1/ Mirrors are usually made out of metals, which don't allow electric field inside their bulk and therefore reflect light. But can you make a mirror out of transparent materials?

It turns out that you can, and the physics of "transparent mirrors" has an analog in graphene.

2/ The idea is to make a stack of different transparent materials. At each interface, there is a finite probability that the light will be transmitted vs. reflected. The more interfaces there are, the harder it is for light to pass through them all.

3/ A common situation is that one of the two materials is just air: at each air/material interface the light can be reflected back. This is why a roll of saran wrap is shiny like a mirror (it has many plastic-air interfaces), even though each layer is transparent.

4/ I will say that the math of this problem is far from obvious! There are infinitely many paths that a ray of light can take through a material, reflecting back and forth at each interface. And light is a wave, which can exhibit phase cancellation at each interface.

5/ This problem is actually over 200 years old, but it wasn't until the 1990s that it was really solved, by Berry and Klein:
https://iopscience.iop.org/article/10.1088/0143-0807/18/3/017/meta

The key is to make an analogy with Anderson localization, which describes the non-propagation of waves in a disordered landscape.

6/ In graphene, electrons behave like light: their energy is proportional to the inverse of their wavelength. Disorder in their velocity (as happens, for example, when two graphenes have a spatially-random twist between them) acts like a disordered index of refraction.

7/ The key difference is that in graphene an electron cannot reflect when it comes to an interface at a perfectly perpendicular angle.

This is called Klein tunneling, or the "Klein paradox". It arises because the backscattered wave function is orthogonal to the transmitted one.

8/ What this means in graphene is that if you have a line-like pattern of disorder, electrons at normal incidence are transmitted, while light at an angle is reflected.

You've made a "mirror" that is shiny when seen at an angle and transparent when you look directly at it!

9/ Together with my student @The_Correlator, we have examined this effect in twisted bilayer graphene in a new paper: https://arxiv.org/abs/2008.05481 

It turns out that the physics is much richer still, and may lead to a new method for filtering electrons by energy or by spin.

10/10 Congratulations to @The_Correlator on his first paper!