  The speed of light ($c$) in a vacuum is independent on a choice of frames to
describe the propagation, according to the theory of relativity. We consider
how light is characterised in a material, where the speed of light is different
from that in a vacuum due to the finite dielectric constant. The phase velocity
in a material is smaller than $c$, such that the speed of a moving frame can be
larger than the phase velocity, such that the frame can move faster than the
speed of light in a material. Consequently, an unusual Doppler effect is
expected, and the wavelength in the moving frame changes from the red-shift to
the blue-shift upon increasing the speed of the frame. The corresponding energy
of the light also changes sign from positive to negative, while momentum is
always positive, leading to the changes of sings for the phase velocity and the
helicity. In a graded index fibre, where the exact solution is available, even
more complicated phenomena are expected, due to the finite effective mass of
photons. Upon the increase of the energy gap, generated by optical confinements
and optical orbital angular momentum, the effective mass of photons increases.
If the gap is large enough, momentum starts to change the sign upon increasing
the frame velocity, while the energy of photons is always positive. In this
case, the phase velocity diverges if momentum is in agreement with the fame
velocity. Contrary to the unusual behaviours of the phase velocity, the group
velocity is always below $c$. This thought-experiment might be useful to
consider the insight for the polarisation sate of light.
