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?

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i know the answer is yes (trivially by definition), i guess I'm looking for microscopic insight--- seems like different materials would do different things to the wavelength--- but i guess it's just a factor of how much they shrink it –  Timtam Aug 17 '11 at 16:43
What insight are you after? The fact that the speed is lowered should be completely obvious. So do you want to know how speed of propagation corresponds to wavelength? Or do you want understand why the speed is lowered microscopically (this has to do with the fact that photons interact with matter and thereby slow down)? –  Marek Aug 17 '11 at 17:08
i think i understand the fact that this is trivially true, i'm basically interested in various different microscopic examples: e.g. two different materials, in different phases having the same effective refractive index, how would a gas compare to a solid to a liquid etc. sorry if this is too general a question and also interested in what dielectrics would change the frequency of a light wave and how that would work –  Timtam Aug 17 '11 at 17:10
well, there are many possible effects but every calculation eventually boils down to one complex number that tells you how long the photon spends at a given atom (or molecule, or lattice, or whatever) and how much it scatters. So that real part is related to index of refraction and complex to the attenuation of the signal. To obtain the full picture one then integrates over all photons and all scattering places in the material. –  Marek Aug 17 '11 at 17:19

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.

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As for your second question, dielectrics which change the frequency, any medium which changes the frequency must be a nonlinear medium. There are many of them. They do not change the frequency continuously, but rather they generate higher order harmonics, an integral number times the original frequency.

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