A combination of helium- and gallium-ion beam milling together with a fast and reliable sketch-and-peel technique is used to fabricate gold nanorod dimer antennas with an excellent quality factor and with gap distances of less than 6 nm. The high fabrication quality of the sketch-and-peel technique compared to a conventional ion beam milling technique is proven by polarisation-resolved linear dark-field spectromicroscopy of isolated dimer antennas. We demonstrate a strong coupling of the two antenna arms for both fabrication techniques, with a quality factor of more than 14, close to the theoretical limit, for the sketch-and-peel–produced antennas compared to only 6 for the conventional fabrication process. The obtained results on the strong coupling of the plasmonic dimer antennas are supported by finite-difference time-domain simulations of the light-dimer antenna interaction. The presented fabrication technique enables the rapid fabrication of large-scale plasmonic or dielectric nanostructures arrays and metasurfaces with single-digit nanometer scale milling accuracy.
Plasmonic nanostructures such as metallic nanoantennas act as functional elements in a wide range of applications. The intense evanescent electric field at the air–metal interface allows for the generation of 2-photon-photoluminescence , , , second , , , , , or third , , ,  harmonic and of femtosecond electron pulses , . The spatial localisation of the electric field down to dimensions of the plasmonic nanostructure, i.e. to subwavelength dimensions, is accompanied by a dramatic field enhancement, which can then be employed in single-molecule spectroscopy , , scanning near-field microscopy , or electron steering processes , . When arranged in groups of plasmonic particles such as bowtie , , ,  or rod , ,  dimer antennas, both the enhancement and the spatial confinement of the electric field can be boosted significantly, thanks to the growth of near-field–mediated optical interactions , , . The electromagnetic coupling of the dimer antenna results in the hybridisation of the surface plasmon resonances (SPRs) of the monomers, forming an optically bright bonding mode and a dark, antibonding mode , , , , . In this hybridisation model, the coupling strength and therefore the energy splitting of the two modes strongly increase with decreasing gap size , . When this resonance shift δλ exceeds the linewidth Δλ of the linear optical resonance, i.e. δλ/Δλ>1, the two antennas are referred to be strongly coupled. Here, the energy dissipation is smaller than the coupling, resulting in a periodic transfer of energy between both antenna arms , . In order to reach this strong coupling regime in the field of plasmonics, special emphasis was put on the precise fabrication of the few-nanometer feed gaps between the metallic nanostructures. For bottom-up fabrication processes, single-molecular layer gaps between a gold nanosphere and a metallic substrate could be achieved, recently reaching an anomalously large spectral shift near the quantum tunnelling regime , . However, the ability to precisely shape the nanostructures is limited with this approach. Top-down lithography processes such as photo-resist–based electron-beam lithography (EBL) , ,  provide high flexibility in controlling the shape of the nanoparticles and typically achieve gap sizes of slightly below 20 nm, limited by the proximity effect due to the backscattered electrons interacting with the sample . Under optimized conditions ,  or with overexposure techniques , gap sizes down to the single-digit nanometer level were accomplished. However, EBL is a sequential process, requiring several processing steps before the final device is created. Ga-based focused ion beam (Ga-FIB) milling procedures , , , in contrast, are limited by the finite focusing properties of the Ga+-ion beam. Furthermore, undesired Ga+-ion implantation into the sample as well as amorphisation of the nanostructure edges can furthermore significantly increase the linewidth of the longitudinally polarized SPR (LSPR) , , , . As a consequence, the quality factor Q of the antenna, defined as the ratio between the centre wavelength λ0 and its linewidth ΔλFWHM taken as the full width at half maximum (FWHM), decreases considerably compared to the theoretical limit given by the frequency-dependent dielectric function of the structured materials. Hence, strongly coupled metallic nanostructures are difficult to realize using top-down processes such as EBL or Ga-FIB.
