What is the limit to how reflective a mirror can be? Wikipedia states the reflectivity of a Bragg mirror can be "99.999% or better" given you can control the wavelength, incident angle, and light polorization. My question is how much better?
This question is in the context of something that could be built with modern day technology. The formula for reflectivity in a Bragg mirror given here, seems to indicate that an linear increase in the number of layers results in an exponential increase in reflectivity. Specifically, if the reflectivity is R and the number of layers is N, and $n_1$, $n_2$ are the refractive indices of the 2 materials with $n_1\gt n_2$ then
$$\frac{1}{1-R} \propto \left( \frac{n_1}{n_2} \right) ^{2N}$$
but I guess this relationship would break down at some point. There is nothing stopping us from making a mirror 1000 layers thick, but I doubt it would have $R \approx1-10^{-1000}$. So why specifically can we not build a mirror this reflective, and where is the limit? Surface imperfections, something quantum level?
 A: All such limits are due to engineering; MIT has announced a perfect mirror!
With the introduction of meta-materials and photonic crystals, new opportunities are available for exploitation.
Here is the abstract from the (pay walled) scientific paper, Observation of trapped light within the radiation continuum, Nature, 2013:
The ability to confine light is important both scientifically and technologically. Many light confinement methods exist, but they all achieve confinement with materials or systems that forbid outgoing waves. These systems can be implemented by metallic mirrors, by photonic band-gap materials, by highly disordered media (Anderson localization) and, for a subset of outgoing waves, by translational symmetry (total internal reflection1) or by rotational or reflection symmetry. Exceptions to these examples exist only in theoretical proposals. Here we predict and show experimentally that light can be perfectly confined in a patterned dielectric slab, even though outgoing waves are allowed in the surrounding medium. Technically, this is an observation of an ‘embedded eigenvalue’9—namely, a bound state in a continuum of radiation modes—that is not due to symmetry incompatibility. Such a bound state can exist stably in a general class of geometries in which all of its radiation amplitudes vanish simultaneously as a result of destructive interference. This method to trap electromagnetic waves is also applicable to electronic and mechanical waves.
A: When a photon interacts with an atom three things can happen:
1.elastic scattering, the photon keeps its energy and changes angle


*inelastic scattering, the photon gives part of its energy to the atom and changes angle

*absorption, the photon gives all its energy to the atom and the atom gets excited, the valence electron that absorbs the photon will move to a higher energy level as per QM
It is not absorption in case of a mirror. 
It is elastic scattering, Rayleigh scattering. That is the only way the photons' energy and phases does not change. This is how a mirror image is built. 
Even though the surface is rough on the quantum level, if the wavelength of the light (visible) is big enough relative to the hills and valleys in the surface, the surface will seem smooth (to the light).
That is how the surface will act as smooth and will reflect. 
The reason why all the photons are reflected to the same angle is the molecular structure and atomic structure on the QM level.
Of course the reflections will not be 100% perfect because there is always destructive interference. 
Now in case of a Bragg mirror, the layers are so that there is a thin layer and then a thick layer and then a thin and a thick.

The difference between the layer's thickness has to be so that when the photon passes through them, its path will have a difference between the layers exactly of one wavelength.
The reason for that is this way the reflected photons create constructive interference, and the mirror will work more effectively.
Now the wavelength of visible light is 400-700nm. So you say we should add 1000 layers, that should be in this case 2000 layers (thin, thick), and so that mirror would be more than 10^3*4*10^-7m=4*10^-4m (more than the thick layer 1000 times with the lowest visible wavelength).
That is 0.04cm.
Now the problem with metal is that it is a conductor, that is why it cannot so much refract, but it always reflects.
There is a limit how deep light can go into metal.
0.04cm is really too much for the visible light to go into the metal, but if you wanted even more perfect level of constructive interference, you would have to increase the layers and light could not go into those layers because the metal is too thick.
