The system under study consists of a dimer composed of two spherical nanoparticles of radius *R*=100 nm separated by a centre-to-centre distance *d*=250 nm, as shown in Figure 1A. One of the particles (blue) has a passive metallic response that we describe using the Drude dielectric function [1]

${\epsilon}_{\text{L}}\mathrm{(}\omega \mathrm{)}={\epsilon}_{\infty}-\frac{{\omega}_{\text{p}}^{2}}{\omega \mathrm{(}\omega +i{\gamma}_{\text{p}}\mathrm{)}}\mathrm{,}$

with *ε*_{∞}=2, ℏ*ω*_{p}=2 eV, and *γ*_{p}=0.05*ω*_{p}, which are realistic values for transparent conductive oxides at frequencies below their band gap [51]. The other particle (red) is made from the same metallic material but has an active response arising from being doped with a gain material, which we model by adding a Lorentzian gain term to the dielectric function,

${\epsilon}_{\text{G}}\mathrm{(}\omega \mathrm{)}={\epsilon}_{\text{L}}\mathrm{(}\omega \mathrm{)}+F\frac{{\gamma}_{0}}{\omega -{\omega}_{0}+i{\gamma}_{0}}\mathrm{.}$

Figure 1: Analysis of the absorbed and radiated power.

(A) Schematics of the system under study, consisting of two spherical nanoparticles with a radius *R*=100 nm separated by a centre-to-centre distance *d*=250 nm, which are excited by a dipole source placed in the centre of their gap. One particle has a conventional passive plasmonic response (blue), whereas the other displays optical gain (red). (B, C) Total power radiated (red curve) and absorbed (blue curve) by the dimer when excited by the dipole. These quantities are normalised to the power radiated by a dipole **p**_{0} in vacuum *P*_{0}=4*ω*^{4}|**p**_{0}|^{2}/(3*c*^{3}). In B, we assume a fully passive dimer (i.e. *F*=0), whereas in C, the pumping parameter is set to *F*=0.71. Notice that in B the red curve is multiplied by 15 to improve visibility.

This Lorentzian gain term represents the optical gain provided by optically pumped quantum emitters, such as dye molecules or rare earth ions [39], [41], [50]. This approach is justified for systems in which quantum effects are not expected to be relevant [52] and provides a good approximation to the steady-state solution of a time-dependent semiclassical description of a gain medium [53], [54]. Here, we assume *ℏ**ω*_{0}=0.94 eV for the gain frequency and *γ*_{0}=0.01*ω*_{0} for the gain bandwidth, while *F*, which describes the population inversion of the gain material, is left as the parameter that controls the level of gain in the system and is determined by the level of external pumping. These values are compatible with experimental measurements (see [39], [50], [55]) and references therein).

We begin our study by analysing the response of a fully passive dimer, i.e. with no external pumping, *F*=0. We calculate the total power radiated and absorbed by the dimer when excited by a dipole placed at the centre of the gap between the two nanoparticles, as shown in Figure 1A. We neglect any possible back-action of the dipole source on the material response of the particles. The corresponding results, obtained from the rigorous solution of Maxwell’s equations using a finite element method, are shown in Figure 1B. We assume that the exciting dipole, which oscillates at frequency *ω*, is oriented perpendicular to the axis of the dimer and normalise the results to the power radiated by a dipole of amplitude **p**_{0} in vacuum *P*_{0}=4*ω*^{4}|**p**_{0}|^{2}/(3*c*^{3}). Examining the results, we observe that, for *F*=0, the power absorbed by the dimer (blue curve) is more than 15 times larger than the radiated power (red curve, notice the scaling factor), and therefore the system behaves as a poor nanoantenna. However, this behaviour is dramatically changed when the pumping parameter is set to *F*=0.71, a value chosen to ensure that the absorption of the system still remains positive, and therefore the system does not reach a lasing threshold. In this case, as shown in Figure 1C, the radiated power increases, and simultaneously, the absorbed power is reduced, almost vanishing near *ω*_{0}. This results in a frequency range around *ω*_{0} in which the radiated power dominates over the absorbed power, and hence, the dimer acts as an efficient nanoantenna. Interestingly, both the radiated power and absorbed power display a monotonic behaviour with the increase in the gain, as shown in Figure 2. Furthermore, it is worth noting that the normalised radiated and absorbed power shown in Figures 1 and 2 are, respectively, equal to the radiative and nonradiative components of the local density of optical states (LDOS) when normalised to the LDOS of vacuum.

