Surface Plasmon Polaritons and V-Grooves

1. Surface Plasmon Polaritons and V-Grooves Adam Blake August 29, 2014

2. Simple Harmonic Oscillator • The motion is repetitive in time • Something might oscillate once in an amount of time T. Angular frequency w makes it so that when time advances by T, the sine function describing the oscillation advances by 2p.

3. Waves • Waves travel and therefore oscillate both in time and in space • In addition to oscillating in time, waves also oscillate in space. The wavenumber k makes it so that when you advance in space by l, the sine function describing the wave advances by 2p.

4. Dispersion Relation • w and k are related via a dispersion relation • w is written as a function of k • A very simple dispersion relation is that for light propagating in a vacuum: c = lf or w = ck

5. Surface Plasmon Polaritons • Maxwell’s Equations can be solved in the case of a metal – dielectric interface (see Raether) • The solution gives standing waves of charge oscillations along the interface: surface plasmonpolaritons (SPPs) • Name derived from more accurate quantum formulation: a quantum of surface charge oscillation (a plasmon) couples to a quantum of light oscillation (a photon) and the result is a hybrid of each • Intense, highly localized (evanescent) fields result. These fields decay exponentially with distance from the interface. • The resulting dispersion relation, which relates the wave number of an SPP to its angular frequency is: k = (w/c)[e1e2/(e1+e2)]1/2

6. Surface Plasmon Polaritons • In order to excite SPPs with light, both the wavenumberand frequency of the light must be matched to those of the SPP • This corresponds to the light line intersecting the SPP curve – but it doesn’t. For a given frequency, the SPP always has a larger wavenumber (smaller wavelength) than light. • If we can get light to couple to the SPP, this means that light of a given frequency can excite an effect (an SPP) of a smaller wavelength, thereby sidestepping the diffraction limit.

7. Surface Plasmon Polaritons • Two methods for coupling light to SPPs: • Attenuated Total Reflection (ATR): couple light on one side of a metal film to an SPP on the other side of the film • Use a grating to reduce the wavenumber of the SPP • The second method will be of interest in this presentation.

8. V-Grooves • Three types of resonances are observed • Geometric resonance: the groove acts like a “leaky” resonantor • SPP resonance: oscillating charge density waves travel and reflect from the adjacent groove, creating a standing SPP mode • Channel plasmonpolaritons propagate down the channel, which acts like a waveguide

9. V-Grooves • Channel plasmonpolariton is investigated by placing fluorescent beads (100 micron diameter) in the channel

10. V-Grooves • The SPP resonance is understood in terms of round-trip distance of the charge density waves • SPP resonance should therefore depend on the period of the grooves • It should not depend on the angle of incidence of incoming light (and it doesn’t)

11. V-Grooves • The SPP resonant frequency has a complicated dependence on groove structure • Hard to say exactly where the reflection of the charge density wave take place

12. V-Grooves • As depth increases, SPP mode shifts to lower energy • As groove angle increases, SPP mode shifts to higher energy

13. V-Grooves • As incident angle increases, SPP mode doesn’t move appreciably • As period increases, SPP mode moves to lower energies

14. Coupled Oscillators • Consider two mass spring systems with a spring connecting each mass • If we displace one and let it go, it will oscillate. Over time, its oscillation will decrease and the other’s will increase, and then vice versa later on. Energy exchanges back and forth between the oscillators. • There are normal modes for which each one oscillates with a constant amplitude and frequency. Can you describe the two normal modes of the coupled masses? • The frequencies of the normal modes are different than the frequency of each uncoupled oscillator. The stronger the coupling, the greater the separation in frequencies.

15. Coupled Oscillators • Two-level emitters are oscillators as are SPPs. They can couple to each other via the electromagnetic fields that they emit. • The resulting system is a hybrid of the two. When driven near where the uncoupled resonances overlap, they demonstrate normal mode splitting. • The minimum splitting is called the Rabi splitting. The horizontal line is the emitter transitionfrequency. The diagonal line is actually partof the SPP dispersion curve. The dashed linesare the upper and lower polaritons.

16. Coupled Oscillators • The stronger the coupling, the greater the Rabi splitting between the upper and lower polaritons • These curves demonstrate avoided crossing. The horizontal line is the emitter transitionfrequency. The diagonal line is actually partof the SPP dispersion curve. The dashed linesare the upper and lower polaritons.

17. V-Grooves: Our Objective • We know that the SPPs will couple to the emitters that we place in the grooves • Strong fields in the grooves • Localized or spread out? • Can we couple one groove to another? • Would this introduce additional structure in the upper and lower polaritons?