Plasmonics Part II

Sorry for the long wait for this second post in the plasmonics series, but I got extra busy with school and the lab.

Anyways, before I start talking about what plasmons are and what they are good for, I want to thank “plasmonicsfocus” for pointing out a fact I failed to mention in my previous post.  The 10-6 transmission factor does not exactly represent the light passing through the tip.

See, classical diffraction theory predicts that the transmission of electromagnetic radiation through an aperture, of radius d, in a perfectly conducting plane obstacle should be given by,

T = (8/27)(pi)2(kd)4 + O[(kd)6]

So when the radius of the aperture is much smaller then the wavelength of the incoming wave, (d<<lambda) the transmission of energy through the aperture dies like (d*lambda)-4.

The catch here is that this analysis assumes the transmission is measured far from the aperture (far field diffraction). Far-field diffraction theory neglects those terms in the radiation field that die off faster than x-2 in the distance from the aperture.  So there is in fact, more energy that is interacting with the sample before the wave disperses to the far field.

For example, part of the transmitted energy emerges in an evanescent wave that travels along the reflecting interface between the media. Now it is possible to couple to this wave, despite the fact that its intensity diminishes exponentially with distance from the interface, and this allows one to create surface plasmons in thin metal films.

However I get ahead of myself.

Plasmons are plasma oscillations, or you could think of them as charge density waves.

Simply imagine a sphere comprised of many discrete, evenly-spaced positive charges, which we can approximate this as a positive charge distribution.  Now imagine a free electron cloud hovering just on top of the positive charge distribution, but not in contact.  The electrons are interacting with each other, and can be approximated as a negative charge distribution.  You have arrived at what is known as the Jellium model in metals.  These are plasmons in metal, and when the electrons move around, they are pulled back toward the positive charge distribution.  We assume they are not energetic enough to escape the E-field by the positively charged nuclei.  Thus they end up oscillating over/about the positive charge distribution.

In case you didn’t want to read all that, suffice it to say that plasmons in metal are basically vibrational modes of the electron gas density oscillating about the metallic ion cores.

The oscillation frequency of the plasmon (wp), can be derived fairly easily after making the assumption that the E-field confining the electrons is harmonic.  Note how in the bottom right of the following picture, the frequency/waveguide (w/k = w*lambda) asymptotically approaches c.

Now surface plasmons describe the special case in which the charges are confined to the surface of the metal. In this case, the electric field is strongest in the plane of the metallic surface.  This is because the strength of the E-field dies off exponentially as you get further away from the charge distribution.

Plasmons confined to a plane do not radiate light, but when the local planar symmetry is disturbed, they can.  So unless you have a perfectly planar sample, and I really mean perfectly planar, the plasmons can radiate.  Surface plasmon-mediated emission from defects in metal surfaces has been observed, for example here.  These features have been described as “plasmon flashlights”; they are a localization of photon emission in a region generally experiencing non-radiative, collective oscillation of the surface electron density. Hence, It is possible to design structures that take advantage of this emission of surface plasmon radiation.

Lets take a look at a little more rigorous derivation of plasmon-light coupling:

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NOTE: The slides in this post were made by Jim Shuck, LBL.

That is really the main point to take from this discussion.  Surface plasmons can oscillate at relatively low frequencies for relatively small wavelengths.  That’s why if you can turn light into plasmons (couple them), you can do sub-diffraction imaging.

Next time I’ll wrap all this up by talking about plasmonic nano-antennas.  Until then, if you’re interested you might want to look at these papers.

Toward Nanometer-Scale Optical Photolithography: Utilizing the Near-Field of Bowtie Optical Nanoantennas

1. shai says:

Hi Mitch,
We are interested in plasmons and diffraction limit.Did you post Plasmonic part 3 anywhere? If not, where can we read about it?

Thanks alot

2. Dima says:

Hey Mitch,

I didn’t understand why do you need local planar symmetry to get plasmons radiated? Could you please explain it? It’s something with destructive interference of plasmons radiation when we have planar symmetry?

Thanks a bunch 🙂