# From Far-Field to Near-Field Micro- and Nanoparticle Optical Trapping
## 3 Plasmonic Optical Tweezers
### Self-Induced Back Action Effect  
The diffraction and transmission of light through a single, subwavelength-sized circular hole on a metallic surface was first theoretically studied in 1944 by [PERSON] [55]. Assuming a perfectly conducting and infinitely thin material, [PERSON] calculated that the transmitted light would scale as \(T\propto(d/\lambda)^{4}\), where \(d\) is the radius of the hole and \(\lambda\) the wavelength of the incident light, as illustrated in Figure 7. In 1998, the remarkable phenomenon of the _extraordinary transmission of light_ was experimentally observed by [PERSON] et al. [56], when they studied the effects of the geometrical characteristics of an array of tiny holes, drilled on different metallic films, on the transmission of light. In that work, UV-Vis-NIR spectrophotometry of the array was performed and it revealed the existence of maxima in the transmission intensity, see Figure 8a, with much higher values than predicted from [PERSON]'s theory. These maxima could not be explained simply by diffraction theory and they were associated with the resonant frequencies of the excited surface plasmons of the metal. A detailed theoretical explanation of this observation was provided a few years later [57; 58], where the calculated positions of the transmission maxima were in very good agreement with the experiment, confirming that the mechanism of extraordinary transmission was taken into account in their model (Figure 8b). In addition, in his theoretical work, [PERSON] investigated the case whereby the nano-hole is filled with a dielectric material of high refractive index (Si). He found that the transmission cross-section at the resonance frequency was almost three times higher than without the filling, leading to increased transmission [58]. The increase of the refractive index at the nano-hole causes the wavelength of the light to decrease:  
\[n_{fill}=\frac{c}{c_{fill}}>1\Leftrightarrow n_{fill}=\frac{\lambda}{\lambda_{ fill}}>1\Leftrightarrow\lambda_{fill}<\lambda. \tag{16}\]  
A consequence of this effect is that, according to [PERSON]'s theory, the transmission of light through the subwavelength aperture increases, a phenomenon also known as dielectric loading. Figure 9 shows how the wavelength shift causes a significant increase in the transmission of light \(T_{2}\), compared to transmission of light \(T_{1}\) through a non-filled aperture. Figure 8: (**a**) Experimental and (**b**) theoretical plots of the normalised transmittance as a function of the incident light wavelength, for a square array of holes on a Ag thin film with thickness \(h=\)320 nm. The diameter of the holes was \(d=280\) nm in the experiment and \(d=240,280,320\) nm for the calculations and the lattice constant 750 nm. Inset in the left plot shows an SEM image of the nano-hole lattice. Image reproduced with permission from [57]. Figure 7: (**a**) Diffraction and transmission of visible light through a subwavelength circular hole of radius, \(d\), on a perfectly conducting and infinitely thin metallic film. (**b**) According to [PERSON]’s calculation the transmission of incident light \(\lambda\) scales as \((d/\lambda)^{4}\). Figure inspired by [59]. As mentioned in the previous section, at around the same time, the use of plasmonics for enhanced optical trapping had started to attract attention and, despite how promising they might seem for trapping subwavelength particles, it became apparent that their use is limited to particles with a minimum diameter around 100 nm due to photothermal effects [60]. It was the combination of these two different studies on plasmonics that brought the realisation that the resonance frequency of the excited plasmonic field is very sensitive to changes of the local refractive index. Thus, by proper engineering of the plasmonic structure, the trapped particle itself could actively contribute to its own trapping potential in a dynamical way [61]. This plasmonic structure - particle interaction promised high tunability of the trapping potential, which was no longer a static.