Designing Plasmonic Gratings with Transformation Optics

Plasmonic gratings that support both localized and propagating plasmons have wide applications in solar cells and optical biosensing. In this paper, we report on a most unusual grating designed to capture light efficiently into surface plasmons and concentrate their energy at hot spots where the field is resonantly enhanced. The dispersion of the surface plasmons shows degeneracy points at $k=0$, where, despite a strongly modulated grating, hidden symmetries forbid hybridization of plasmons traveling in opposite directions.

Popular Summary

Graphene, one of the most promising materials for next-generation electronics applications, owes much of its extraordinary electrical properties to its vanishing band gap at certain degeneracy points called “Dirac points.” Here, we design a one-dimensional thin metallic grating whose plasmon dispersion also exhibits “degeneracy points” where the band gap vanishes. We are able to concentrate light energy at hot spots, effectively enhancing the photocurrent in specific locations.

Transformation optics is a new tool in the study of the laws of electromagnetism that has been used for designing invisibility cloaks. One of the great virtues of transformation optics is that it relates seemingly unrelated geometrical objects via a strict mathematical prescription. In our case, this ability allows us to relate a metallic structure, which naturally features a gapless spectrum (a metal slab), to a very special class of periodic metallic gratings. While periodic metallic gratings are not generally expected to exhibit degeneracy points, our special class of periodic gratings inherits all of its properties from the slab, including the degeneracy points. We study how these gratings responded to plane-wave illumination, and we find that light can be preferentially concentrated (20–30 times) in hot spots. The enhanced photocurrent at these locations enables studies of how the refractive index changes locally. We additionally show that our analytical solutions and simulations agree remarkably well.

We expect that our results will pave the way for a better understanding of plasmonic gratings, which are critically important for the design and improvement of ultrathin solar cells and plasmonic biosensors.

Figure 1

Schematic of the transformations taking an infinitely long slab to a periodic grating. Each step gives the conformal transformation relating two neighboring structures. We use a complex number notation for the transformations, e.g., $w^{\prime }=u^{\prime }+i{v}^{\prime }$, $z=x+iy$, etc. An exponential map transforms a slab to an annulus, and an annulus is transformed to a nonconcentric annulus via a Möbius transformation [25]. Finally, a shifted logarithm takes a nonconcentric annulus to a grating with one flat and one modulated side. Note that $\gamma$ determines the periodicity and the overall dimension of the grating, while $w_0$ determines the modulation depth.

Figure 2

Contour plot of the conformal transformation in Eq. (2). The solid lines show how the Cartesian coordinate grid in the $x,y$ frame is distorted under the transformation in the $u,v$ frame. The dashed lines show the branch cuts of the transformation running from the branch points to infinity. The parameters were $\gamma =1$, $y_0=0.033$, and $w_0=0.6$.

Figure 3

Dispersion relation of two gratings in the first Brillouin zone. The magenta triangles and green open circles are COMSOL simulations for a grating with $\gamma ={10}^{-8},w_0=2.5$ and $\gamma ={10}^{-8},w_0=1.5$, respectively. $a=2\pi \gamma$ in both cases, we start with a slab at position $x_0=1$ and with thickness $d=0.5$. The solid lines are analytical calculations. The dashed line corresponds to the light line.

Figure 4

Reflectance (top) and transmittance (bottom) for the weakly (left) and strongly (right) modulated gratings (shown in Fig. 3. Blue lines correspond to analytical calculations, whereas the red lines with circles are COMSOL simulations. In COMSOL, the wave is incident on the modulated side of the grating. The insets show the electric energy density in and close to the grating at the first transmission peak and dip (same color scale). For both analytics and simulations, the permittivity of the grating is ${ϵ}_m=1-{\omega }_p^2/[\omega (\omega +i\gamma )]$ with ${\omega }_p=8\mathrm{eV}$ and $\gamma =0.032\mathrm{eV}$. The surrounding medium has ${ϵ}_d=1$.

Figure 5

Mode profile of the plasmon resonance at approximately 4.5 eV. We show the imaginary part of the electrostatic potential for the weakly modulated grating (top) and for the strongly modulated one (bottom). The color scales and the incident field are the same in both cases. The distributions have been obtained from the transformation optics approach using magnetic sources as described in the Supplemental Material [26]. We used the Drude model given in Eq. (1) for the permittivity. $\mathrm{Re}({ϵ}_m)=-2.16$ and $\mathrm{Im}({ϵ}_m)=0.022$ at $\omega \approx 4.5\mathrm{eV}$.

Figure 6

Maximum field enhancement normalized by the incident electric field for the weakly (top) and strongly (bottom) modulated gratings. The data shows a 1D slice through the grating parallel to the $x$ axis. The $y$ position is such that the cut line is just below the lowest point of the modulated interface, i.e., always inside the grating. The red open circles show COMSOL simulation for a plane-wave incident on the modulated interface, and the blue solid lines are analytical calculations based on the transformation optics approach. In both cases, we use permittivity data of silver [43] with $\mathrm{Re}({ϵ}_m)=-3.96$ and $\mathrm{Im}({ϵ}_m)=-0.16$ at $\omega \approx 3.18\mathrm{eV}$.