Recently, He+-ion–based lithography (He-FIB) was introduced as a fabrication method to achieve gap sizes well below the 10-nm regime , , , . Here, He+ ions instead of Ga+ ions are applied for the precise milling of plasmonic nanostructures in a smooth, single-crystalline metallic film , , . As a distinct advantage, a much reduced line broadening of the plasmon resonances is observed when using He-FIB instead of Ga-FIB. The improved milling accuracy, however, comes at the cost of a significantly reduced milling speed due to the smaller ion mass compared to gallium. At higher ion doses, the ion bombardment can lead to an undesired implantation of helium at the substrate–metal interface . To overcome this issue, Chen et al.  introduced a simple, fast, and reliable fabrication technique called “sketch and peel”–based FIB milling (SaP). In this process, a FIB at an appropriate dose is used to sketch only the outline of the desired plasmonic nanostructure to completely separate the particle and the rest of the film. In a second step, an adhesive tape is adhered onto the metallic surface and then peeled off from the substrate, resulting in dimer metallic particles with gap distances down to 20 nm. The key of this approach is that the surrounding metallic film can be selectively removed while the isolated plasmonic structures remain on the substrate during the peeling process.
In this work, we further optimize this approach and propose a combination of Ga-FIB, for the rough outlining, with He-FIB lithography, for the precise fabrication of the gap region of the dimer antenna, together with the SaP technique for the lift-off process. By using this combination, a reduction of the minimum gap size of down to 5 nm was achieved. We find a significant red shift of the LSPR of more than 150 nm for gap sizes of down to 5 nm accompanied with an outstanding quality factor of up to 14, proving the fabrication of low-loss plasmonic nanoresonators with performances approaching the theoretical limit.
The proposed SaP technique can fabricate large-size arrays of plasmonic structures without losing the fabrication speed or the nanometer-milling accuracy. Hence, this technique enhances the current capability of FIB lithography for the fabrication of metallic structures and therefore can bring the fabrication of large structures or large arrays of nanostructures with nanometer precision in a resistless and solvent-free way into reach.
2 Results and discussions
2.1 Sample preparation
Gold films with a thickness of 30 nm were deposited on a fused silica substrate using an electron-beam physical vapour evaporation process at a rate of 0.26 nm/s at a pressure of 8×10−6 mbar. The grain size of the polycrystalline gold films was increased in an annealing process at a temperature of 350°C for 8 h. The annealing process enlarges the typical size of single-crystalline grains from about 50 nm to 500 nm in diameter. Ion beam milling was performed by using both focused Ga+ ions (Ga-FIB) in a dual-beam microscope (FEI Helios Nanolab 600i) and focused He+ ions (He-FIB) in a He/Ne ion microscope (Zeiss Orion NanoFab).
Figure 1 depicts the three different strategies applied to fabricate pairs of gold nanorods with dimensions of 90×35×30 nm3 and with gap distances ranging from 5 to 35 nm. The rod length was chosen to match the LSPR of the long axis of the particle with the measuring range of our laser source and spectrometer. The rod width of 35 nm is limited by the fabrication process of the Ga+-ion beam. The distance between two dimers was set to 3 μm, large enough to selectively image isolated dimer antennas in the confocal dark-field microscope. In the first two, gallium-based lithography strategies, the Ga+-ion beam (blue) with a current of 24 pA, was used to remove most of the gold film (A), followed by a second step of Ga+-ion beam writing at a current of 7.7 pA to outline the still-connected dimer nanorods (B). The dimers were finalized in a third step by cutting through the nanorods using either the Ga-FIB (“gallium lithography”) or the He-FIB (“gallium/helium lithography”) at a current of a few pA (C). Figure 1D shows a scanning electron micrograph of a prototypical nanorod dimer antenna with a gap distance of 6 nm. The three different fabrication steps (A–C) are visible by the different grey scales of the outer rim, the inner rim, and the gap region. Interestingly, charging effects of the dimer during the scanning electron microscope (SEM) image acquisition process were negligible, indicating that the first and second fabrication steps indeed induced doping of gallium or gold in the substrate.
