle, for any given glass, at which all of the light that strikes the surface from the inside is reflected back: producing, in the ideal, total internal reflection. 
(Total internal reflection is useful not only in prisms. It is what makes light pipes and optical cable work as well.)
A porro prism erector system works, in theory, very efficiently because all of the light that will form the image strikes each air glass boundary at an angle that is shallow enough to produce total internal reflection. No light is lost to the image.



In fact, inexpensive porro prism erecting systems use a glass (Bk7) which has a index of refraction that is near the critical limit for what will work…so that light from the center of the prism is reflected back properly, while a small portion of the light from the edges of the prism leaks out through the glass-air boundary. If you look at the exit pupil (the small circle of light that floats in the center of the eyepiece when you hold the eyepieces away from your eyes) of a binocular that uses Bk7 glass prisms, you can see that edges are shadowed, with an arc of the perfect circle being cut off on two sides.
Increasing the density of the glass used (generally using a glass called BAK-4) cures that problem, and BAK-4 porro prism erecting systems are among the most efficient available, passing well over 90-95% of the light that enters them.

Porro prism erecting systems have other unique features, of course, primary among them the fact that the light entering the system is “offset” from the light leaving by the width of the system. This gives porro prism glasses, and most spotting scopes, their characteristic dog-leg, or “z” shape. Because the objectives of porro binoculars are always either further apart or closer together than the human eyes looking through them, they alter the perspective of the view, producing either an increased sensation of three dimensionality (further apart), or a decreased sense of dimensionality (closer together). Porros also alter our perception of the size of the object, and our distance from the object, for the same reasons.
Very early in the development of binoculars, roof-prisms, which keep the exiting light in line with the entering light, were developed. The barrels of roof prism binoculars are straight, and the sense of dimensionality, size, and distance relationships are much more natural.
(Besides the handling advantages that roof-prisms offer, about with more in a future article, this difference between in the apparent size of the object being observed for porros and roof binoculars of equal power probably did more to contribute the popularity of roofs among birders than any other factor. The bird simply looks bigger through roofs. It is not bigger. Projecting the images of an 8x roof and an 8x porro and measuring the size of the object shows that they are identical…however it is very difficult to convince our brains that we are seeing the same size bird through both glasses. I have a friend who has quantified the effect, associating the perceived difference in magnification with how far apart the objectives are. Inverted porros, by the way, with the objectives closer than the human eyes, show this effect at its most dramatic.)



A roof prism takes some special attention if it is going to work at all. In the most common roof prism, the Schmidt Pechan, one of the glass-air boundaries is at an angle such that it can not produce total internal refection for any of the light that strikes it. Most of the light would pass right through. This problem is solved by “coating” that surface of the prism with a mirror coating: a thin layer of reflective metal which turns all the light that strikes it (producing the “roof” in the “roof prism” design). Silver roofs were used until aluminum became readily available (aluminum has the advantage of not tarnishing or loosing its reflectivity as quickly as silver). Most high quality roof-prisms have returned to the use of silver because it reflects light more efficiently, and because the silver does not tarnish in a binocular that is waterproof and filled with inert nitrogen gas anyway.
Unfortunately, even the best silvered surface is not as efficient as a proper glass-air bounday at the correct angle. Some of the light that strikes it always leaks through: up to 15% for an aluminized prism. Therefore, until recently, no roof-prism binocular was as bright as a porro prism binocular of equal quality in side by side testing.
Then too, when light reflects back from a mirror, its phase is changed. One way of picturing this is to say that light waves come in bundles, with the waves arrayed in all directions, and crossing at the center of the bundle. When light reflects from a mirror, it becomes partially polarized, with more of the waves aligned, shall we say, horizontally than vertically. Some energy (brightness) and some information (resolution) is lost, and even more is lost when these partially polarized rays interact (interfere) as they are recombined in the image.


Untreated roof prisms, therefore, are slightly dim, and slightly soft, when compared to porros of equal quality.
As mentioned above though, many birders were willing to live with the difference in optical performance for the handling and perspective advantages of roofs. (And many more were convinced there could not be an actual difference in performance, or if there were one, it had to favor the more expensive roof prism design.)
There is another type of roof prism, called the Abbe-Konig, which can be used in binoculars. The Abbe-Konig is typical of designs from Zeiss, in particular, and, because of its size (primarily length) compared to a Schmidt Pechan, produces binoculars that are noticeably taller than other roofs (think of the 7x42 Zeiss Classics).



The Abbe-Konig has the advantage of not having a surface that requires mirroring: all the light is turned by total internal reflection. That means that the Abbe-Konig design is inherently brighter than any normal Schmidt Pechan system of equal quality in side by side comparisons.
Abbe-Konig systems still suffer from phase distortion however. The cause of the distortion and phase shift is slightly different than in the Schmidt Pechan, but, as the light from various paths in the prism recombines in the image, phase issues are no less troublesome, as they limit both brightness and resolution.
In the 1980s, Zeiss optical engineers developed a coating for prism surfaces. It is similar to the anti-reflective coatings used on lenses in that it consists of multiple thin layers of high index of refraction minerals, and it eliminates the phase distortion in the Abbe Konig design. Other manufacturers soon applied similar coatings to the Schmidt Pechan design to improve both the brightness and the resolution of images formed with their erector systems.
This is generally called “phase coating” and allows at least the Abbe-Konig design (which has, remember, no mirror leakage) to equal, for the first time, the efficiency and performance of the best porros (above 90%). The improvement in the Schmidt Pechan systems was more subtle, primarily, in my experience, consisting of increased contrast and sharpness.


Finally, within the past few years, high end makers have experimented with replacing the mirrored surface of the Schmidt Pechan system with very complex multi-layer coatings of high and low index of refraction minerals. In order to accomplish something approaching total internal reflection, the spectrum of visible light must be broken up into a high number of sub-spectrums, and coating layers developed for each division.
The first efforts used over 30 layers of coatings for a 2-3% increase in efficiency. More recent coating systems employ over 70 layers, in multiple applications, to raise the efficiency of Schmidt Pechan systems to equal or exceed that of the best Abbe-Konig and porro systems.

So there are now three choices for high efficiency prism erector systems: Porro prisms, Phase coated Abbe-Konig systems, and Phase coated Schmidt Pechan systems with dielectric “mirror” coatings.
Undoubtedly, just as multi-coating became a standard, and then Phase coating spread to cover the majority of roof prisms, dielectric mirror coatings for Schmidt Pechan prisms will proliferate, and will work their way down into less and less expensive binoculars over the next 5 years.
Remember though, that just as there are more and less efficient multi-coatings, and less and more efficient phase coatings, there will be, for some time to come, real differences in what manufacturers are calling dielectic mirror coatings. The key, as I understand it, is the number of segments the spectrum is broken down into, and the resulting number and tuning of the individual layers of coating. Not all dielectric mirrors are created equal nor will they be anytime soon. Performance will vary between manufacturers.
Which prism a designer uses depends on both price and performance goals. The best performance at the lowest cost can always be produced using simple, high-quality porro systems. Phase coated Abbe-Konig systems are appropriate for full sized binoculars in the upper price range (reaching down into the mid-range with the Zeiss Conquest 40s), and phase coated Schmidt Pechan systems with dielectric mirrors are currently appropriate for pricy full-sized, mid-sized, and even compact (pocket sized) binoculars.
In the end, the type of prism system is not as important today as its efficiency.
Or at least that's the way I see it....