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

• Karthiksrao
So, maybe the reason is that you cannot excite the required charge distribution with TE waves.In summary, the conversation discusses the creation of evanescent waves on a surface and how the presence of periodic structures can "free" these waves and allow them to propagate to the surrounding media. The effect is similar to sampling a signal below the Nyquist criterion. The conversation also touches on the excitation of surface modes, specifically surface polaritons, and how they can only be activated by TM waves due to the required charge distribution.

#### Karthiksrao

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.

## 1. What is the significance of periodic structures in wave propagation?

Periodic structures are important because they can cause the coupling of evanescent waves (waves that decay exponentially with distance) to propagating waves (waves that travel indefinitely). This phenomenon can be used to control and manipulate the propagation of waves, which has many practical applications in fields such as optics, acoustics, and electronics.

## 2. How does the coupling of evanescent waves to propagating waves occur in periodic structures?

The coupling of evanescent waves to propagating waves in periodic structures is due to the presence of a band gap - a range of frequencies or wavelengths in which waves cannot propagate. When an evanescent wave with a frequency or wavelength within the band gap encounters a periodic structure, it can couple to a propagating wave with a frequency or wavelength outside of the band gap, allowing it to travel through the structure.

## 3. Can the coupling of evanescent waves to propagating waves be controlled?

Yes, the coupling of evanescent waves to propagating waves can be controlled by adjusting the properties of the periodic structure. Factors such as the spacing and size of the periodic elements, as well as the refractive index of the material, can affect the band gap and thus the ability of evanescent waves to couple to propagating waves.

## 4. What are some applications of the coupling of evanescent waves to propagating waves in periodic structures?

The coupling of evanescent waves to propagating waves in periodic structures has many practical applications. For example, it can be used to create photonic and phononic crystals, which can manipulate light and sound waves, respectively. It is also used in sensors, filters, and other devices that rely on wave propagation.

## 5. Are there any limitations to the coupling of evanescent waves to propagating waves in periodic structures?

One limitation of this phenomenon is that it only occurs within a specific range of frequencies or wavelengths, determined by the band gap of the periodic structure. Additionally, the coupling efficiency can be affected by factors such as imperfections in the structure or the angle at which the wave is incident. These limitations must be considered when designing and using devices based on the coupling of evanescent waves to propagating waves in periodic structures.