the local electric field vector), as symmetry considerations forbid this. Rather, circular birefringence only appears when considering nonlocality of the material response, a phenomenon known as spatial dispersion. Nonlocality means that electric fields in one location of the material drive currents in another location of the material. Light travels at a finite speed, and even though it is much faster than the electrons, it makes a difference whether the charge response naturally wants to travel along with the electromagnetic wavefront, or opposite to it. Spatial dispersion means that light travelling in different directions (different wavevectors) sees a slightly different permittivity tensor. Natural optical rotation requires a special material, but it also relies on the fact that the wavevector of light is nonzero, and a nonzero wavevector bypasses the symmetry restrictions on the local (zero-wavevector) response. However, there is still reversal symmetry, which is why the direction of natural optical rotation must be 'reversed' when the direction of the light is reversed, in contrast to magnetic Faraday rotation. All optical phenomena have some nonlocality/wavevector influence but it is usually negligible; natural optical rotation, rather uniquely, absolutely requires it. The phase velocity of light in a medium is commonly expressed using the index of refraction n, defined as the speed of light (in free space) divided by its speed in the medium. The difference in the refractive indices between the two circular polarizations quantifies the strength of the circular birefringence (polarization rotation), . While is small in natural materials, examples of giant circular birefringence resulting in a negative refractive index for one circular polarization have been reported for chiral metamaterials. The familiar rotation of the axis of linear polarization relies on the understanding that a linearly polarized wave can as well be described as the superposition (addition) of a left and right circularly polarized wave in equal proportion. The phase difference between these two waves is dependent on the orientation of the linear polarization which we'll call , and their electric fields have a relative phase difference of which then add to produce linear polarization: where is the electric field of the net wave, while and are the two circularly polarized basis functions (having zero phase difference). Assuming propagation in the +z direction, we could write and in terms of their x and y components as follows: where and are unit vectors, and i is the imaginary unit, in this case representing