Photonic and phononic modes in acoustoplasmonic toroidal nanopropellers

1. Materials and Methods

The geometry of the TnP used in this study is based on a three-lobed structure of identical rings, with an inner diameter of 65 nm and a wall thickness of 20 nm, as presented in Figure 1. The centers of the rings are disposed in the vertices of an equilateral triangle, as a result, the whole structure is circumscribed in a circle with a diameter of 200 nm. The twisted TnP is subsequently obtained by vertically sweeping this geometry along a helical path (L) with a varying $\alpha$. That sweep, together with the vertical dimension (H=60 nm), defines the pitch of the propeller ($P=2\pi H/\alpha$). TnP is a singularly connected structure and with a topological genus 3.

As mentioned, to simulate the optical and acoustic response, we assume the TnP is made of gold, following the dispersive optical constants given in Ref. [23] provided in the Lumerical® software, and the thermal expansion, sound speed and Young modulus embedded in the Finite Element Method COMSOL®.

Lumerical® software suite has been employed to solve the scattering and absorption cross section. The solutions are based on the Finite Difference Time Domain (FDTD) method. We use an impinging circularly polarized plane wave with the proper circular polarization along the z axis, with a normalized amplitude in the whole simulation cell. The cell geometry (4 $\mu$m × 4 $\mu$m × 7 $\mu$m) ensures that perfectly absorbing boundary conditions do have a negligible effect on the electromagnetic fields obtained. We use a refined mesh in the nanostructures and in the near field region (0.3 $\mu$m × 0.3 $\mu$m × 0.9 $\mu$m) of 2 nm × 2 nm × 2 nm, dx-dy-dz, respectively, growing uniformly up to a maximum of 40 nm out of the near field close to the simulation boundaries, so that convergence (to the best of our numerical capabilities) is attained. The total and scattered fields are then collected to give rise to intensity color maps and cross sections.

The Finite Element Method (FEM) within COMSOL® provides a suitable environment to model the acoustic response upon isotropic heat expansion of a metallic nanostructure. In our simulations the toroidal propeller is, as mentioned above, assumed to be made of gold, standing on a layer of silicon dioxide semi-spherical substrate. The material parameters used for FEM simulation are the ones built-in the materials library of COMSOL®. The SiO₂ substrate is a 800 nm diameter semi-spherical region surrounded by perfectly matching layers (PML) that truncate the physical domain, see e.g. Ref. [20].

2. Results

2.1. Optical Characteristics

As it can be seen, in a twisted TnP geometry like the one depicted in Figure 1b, there is a clear handedness. Therefore, it is expected that the optical response will present an helicity dependence that should, additionally, depend on the twisting angle $\alpha$. To explore that possibility, the optical response of this structure is interrogated by impinging with a plane wave whose propagation vector is perpendicular in the x-y plane (i.e. in the z direction, from $z=+\infty$), and its polarization is either left-circular (LCP) or right-circular (RCP) as

$$|LCP\rangle =\frac{|x\rangle +i|y\rangle }{\sqrt{2}};|RCP\rangle =\frac{|x\rangle -i|y\rangle }{\sqrt{2}},$$

(1)

where $|x\rangle$ represents a linearly polarized wave along the x-direction while travelling along the z-direction, and $|y\rangle$ is travelling in the same direction but linearly polarized along the y-direction.

Let us first describe the optical response, in the form of scattering and absorption cross sections, for the untwisted nanostructure as a function of the wavelength in the visible range of the optical spectrum. Similarly to a system resembling a metallic gold disc [24], the system present a plasmon resonance in the range of 0.8 $\mu$m (see Figure 2a). Another smaller resonance is also observed around 0.6 $\mu$m, expected as long as the dielectric constant of Au stays in the metallic range. It is worth noting that for this case there is no difference between LCP and RCP, as foreseen for a structure with no handedness.

In Figure 2b,c we present the scattering and absorption cross sections for the resonance at wavelength ${\lambda }_{\mathrm{res}}=0.825\mu$m, as a function of the twisting angle. Noticeably, in this case there are big differences between LCP and RCP incoming waves, signaling the presence of chirality. However, the most prominent feature is the oscillatory pattern in optical response. In this structure, the only relevant parameter, from the point of view of the optical resonances, that changes with the twisting angle is the length of the pores forming the lateral surface of the propeller. The length of the helix used for generating the TnP (see Figure 1(b)) can be expressed as

$$L=\sqrt{{\left(\frac{\alpha D}{2}\right)}^2+H^2},$$

(2)

where $\alpha$ is in radiands, D is the radius of the helix, and H the height as mentioned above (see Figure 1b). Notice that the radius of the helix, optically speaking, will have a different dimension depending on the nature of the resonance, i.e. weather the field is predominantly in the air or in the metal.

In order to complete the optical analysis, it is apparent that the complete spectrum of the scattering and absorption cross sections needs to be obtained. This is shown in Figure 3, where we present the scattering and absorption cross sections as a function of the wavelength, for twisting angles bellow 360 degrees.

