Optical Nonlocality in Polar Dielectrics

Phonon polaritons localized in polar nanoresonators and superlattices are being actively investigated as promising platforms for midinfrared nanophotonics. Here we show that the nonlocal nature of the phonon response can strongly modify their nanoscale physics. Using a nonlocal dielectric approach, we study dielectric nanospheres and thin dielectric films taking into account optical phonons dispersion. We discover a rich nonlocal phenomenology, qualitatively different from the one of plasmonic systems. Our theory allows us to explain the recently reported discrepancy between theory and experiments in atomic-scale superlattices, and it provides a practical tool for the design of phonon-polariton nanodevices.

Popular Summary

Nanophotonics relies on the ability to control and concentrate light across distances well below the diffraction limit of light. However, expanding this ability to systems in which the light interacts with vibrations of the crystalline structure has been hampered because it is extremely difficult to predict their optical response at the nanoscale. We have developed an analytic theory to solve this problem without requiring any substantial computational power. Tests using recently published experimental data show that this theory can provide quantitative predictions for technologically relevant applications.

The interaction of an electromagnetic field with a crystal lattice is usually modeled using a century-old approach: The crystal is described as a homogeneous material in which the speed of light depends upon its frequency. The simplicity of such an approach has underlined many of our recent advances in the ability to manipulate light at the nanoscale.

However, one of the main limitations of this simplified approach is that electromagnetic-induced vibrations of a crystal lattice propagate for a few nanometers before dying out, decisively affecting the physics over this short distance. This phenomenon, called nonlocality, is the bane of our efforts to design truly nanoscopic crystalline structures. It obliges us to employ approaches requiring important computational resources while providing little physical insight. Our theory, on the other hand, provides quantitative results and a more transparent physical interpretation while requiring little numerical manipulation beyond basic algebra.

Using this theory, a variety of nanoscopic materials and geometries can now be explored, improving our understanding of the consequences of nonlocality for nanotechnology. Eventually this approach will empower the design of nanostructures with bespoke crystal structures and optical responses.

Figure 1

(a) Illustration of nonlocal effects in gold and SiC nanospheres. The two panels illustrate the dispersion of the longitudinal plasma (left) and longitudinal optical phonon (right) as a function of frequency. Regions with real wave vector solutions are shaded gray. Green shading illustrates instead the region where the dielectric function is negative and can sustain surface modes, plasmonic or phononic. The Fröhlich resonance frequency, where ${\epsilon }_{\mathrm{LRA}}({\omega }_F)=-2$, is shown by the solid blue line, and the frequency axis is normalized to ${\omega }_F$ in each case. The two spheres in the center sketch the longitudinal mode electric field distribution in the case of imaginary (top) and real (bottom) wave vectors, relevant, respectively, for the metallic and dielectric cases. (b),(c) Extinction cross section for a 10-nm-radius (b) gold and (c) $3C$-SiC nanosphere. Results calculated with a local response approximation (LRA) are shown by red dashed curves, and those calculated with a full nonlocal model by blue solid curves. Parameters used in the calculations for both gold and $3C$-SiC are reported in Appendix pp3.

Figure 2

Panels (a) and (b) show a comparison of nonlocal extinction efficiencies for $3C$-SiC spheres of different radii. (c) Detailed comparison of the extinction (${\sigma }_{\mathrm{ext}}$) and scattering (${\sigma }_{\mathrm{sca}}$) cross sections for a 4-nm-radius sphere in the local and nonlocal cases. Both curves have been normalized to unity so they may be presented on the same vertical scale. The inset is an enlargement of the resonance near $905{\mathrm{cm}}^{-1}$. The vertical dashed lines indicate the frequencies of the bare, discrete longitudinal resonances.

Figure 3

(a) Absolute nonlocal frequency shift of the main Fröhlich peak in the extinction cross section as a function of the absolute value of the parameter quantifying the nonlocality. (b) Local and nonlocal quality factors for the Fröhlich resonance of a 3C-SiC sphere, the dots represent the analytical model in Eq. (9).

Figure 4

Comparison of the local (blue solid line) and nonlocal (red dashed line) ENZ mode frequencies in an AlN film as a function of inverse film thickness $d^{-1}$. The zone-center longitudinal optical phonon frequency ${\omega }_L$ is labeled on the axis and the $n=1$ longitudinal mode energy from Eq. (11) by green circles. Above the plot a sketch of the Otto configuration considered is shown.

Figure 5

Comparison of experimental reflectance data [31] (blue solid line), a local dielectric description (green dot-dashed line), and a nonlocal description (red dashed line) of nitride superlattices. Panel (a) shows results for heterostructure A and (b) for heterostructure B as described in the text.