Figure 2: Detailed analysis of the absorbed and radiated power.

Total power radiated (A) and absorbed (B) by the active–passive dimer when excited by a dipole source, calculated for increasing values of *F*. Both quantities are normalised to *P*_{0}.

The increase in *F*, in addition to mitigating the absorption losses of the system and therefore enhancing the power radiated, also serves to introduce a phase difference in the optical response of the nanoparticles. This phase difference, which ultimately arises from the negative imaginary part of the dielectric function of the gain material, can be exploited to achieve directionality in the radiated power [56]. From the perspective of PT symmetric systems, the increase in the gain brings the system closer to the point in which the gain compensates the losses and therefore is expected to produce an asymmetric response [30], [57], [58]. To analyse this possibility, in Figure 3A, we calculate the power radiated by the dimer along its axis, both in the direction of the gain side (*θ*=0°) and in the direction of the loss side (*θ*=180°), as depicted in the inset. The corresponding results, normalised to *P*_{0}, are plotted using red and blue curves, respectively. We perform this calculation for increasing values of the pumping parameter *F* below *F*=0.71, for which, as discussed before, the absorbed power remains positive, thus ensuring we are far from any lasing threshold, and for a frequency detuning *ω*−*ω*_{0}=−0.19*γ*_{0}. As expected, for *F*=0, the radiated power is completely symmetric; however, as this parameter grows, the power radiated becomes increasingly asymmetric. This is illustrated by the black dashed curve, which represents the ratio between the power radiated along the two directions. Clearly, this quantity grows monotonically from 1, for the completely passive dimer (i.e. *F*=0), to almost 3 when the pumping parameter is increased to *F*=0.71. The lack of a threshold behaviour indicates that the asymmetry does not directly result from an exceptional point [59]. For *F*=0.71, the power radiated by the dimer displays a strong imbalance toward the direction of the gain particle, as can be seen from the analysis of the full angular distribution plotted in Figure 3B. In this plot, the red curve represents the angular distribution of the power radiated by the active–passive dimer with *F*=0.71, whereas the black curve represents the results for the completely passive dimer (*F*=0). Therefore, the presence of gain, in addition to increasing the power radiated by more than one order of magnitude as compared with the passive dimer (note the scaling factor), directs the energy predominantly toward the gain side of the dimer.

Figure 3: Analysis of the directionality of the power radiated.

(A) Power radiated by the dimer along its axis in the direction of the gain side, *θ*=0° (red solid curve), and the loss side, *θ*=180° (blue solid curve), normalised to *P*_{0} (left axis). This quantity is plotted as a function of the pumping parameter *F* for a detuning *ω*−*ω*_{0}=−0.19*γ*_{0}. The black dashed curve (right axis) shows the ratio between the power radiated along the two aforementioned directions. (B) Angular distribution of the power radiated by the system for *F*=0.71 (red curve) compared with that of the passive dimer, *F*=0 (black curve). Note that the latter is multiplied by 5 to improve the visibility.

We can gain a deeper insight into the optical response of the active–passive dimer by using a simple dipolar approach, in which each of the nanoparticles is modelled as a point dipole. Within this approximation, the optical response of the passive and active particles is described using the frequency-dependent polarisabilities *α*_{L} and *α*_{G}, respectively. We assume that these polarisabilities already incorporate the corrections arising from the mutual interaction between the nanoparticles. Then, as shown in Appendix B, the power radiated at an angle *θ* by the dimer and the dipole emitter can be written as

$$\begin{array}{l}\frac{P\mathrm{(}\theta \mathrm{)}}{{P}_{0}}=\frac{3}{8}\mathrm{(}1+{\mathrm{cos}}^{2}\theta \mathrm{)}[1\text{\hspace{0.17em}}+|{G}_{0}{\alpha}_{\text{G}}{|}^{2}+|{G}_{0}{\alpha}_{\text{L}}{|}^{2}\\ +2|{G}_{0}{|}^{2}\text{R}e\left\{{e}^{-ikd\mathrm{cos}\theta}{\alpha}_{\text{G}}{\alpha}_{\text{L}}^{\ast}+{e}^{-ik\frac{d}{2}\mathrm{cos}\theta}\mathrm{(}\frac{{\alpha}_{\text{G}}}{{G}_{0}^{\ast}}+\frac{{\alpha}_{\text{L}}^{\ast}}{{G}_{0}}\mathrm{)}\right\}]\mathrm{,}\end{array}$$(1)

where *G*_{0}=*e*^{ikd}^{/2}[2*k*^{2}/*d*+4*ik*/*d*^{2}−8/*d*^{3}] is the dipole–dipole interaction tensor connecting the dipole source with the dipoles induced in either of the two particles. Using this expression, we can write the ratio of the power radiated along the *θ*=0° and *θ*=180° directions as (1+*β*)/(1−*β*) with

