Is periscope window mathematically possible? I was always wondering, why don't we have periscope windows?
What I imagine is a "light intake" on a roof, from which the light is concentrated into a straight long narrow tube that takes it to an underground flat where the light is dispersed into a fake window.
Is such design mathematically possible? What would determine the viewing angles of the fake window?
Maybe it could use Fresnel lenses to save on cost and weight. We would probably not get a clear picture of the outside world but at least lot of natural daylight.
 A: While the idea is quite cool there are several issues with it:


*

*I pictured the principle below. Only light nearly paralell to the periscop opening can pass through the periscope windows. All other light is absorbed in the walls. You could of course use mirror walls which would be very costly and

*These mirrors would become dirty quite quickly, just like normal windows. So you would need special shaft crawlers that clean the periscope from time to time.

*You would need normal windows on the outside of a building in order to capture light which would reduce the space for normal windows. You could place the openings on the roof but this would only suffice for one or two floors at maximum. And one story buildings can have normal windows in the roof. No need for periscopes.

*At last, as a rule of thumb, every optical part that light passes through reduces the light intensity by about 15 %. So even with the simple periscope pictured below you loose 30% of the light just by passing through the mirrors.


So to summarize: Classical daylight simulating lightbulbs are a much cheaper and practical solution. 

A: Depends on how narrow this tube is. Without moving optics that can track the sun, the tube must be as large as the collection area, and no kinds of tricks with mirrors or lenses can make it better. If sun-tracking optics are allowed then you can do much better, provided it's not cloudy.
Let's start with a practical example: sun tunnels which transmit light from a skylight into a fixture below:

The light is diffuse, so you get natural light, but no "picture".
The technical data gives a visual transmittance of 0.36 for the most efficient models, meaning 36% of the light hitting the collector on the roof ends up coming out the fixture at the bottom. That may not seem like much, but since our visual perception is logarithmic it's not nearly as bad as it seems.
As an example, the 14" model has a collection area of approximately 0.1 square meters. The illuminance outside on a sunny day is around 150,000 lux. That means the luminous flux at the collector is:
$$ 150000\:\mathrm{lx} \cdot 0.1 \:\mathrm{m^2} = 15000 \:\mathrm{lm} $$
15000 lumens. The visual transmittance coefficient of 0.36 means the luminous flux of the fixture, after the light lost in the optical system between the collector and fixture is:
$$ 15000 \:\mathrm{lm} \cdot 0.36 = 5400 \:\mathrm{lm} $$
That's a lot of light. For comparison, a typical "60 watt equivalent" LED is only about 800 lumens.
Could we improve on the performance of this sun tunnel with some optical system? Perhaps we'd like to collect more light from a larger area, while keeping the tube small. Can some arrangement of mirrors or lenses help?
There's an optical law called the conservation of étendue which is relevant. The best explanation I've found is in an XKCD what-if on starting a fire with moonlight:

Maybe you can't overlay light rays, but can't you, you know, sort of smoosh them closer together, so you can fit more of them side-by-side? Then you could gather lots of smooshed beams and aim them at a target from slightly different angles.

Nope, you can't do this.
It turns out that any optical system follows a law called conservation of étendue. This law says that if you have light coming into a system from a bunch of different angles and over a large "input" area, then the input area times the input angle equals the output area times the output angle. If your light is concentrated to a smaller output area, then it must be "spread out" over a larger output angle.

In other words, you can't smoosh light beams together without also making them less parallel, which means you can't aim them at a faraway spot.

So your first attempt at improving the sun tunnel might be to put a big lens at the top which focuses light on the smaller entrance to the tunnel.
If the sky is equally bright all over (it's cloudy), you have a problem: you're already collecting light from a 180 degree hemisphere. If you attempt to focus light down on a smaller point it must be spread out even more, beyond 180 degrees. But that means some of the light is turned around, going back at the sky, which doesn't help your objective of lighting the room below. So in this case, you simply can't cram any more light in the tube.
If it's sunny, maybe the optics should focus the brightest part of the sky, the sun's disk? This is a good idea, because now the input angle is only about 0.53 degrees. This means you have some margin to focus the incoming light without spreading it out beyond 180 degrees. But while physically possible, I'm not sure it's economically feasible since it would require expensive sun-tracking optics, and on a cloudy day it wouldn't work any better than the cheap variety with no optics at all.
A: What you describe is called a light pipe and they are already on the market. They are capped with a clear plastic dome and lined along their entire length with very shiny foil (which tends to minimize light loss), the outer surface of which is wrapped in insulation. The shiny lining means that they will reflect light along a curved path without needing flat mirrors. 
They do a pretty good job of bringing light through an attic space (one or two meters distance) and into a dark part of the living space below. However, if you want lots of light in the ground floor of a multi-story building, you need a large diameter tube, which intrudes into the living space of the rooms it traverses on its way down. 
This makes it practical only for applications where the tube extends down through an otherwise unoccupied space. 
