Tailoring Optical Fields Emitted by Subwavelength Nanometric Sources

Abstract

In this work, we study a simple way of controlling the emitted fields of subwavelength nanometric sources. The system studied consists of arrays of nanoparticles (NPs) embedded in optical active media. The key concept is the careful tuning of NP’s damping factors, which changes the eigenmode’s decay rates of the whole array. This inevitably leads, at long time, to a locking of relative phases and frequencies of individual localized-surfaces-plasmons (LSPs) and, thus, controls the emitted field. The amplitude of the LSP’s oscillations can be kept constant by embedding the system in optical active media. In the case of full loss compensation, this implies that not only the relative phases, but also the amplitudes of the LSPs remain fixed, leading us, additionally, to interpret the process as a new example of synchronization. The proposed approach can be used as a general way of controlling and designing the electromagnetic fields emitted by nanometric sources, which can find applications in optoelectronic, nanoscale lithography, and probing microscopy.

Appendix: Dissipative Couplings and Dynamical Phase Transitions

We mentioned that in the case of coupled piano strings, there is a dissipative coupling between the strings which can be modeled by an imaginary coupling term in the dynamical matrix 𝕄. Pure imaginary, or at least complex, couplings have interesting effects on the properties of the eigenvalues of 𝕄. At some critical values of the system’s parameters, there can be a collapse of the real part of the eigenvalues of 𝕄 and a bifurcation of their imaginary part at points called “exceptional points.” There, among other effects, 𝕄 becomes singular and the system’s eigenvectors behave oddly in their surroundings [39, 40]. Since the dynamical observables have a nonanalytic dependence on the system’s parameters, this results in what is called a dynamical phase transition, DPT [28, 29, 39, 41].

In the case of plasmonics systems, as those showed in this work, the complex coupling can be seen as just the consequence of the effective interaction between two parts of a system connected through a bridging dissipative subsystem. For example, if we have three NPs aligned, one can always calculate an effective coupling between the NPs at the ends [43]. The result of this is a complex effective coupling, consequence of the damping term of the NP in the middle [28, 29].

Figure 5 shows that the eigenvalues of 𝕄 present a collapse of their real part accompanied by a splitting of their imaginary part. Just as in the example of the coupled piano strings. This case corresponds to a very large value of the damping term of the middle NP and a mistuning parameter, δ, below a critical value. Here, it should be mentioned that what really sets the decay rates are the imaginary part of the poles, Im(ω _pole), of the response function χ(ω), and not the imaginary part of the eigenvalues of 𝕄. In the wide band approximation, these last coincides with $\text {Im} \left ( \omega _{\text {pole}}^{2} \right )$. This distinction can be quite irrelevant in some situations but becomes fundamental in others. In Fig. 6, we consider the case of two interacting NPs. We can see that although the eigenvalues of 𝕄 have exactly the same imaginary part, which would preclude the synchronization mechanism depicted in the main section of the article, there is a difference in the imaginary part of ω _pole. Although this difference is very small, as compared with the case shown in Fig. 5, it is enough to give rise to a characteristic asymptotic state and, thus, it can be used to induce a phase and frequency locking. In this example, the mode with the longest lifetime will be the antisymmetric one. This, at sufficiently long times, implies that the LSPs of both NPs will end oscillating in anti-phase.

Fig. 5

Panels a, b are, respectively, the real and imaginary part of $\omega ^{2}_{\text {pole}}$, the eigenvalues of 𝕄; while panels b, d are, respectively, the real and imaginary part of ω _pole, the poles of χ. The system consists of three aligned NPs with $\delta =\omega _{_{\mathrm {SP1}}}^{2}-\omega _{_{\mathrm {SP3}}}^{3}$, where 1 and 3 stand for the NPs of the edges. Only nearest neighbor couplings are considered, ${\omega _{X}^{2}} = 0.2$, $\omega _{SP}^{2}=1$ (for δ = 0), and Γ = 0.03 for all NPs except for the middle one where Γ = 0.7 (Color online)

Fig. 6

The same as Fig. 5 but for a system of two NPs with equal damping, Γ = 0.03. Notice that in spite of the small differences in decay rates as compared to those in Fig. 5, they can be enough to produce an observable phase locking through the use of an active medium (Color online)

In general, systems with dynamical phase transitions are expected to have large differences in the imaginary parts of the eigenfrequencies, as in the case of coupled piano strings or in the example shown in Fig. 5. However, phase and frequency locking is not an exclusive phenomenon of this situation. For the particular case of metallic nanoparticle arrays, the value of the damping terms needed to achieve the DPT described here are far from the realistic situation, at least for metallic NPs. Thus, the cases discussed in the main section of the article correspond to systems that do not present a DPT.