The third fabrication strategy includes the much faster SaP strategy. Here, the contour of the desired nanostructure was milled into the metal film for 600 ms using the Ga-FIB (blue) at a current of 1.1 pA (Figure 1E). Subsequently, the He-FIB (red) was used at a current of 1 pA for 200 ms to split the antenna in half and fabricate gaps with gap sizes between 5 and 35 nm (F). Finally, a carbon-free, adhesive tape attached to the metal film was used to selectively peel the gold film outside the contoured area (G). In Figure 1H, a helium-ion micrograph of a prototypical antenna with a gap size of 5 nm is displayed, in which the high uniformity of the gap region becomes visible. Here, the gap size was taken as the FWHM of cross sections along high-resolution scanning electron and helium-ion micrographs (cf. Chapter S2 of the Supporting Information). The dimer antennas remained on the substrate because of the formation of a sidewall coating . Here, charging effects of the sample during the image acquisition precluded high-resolution SEM images of isolated, stripped dimer antennas. However, low-magnification SEM and far-field optical microscope images of arrays of dimer antennas indeed prove the successful peeling process (cf. Chapter S3 of the Supporting Information).
2.2 Optical modelling
The linear optical response of the dimer antennas was simulated using a commercial software package, Lumerical FDTD Solutions. The 3D model system consisted of the gold dimer antenna on a fused silica substrate with the dimensions matching the same geometrical arrangement as the sample. The dielectric functions for gold were taken from Olmon et al. . The fused silica substrate was modelled with an index of refraction of n=1.45, and the whole structure was surrounded by vacuum. Perfectly matched layer boundary conditions were used on all sides of the calculation domain. A linearly polarized incident pulse with a Gaussian frequency spectrum was used to excite the sample. The polarisation was set along the long axis of the antenna. The total simulation region was set to (750 nm)3 with a spatially variable mesh size of down to 1 nm in the gap region The simulation region was chosen sufficiently large to avoid a near-field absorption at the boundaries of the simulation box.
2.3 Optical dark-field spectroscopy
Linearly polarized, ultra-broadband laser pulses from a coherent white light source (WLS) (Fianium SC-450-4) with a spectrum ranging from 450 nm to 2 μm with few-picosecond pulse duration and an average pulse energy of 60 pJ at 78 MHz repetition rate were collimated, spatially filtered, and focused to a rather large diameter of approximately 1.1 mm×120 μm (FWHM) onto the sample at an angle of approximately 75°. The polarisation of the light was controlled using an ultra-broadband linear polarizer in combination with an achromatic half-wave plate. The incident polarisation is chosen along the long axis of the dimer in order to optimize the coupling to the optically bright bonding mode of the nanorod dimer , , . The scattering spectrum of an isolated dimer was collected in reflection geometry using an extra-long working distance glass objective with a NA=0.7 (Nikon, BD Plan 60x ELWD) and sent to a monochromator (Princeton Instruments, Acton SP2500) attached to an LN2-cooled CCD camera (Princeton Instruments, SPEC10 100BR). By imaging the array of nanorod dimer antennas onto the CCD of the spectrometer, individual dimer antennas were selected. The position of the sample in the laser focus could be adjusted in all three dimensions with a manual three-axis linear actuator. A more detailed scheme of the experimental setup is given in Chapter S1 of the Supporting Information.