In all panels in Figure 3 we observe that the resonances at 0.825 $\mu$m and at 0.6$\mu$m, as a function of the twisting angle, behave in the following manner: for $\alpha$ from 0 to c.a. 50 deg. there is very little dispersion, for values from 50 to ∼110 deg. the resonances strongly shifts to longer wavelengths, and, interestingly, for twisting angles $\alpha \gtrsim$120 deg. they disperse linearly towards high wavelengths running parallel to each other. This happens both for the absorption and the scattering cross sections. Noticeably, for the twisted structure ($\alpha \sim 120$ degrees), the absorption (depicted in (c) and (d)) is over two times bigger than for the untwisted one, but the volume of the system remains essentially the same. There are obvious differences in the intensity for LCP and RCP waves, pointing out COE, but, apart from the actual value of the cross sections, the overall oscillatory phenomenology is virtually equivalent for both helicities. In fact, the plot of Equation 2 (see Appendix A) shows that it accurately describes the trend followed by the resonance peaks, even in the fact that some may have different slopes (that would depend on a varying effective D).

Additionally, from a simple inspection of the panels in Figure 3 there are obvious signatures of a sizeable COE. This COE is specially relevant in the absorption cross section, where the response of the propeller is enhanced for LCP waves respect to RCP ones, reaching differences larger than 100%. For the scattering cross section, the differences are not as marked and occur at shorter wavelengths than for the absorption.

2.2. Acoustic Characteristics

Let us now address the characterization of the acoustic response of the TnP. For that end we study the vibrations, characterized by the displacement of each point in the harmonic regime upon an initial isotropic thermal expansion produced by a $\Delta T=5$K.

In order to find the resonant modes corresponding to the structure, we monitor the average displacement (i.e. the 3D integral of the RMS displacement of each point of the TnP normalized to its volume). The resonant modes appear as distinct peaks as presented in Figure 4. Two natural, low frequency, modes for an untwisted (red curve) TnP appear at 4.6 GHz and 5.5 GHz respectively and correspond to the breathing modes with maximum (low frequency) and minimum (high frequency) displacement at the point marked as A in Figure 1. As soon as the structure is slightly twisted (even just one degree) we clearly observe two different phenomena. The first one is that the resonances associated with the breathing modes shift towards lower energies (i.e. the wavelength associated to the resonances increases), in perfect correspondence with the previously presented optical behavior. The second observation, now at variance with the optical case, is that in the 3 to 4 GHz region two new resonances appear that disperse downshift in frequency as the twist angle increases. For zero twist angle (red curve in Figure 4 and its inset) there are no resonances in this region, but even for a twist of one degree, they appear (black curve inset Figure 4). The evolution with twist angle from $\alpha$=0 to $\alpha$=15 deg. (see inset in Figure 4) shows a further increase in intensity and a downshifting in frequencies. In Figure A2 in Appendix B we present some snap-shots for the breathing modes at $\alpha =0$ deg. and for the low frequency twist-enabled modes at $\alpha =15$ deg..

Similarly to the optical case, we are presenting in Figure 5 the whole evolution of the resonance location for a TnP with a complete twist angle (360 deg.). As already indicated in Figure 4, we can see that the lowest energy resonances for the untwisted geometry (inset (c) of Figure 4) appear for 4.6 GHz and 5.5 GHz and evolve towards lower energies as the twist angle increases. One relevant aspect that can be extracted from this figure is that, although the resonances always evolve towards a lower energy value, there is a change in the trend. Initially, the frequency decreases roughly as 1/L as expected to fit with the increase of helix lenght (see Equation 2). However, this trend changes at $\alpha \sim 120$ deg.. To shed some light into the change in slope, we refer to the features depicted in the insets (a) and (b) in Figure 5, showing the geometry for twist angles of 360 degrees and 120 degrees respectively. If the structure were made by a single, isolated lobe, at one full loop ($\alpha$=360 deg.) the height of the TnP would match the pitch of the helix, and that would be the first occasion in which one lobe in the base would have another one exactly above. However, our structure is made of three lobes, and Figure 5a, resembles a much more twisted structure, not surprisingly it appears to have three twists. In fact, as seen in Figure 5b, it is at one third of a twist ($\alpha =120$ deg.) when a lobe of the base have another one above. It is then when two effects compete: the resonance trend to decrease in energy due to the growth of L against the resistance due to the upper and lower branches of the TnP.

Eventually, for a very twisted geometry, the resonance should converge to that of a disk shape with a twisted void geometry inside, so compressed that the variation of the angle should have a negligible effect.

3. Discussion and Conclusions

Understanding the origin of these optical and acoustic modes is essential to actually use them in optomechanical systems, where the helicity of light can be use as a design parameter. Towards that end, in this work we have presented a complete study of the optical and acoustic properties of a three-lobed twisted toroidal nanoscale propeller-like structure. The presence of the twist in the TnP is indicated by a red-shift in both its optical and acoustic resonances. This red-shift occurs because the wavelength couples with the effective length of the generative helix, which varies with the twist angle of the TnP. Consequently, the twist angle acts as a tuning structural parameter that influences the nature, intensity (which depends on helicity), and spectral positions of the optical and acoustic resonances. Additionally, the twist present in the TnP leads to new low frequency resonant acoustic modes that are not present in the spectrum of an equivalent untwisted structure. These new acoustic modes appearing at non-zero twisting angles enable a new knob in the engineering of opto-phononic interactions at the nanoscale.