electromagnetism - Exciting Surface Plasmon-Polaritons with Grating Coupling

I'm very new the topic of SPPs and have been trying to understand this particular method of exciting surface plasmons using a 1D periodic grating of grooves, with distance $a$ between each groove. If the light incident on the grating is at an angle $\theta$ from the normal and has wavevector ${\bf k}$, then apparently if this condition is met:

$\beta = k \sin\theta \pm\nu g$

where $\beta$ is corresponding SPP wavevector, $g$ is the lattice constant $2 \pi/a$ and $\nu={1,2,3,...}$ then SPP excitation is possible.

I haven't ever really had a formal course in optics, so my question is where this condition comes from. It seems like Fraunhofer diffraction, but only for the light being diffracted at a $90^\circ$ angle to the normal. Most books don't state how they get this result, they just say it's because of the grating "roughness" which really confuses me.

Any help would be greatly appreciated.

Answer

Don't worry, I did research in surface plasmons and even then I was more than a year into it before I truly understood, on an intuitive level, how the light gets a 'kick' from the grating. You are correct that it is diffraction at a 90 degree angle to the normal, but there is an easier way to think about it.

You say you've never taken a formal course in optics so I'll talk a little bit about diffraction gratings in general. You might have come across one before and know that if a beam of light hits it, it is diffracted into several different beams. Transmissive diffraction gratings are what one usually encounters in high school physics so I'll illustrate one below:

The numbers at the end of each beam are known as the order $\nu$ of that beam. The grating equation is $d(\sin\theta_i + \sin\theta_o) = \nu\lambda$, where d is the distance between lines of the grating, $\lambda$ is the wavelength of the light, $\theta_i$ is the angle of incidence, and $\theta_o$ is the angle of the outgoing beam. In the above illustration, $\theta_i$ is zero.

Next we consider a reflective grating (for example a piece of metal with 1D periodic grooves), as in the following illustration:

The same mathematics govern this situation as well. You'll notice the $\nu=+2$ order being very close to grazing the grating surface. Adjusting the angle of incidence a little bit would cause it to do so. In that case, it would have the required wave vector to launch a surface plasmon, which is the phase matching condition that you started out with. You get the $\beta = k\sin\theta\pm\nu g$ when you convert the grating formula to wave vectors (reciprocal space) which I'm too lazy to do right now.

I suppose you could technically say that the light got a momentum 'kick' from the $\nu=-2$ order being launched in the opposite direction, but thinking of it as the light getting a 'kick' is really misleading in my experience.