## Coupling between plasmon and photon ### Comparison between SPPs and LSPs Let's first review the similarities and differences between SPPs and LSPs. - They could both go beyond the diffraction limit. - Which means they have evanescent fields. - And allow fields to be concentrated. - Losses would increase with confinement. But for SPPs, they are: - Essentially propagating electromagnetic waves, and - Non-radiative, or dark (theoretically, in practical there are losses), and - We have the momentum (wavevector) $\beta=k_{x}=k_{0}\sqrt{\frac{\varepsilon_{m}\varepsilon_{d}}{\varepsilon_{m}+\varepsilon_{d}}}$; $\omega_{SP}$ at $\varepsilon_{m}+\varepsilon_{d}\rightarrow 0$. While LSPs are: - Confined electromagnetic modes, and - Radiative, or say leaky, and - We have the polarizability, $\alpha = 4 \pi a^3 \frac{\varepsilon_{\text{m}} - \varepsilon_{\text{d}}}{\varepsilon_{\text{m}} + 2 \varepsilon_{\text{d}}}$; $\omega_{LSP}$ at $\varepsilon_{m}+2\varepsilon_{d}\rightarrow 0$. ### Excite SPPs with photons It is an eternal topic to make particles gets interacted. Momentum match and energy match should always be kept. For nonlinear optics like SHG, one has to maintain phase match to obtain a second harmonic photon. This is exactly the same for SPPs and LSPs. To have a photon interact with polariton, the momentum and energy must be conserved, which means if we want a photon to excite SPPs, we have to have an intersection at the dispersion diagram. (Though for LSPs things are easier since they are confined particles, and we could use a different picture to depict the phenomena.) If photons may excite SPPs, then the at least one intersection should exist in the dispersion diagram in the propagating direction, $k_{x}$. Recall [[Surface plasmon polaritons#Dispersion of SPPs]]. The (lossless) diagram is ![[Drawing 2024-07-27 20.36.35.excalidraw.svg]] There are no any intersections. Indicating that with photons along, the SPPs cannot be excited, and vice versa. This means SPPs are dark. To satisfy the conservation condition, we should create additional momentum, this could be done by periodic structures, or sub-wavelength scatterers. #### Compensate missing momentum by diffraction gratings Diffraction process could make part of the momentum loss at certain direction. With the diffraction angle $\theta$ and the constructive interference requirement $d\sin \theta=\pm n \lambda$, we can define the momentum change quanta $g$ as $g=\frac{2\pi}{d}$ ![[Drawing 2024-07-27 21.52.00.excalidraw.svg]] $\left|\mathbf{k}_{\text{in}}\right| = \left|\mathbf{k}_{\text{out}}\right| = k$ $\left|\Delta k_{x}\right| = k \sin \theta = \frac{2\pi}{\lambda} \sin \theta$ $\lambda = \frac{d \sin \theta}{\pm n}$ So we define $g \equiv \frac{2\pi}{d}$, this gives the wavevector change $\left|\Delta k_{x}\right| = \pm n g \quad \text{with} \quad n = 0, 1, 2, \ldots$ With the diffraction exists, the plasmon dispersion could be shifted a quanta to intersect with the photon dispersion, namely, ![[Drawing 2024-07-27 22.08.50.excalidraw.svg]] We now could achieve energy and momentum conservation. This means gratings could couple photons to plasmons and vice versa. >[!Notice] >This method of using (periodic) structures to satisfy the momentum conservation condition is essentially [[Quasi-phase-matching]]. #### Compensate missing momentum by sub-wavelength features Similarly, this momentum mismatch couple be done by scatterers. In this case, a sub-wavelength structure could do the job. ![[Drawing 2024-07-28 00.00.45.excalidraw.svg]] But for the scatterer, we no longer have the accurate quanta description. Sub-wavelength feature has range of $k$ and can be consider in a Fourier analysis way. Similarly, gap plasmons by MIM (metal-insulator-metal structure) could also make ### Excite LSPs with photons In contrast to SPPs, LSPs are inherently sub-wavelength structures and could couples to photons. Wavevector has no meaning for LSP and the resonance contains many $k$s. This means we do not have to consider momentum matching. However, this also means LSPs are radiative. There are photon emission and quality factor $Q$ is only $~10$. This is not very small but also not big. #### LSPs excitation process Photons shine on a nanoparticle, in the first 1-100 fs, we have Landau damping, electron-hole pairs formed, and some of the photons get emitted; then 100 fs to 1 ps, carrier relaxation happens, recombination starts; the in 100 ps - 10 ns, e-h pairs recombined and thermal dissipation started. This process provides a local and fast heat source in a small scale. Such heating process is called thermoplasmonics. #### Applications of thermoplasmonics - heating tissues for medical treatment. - Photothermal chemistry to localize reaction on surface. - Additive manufacturing. - Tailor thermal emission and applies in solar cells.