$\beta =\frac{2\left|{G}_{0}{|}^{2}\mathrm{sin}\mathrm{(}kd\mathrm{)}\text{I}m\right\{{\alpha}_{\text{G}}{\alpha}_{\text{L}}^{\ast}\}+\text{\hspace{0.17em}}2\mathrm{sin}\mathrm{(}kd\text{/}2\mathrm{)}\mathrm{}\text{I}m\{{G}_{0}\mathrm{(}{\alpha}_{\text{G}}-{\alpha}_{\text{L}}\mathrm{)}\}}{1+{\left|{G}_{0}\right|}^{2}\mathrm{(}\mathrm{|}{\alpha}_{\text{G}}{\mathrm{|}}^{2}+\mathrm{|}{\alpha}_{\text{L}}{\mathrm{|}}^{2}+\text{\hspace{0.17em}}2\mathrm{cos}\mathrm{(}kd\mathrm{)}\text{R}e\left\{{\alpha}_{\text{G}}{\alpha}_{\text{L}}^{\ast}\right\}\mathrm{)}+\text{\hspace{0.17em}}2\mathrm{cos}\mathrm{(}kd\text{/}2\mathrm{)}\mathrm{}\text{R}e\{{G}_{0}\mathrm{(}{\alpha}_{\text{G}}+{\alpha}_{\text{L}}\mathrm{)}\}}\mathrm{.}$

This expression confirms that, for a completely passive system (i.e. *F*=0), in which the two particles are identical (*α*_{G}=*α*_{L}), *β*=0, and hence, there is no asymmetry in the radiated power. It also shows that it is possible to achieve an asymmetric radiation pattern by introducing a phase difference in the response of the two elements of the dimer, such that the numerator of this expression does not vanish. One way of achieving this is by using nanostructures with different geometries [56], as is the case for the Yagi–Uda nanoantenna [4]. Another option is what we discuss in this work, that is, the combination of nanostructures with balanced gain and loss. Specifically, as *F* increases, the dimer approaches the point for which ${\alpha}_{\text{G}}={\alpha}_{\text{L}}^{\ast},$ thus making *β*≠0, in complete agreement with the asymmetric radiation pattern analysed in Figure 3. Incidentally, this mechanism has the added feature that it can be actively controlled through the tuning of the external pumping, thus enabling the possibility of adjusting the level of asymmetry in the radiated power.

So far, we have analysed the behaviour of the active–passive dimer under a dipole excitation and shown that the system is capable of acting as a tunable antenna in transmitter mode, that is, enhancing and directing the power radiated by the dipole along the axis of the dimer in the direction of the gain particle. However, in order to complete the characterisation of this system, we need to analyse its performance when acting in receiver mode. To that end, in Figure 4, we study the field intensity enhancement, FE(*θ*), produced by the dimer at the centre of its gap when it is illuminated with a plane wave. We define this quantity as the ratio between the squared magnitude of the *x* component of the electric field at the centre of the gap and that of the incident plane wave. Furthermore, we assume that the plane wave propagates toward the dimer at an angle *θ* and is polarised parallel to the plane of incidence (i.e. in the *xz* plane), as shown in the inset. The red and blue curves show the results for the active–passive dimer, with *F*=0.71, when illuminated at *θ*=0° and *θ*=180°, respectively. Analysing these results, we conclude that the field intensity at the gap displays a strongly asymmetrical behaviour, being almost 3 times larger when the plane wave impinges from the gain side of the dimer. This asymmetric behaviour is in sharp contrast with the results obtained for the completely passive dimer (black curve), which are identical for the two different illumination angles. Furthermore, as was the case for a dipole excitation, the presence of gain also enhances the response of the system through the mitigation of its losses, thus resulting in a field intensity that is 21 times larger than that produced by the completely passive dimer. It is worth noting that this asymmetric response could have been obtained from the results of Figure 3 by exploiting Lorentz reciprocity [60] (Appendix C).

