# Waves in periodic structures - Coupling of evanescent waves to propagating waves

Hi,

On a surface evanescent waves are created when total internal reflection occurs. However when this surface has periodic structures of appropriate periodicity (gratings, photonic crystals) the evanescent waves are "freed" from the surface and they propagate to the surrounding media. I am not able to intuitively see how a surface feature can change the intrinsic feature of a wave.

By intrinsic feature, I mean the inplane wave vector of the wave. For evanescent waves, the inplane wave vector component will be greater than magnitude of wave vector. How does the presence of a surface periodic feature, decrease the inplane wavevector to a value lower than the magnitude of wave vector, so that it gets converted to a propagating wave ?

Can somebody throw insight to this ?

Thanks a lot!

I am not sure if I understand your question. When you have a guided wave due to total internal reflection, the in plane wave vector is larger in magnitude than the wave vector in free space, so that propagation is not possible.

If you have a periodic perturbation at the surface, you will scatter a little bit at each of them. You can think as if these perturbations sample your guided wave periodically.

The condition for radiation now is not the wave vector of the guided wave, but the "sampled wave vector". Imagine that the periodicity is exactly equal to the wavelength of the guided wave. In this case all the perturbations will be in phase, so their "effective wave vector" is zero.

Is similar to what happens if you sample a signal below Nyquist condition, you recover an alias with a different frequency. In this case it has a different wave vector that can propagate outside.

Yes. I was aware that the 'effective' wave vector needs to be considered. But I wasn't sure why this should be so.

When you say you can view the scattering from the perturbations as sampling, do you mean to say that the scattering causes a separate mode to be generated, in addition to the 'original' guided wave ? (similar to polarization in material medium giving rise to an additional field in addition to the incoming wave)

Thanks

The guided wave will generate fields in the surface with an spatial variation given by the inplane wave vector k_i. If k_i>k_o, the wave vector outside, propagation will not be allowed because you cannot find any direction such that the projection of k_o onto the surface matches the inplane wave vector k_i

If you have a periodic perturbation with period d, you can think of the surface as being composed by an array of "radiators" with a given field distribution, with a separation d from each other and with a phase shift between them given by d*k_i.

If you chose d such that d*k_i=d*k_o*cos(theta)+n*2*pi, then propagation is allowed in the direction theta. What I meant by sampling is that you sample your inplane wave at distances n*d. The effect is similar to what you get when you sample a given frequency below the Nyquist criterion.

In a related query, surface modes (like surface polaritons) can only be excited for TM waves.
Any idea what this could be due to ? What characteristic of an incoming TE wave makes it impossible for activating surface modes ?

Thanks!

I guess that to support the charge distribution in a surface plasmon polariton you need an electric field in the direction of the propagation. TE modes only have transversal electric fields.