electromagnetism - Does the wavelength always decrease in a medium?

I was studying a GRE Physics Test problem where optical light with a wavelength of 500 nm travels through a gas with refractive index $n$.

If we look at the equations for wave motion and index of refraction

$$c=\lambda_0\nu\quad\text{(in vacuum)}$$

$$v = \lambda\nu\quad\text{(in medium)}$$

$$n = c/v$$

we see that, if the frequency is constant, the wavelength decreases in the medium compared to vacuum. Is this a consistent property at all frequencies and for all mediums with refractive index real and greater than 1?

Are there dielectrics which change the frequency (still for n > 1), and is there an example of that?

Answer

The index of refraction of a material can be less than 1 at high frequency, this is called "anomalous dispersion" and it happens as you cross an energy level of certain materials. It means that the phase velocity of light of a certain frequency is higher than c.

If the index of refraction is constant, as it is for long wavelengths, n has to be bigger than 1 to avoid superlumimal communication.

The principle of energy conservation in a static environment forbids a frequency shift for a photon, since this would add energy or take away energy, and nothing in the medium is changing with the right frequency to do that. But light entering a moving medium shifts frequency. photons can combine to make one of double the frequency in a strong light beam in a nonlinear medium, and this corresponds to making higher harmonics of the classical field.