Based on the results of Figures 3 and 4, we anticipate that a pair of active–passive dimers can efficiently mediate the transfer of energy between two emitters separated by a distance much larger than their resonant wavelengths, i.e. in the far-field region. In particular, consider the pair of nanoantennas depicted in the upper panel of Figure 5A, and assume that the dipole in the leftmost dimer acts as the emitter, whereas the one in the rightmost dimer serves as the receiver. Thanks to the combination of (i) the enhancement of the radiated power and (ii) the increased field at the gap of the dimer, which we analysed, respectively, in Figures 3 and 4, we expect the energy transfer between the two dipoles to be greatly enhanced in this arrangement, which we denote as the LGGL configuration. On the contrary, when the orientation of the dimers is reversed, as in the GLLG configuration shown in the lower panel of Figure 5A, we expect a much smaller energy transfer. In order to verify this hypothesis, we compute the normalised energy transfer rate (nETR) [61], [62], defined as

Figure 4: Analysis of the directionality of the field intensity enhancement.

Field intensity enhancement, FE(*θ*), at the centre of the dimer gap when the system is illuminated with a plane wave. As shown in the inset, we assume the plane wave to propagate toward the dimer at an angle *θ* and to be polarised parallel to the *xz* plane. The colour curves display the results for an active–passive dimer with *F*=0.71 when *θ*=0° (red curve) and *θ*=180° (blue curve), whereas the black curve indicates the results for a completely passive dimer (i.e. *F*=0). Notice that, in this case, the results for *θ*=0° and *θ*=180° are identical.

Figure 5: Analysis of the energy transfer.

(A) Schematic of a nanoantenna pair mediating the radiative transfer of energy between two emitters placed at distances much larger than their resonant wavelengths. We consider two different arrangements that we denote as LGGL and GLLG configurations, where L and G represent a passive and active nanoparticle, respectively. (B) Normalised energy transfer rate (nETR) between the two dipole emitters for the LGGL configuration (red solid curve) and the GLLG configuration (blue solid curve). This quantity is plotted as a function of the pumping parameter *F* for a detuning *ω*−*ω*_{0}=−0.19*γ*_{0}. Notice the axis break introduced to improve visibility for small values of *F*. The black dashed curve (right axis) indicates the ratio between the nETR of the LGGL and GLLG configurations.

$\text{nETR}=\frac{|{\widehat{n}}_{\text{r}}\cdot {E}_{\text{e}}\mathrm{(}{r}_{\text{r}}\mathrm{)}{|}^{2}}{|{\widehat{n}}_{\text{r}}\cdot {E}_{\text{e}\mathrm{,0}}\mathrm{(}{r}_{\text{r}}\mathrm{)}{|}^{2}}\mathrm{.}$

Here, ${\widehat{n}}_{\text{r}}$ is a unit vector oriented as the receiver dipole, **E**_{e}(**r**_{r}) is the electric field produced by the emitter dipole at the position of the receiver when each dipole is placed in the gap of an active–passive dimer, and **E**_{e,0}(**r**_{r}) is the same quantity when the dipoles are in vacuum. Assuming that the two dimers are in the far-field region of each other, we can write the following approximate expression for the nETR (Appendix C):

$$\text{nETR}\approx \frac{4}{3}\frac{P\mathrm{(}\theta \mathrm{)}}{{P}_{0}}\text{FE}\mathrm{(}\theta \mathrm{}\mathrm{)}\mathrm{,}$$(2)

where *θ* is equal either to 0° for the LGGL configuration or to 180° for the GLLG one. Figure 5B shows the results of the nETR, calculated using this expression, as a function of the pumping parameter *F* for a detuning *ω*−*ω*_{0}=−0.19*γ*_{0}. Red and blue curves correspond, respectively, to the LGGL and GLLG configurations. As expected, for *F*=0, both configurations result in the same nETR of approximately 10. As *F* grows, the nETR increases significantly with the LGGL arrangement, reaching values of several thousand when *F*=0.71. Simultaneously, the ratio between the two configurations, which is shown by the black dashed curve (right axis), grows and becomes almost 8 when *F*=0.71. These results confirm that the active–passive dimers can operate as efficient nanoantennas, which can be externally controlled through the adjustment of the pumping level.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.