2.4 Linear scattering spectra
In Figure 2A, normalized linear scattering spectra of two prototypical dimer antennas fabricated using the SaP technique with gap distances of 24 and 6 nm are displayed as a blue dashed and a blue solid line, respectively. For light that is linearly polarized along the long axis of the dimer, these spectra display only a single peak, the optically bright, bonding mode of the dimer. This mode results from the coupling to the fundamental, LSPR of each nanorod. For nanorods with very similar geometries, the respective antibonding mode is optically dark. Higher-order LSPR resonances of the dimer lie on the short-wavelength side, outside the probed spectral window, and for highly symmetric samples, the excitation of transverse SPR resonance is suppressed by the choice of incident polarisation. The measured spectral resonances are well described by single Lorentzian lineshapes, shown by a black dashed and a black solid line. The resonance wavelength λ0,S2 of the dimer with a larger gap is 680 nm with an FWHM of approximately ΔλS2=64 nm, whereas the resonance wavelength λ0,S1 for the dimer with a 6-nm gap is 854 nm with ΔλS1=81 nm. Additionally, the linear scattering spectra of two dimer antennas with gaps of 7 nm (red solid line) and 27 nm (green dashed line), fabricated using Ga and Ga/He lithography technique, are presented. Here, the deduced resonance parameters are λ0,C2=745 nm (ΔλC2=160 nm) for the large-gap dimer and λ0,C1=846 nm (ΔλC1=183 nm) for the small-gap dimer. The spectrum of the exciting WLS is shown as a grey-shaded background.
Now, the resonance wavelengths of 46 fully Ga and Ga/He-fabricated and 560 SaP-fabricated dimer antennas of the same dimensions are compared in Figure 2B. Many more SaP-fabricated antennas could be tested as their fabrication is much faster and more accurate than for FIB milling. Here, open green and red squares denote the dimer antennas fabricated using the Ga and the Ga/He lithography, respectively, whereas open blue triangles represent the antenna produced by SaP. The superior milling precision of the He-FIB in the SaP procedure is apparent in the clustering of the dimer antennas around the designated gap sizes of 5 and 10 nm with standard deviations of 0.7 and 0.9 nm, respectively. The difference between the measured and nominal gap size of about 1 nm is independent of the fabrication technique and the nominal size (cf. Chapter S2 of the Supporting Information). The standard deviation of the gap size distributions for He+-ion cutting of about 1 nm is approximately a factor of two lower than that for pure Ga+ lithography.
As can be seen, all three distributions show a very similar behaviour: For gap sizes larger than approximately 20 nm, the resonance wavelength was constant around 750 nm for the Ga and Ga/He-produced and around 685 nm for the SaP-produced dimer antennas. The difference in resonance positions for the same geometries may be explained by a strongly increased Ga+-ion doping and resputtering of Au of the substrate for the pure FIB approach compared to the outline procedure. This doping and redepositing are demonstrated by high-resolution secondary ion-mass spectrometry (NanoSIMS) measurements of the surrounding area of the dimer antenna (see Supporting Information Figure S5). Metal doping of glass leads to a higher refractive index  causing a red shift of the respective localized surface plasmon (LSP) modes . It is seen in Figure 2B that the red shift between the Ga/He and SaP decreases for smaller gap sizes. We believe that this is clear signature of the improved fabrication quality of the SaP antennas. For the same gap distance, we observe a stronger coupling-induced red shift of the dimer resonance for the SaP antennas than for the Ga/He, implying that the improvement in fabrication quality enhances the field localisation in the gap region and thus the coupling between the two antenna arms. Furthermore, the superior fabrication quality of the SaP approach becomes especially visible for smaller gap sizes. Here, the distribution of the resonance positions of the dimers is narrower, suggesting accordingly smaller geometrical imperfections.
For decreasing gap sizes, however, the resonance position red shifted significantly to 895 nm for all three fabrication processes by nearly δλ=150 nm for the Ga and Ga/He-produced and by more than δλ=210 nm for the SaP-produced nanorod dimer antennas. The optical near-field coupling between the two LSPR resonances of the two nanorods results in the formation of a red-shifted, delocalized bonding mode of the dimer. The fractional red shift of the LSPR resonance that is seen in Figure 2B amounts to up to δλ/λ0=0.31, even larger than values reported for chemically synthesized nanorods with gap distances around 3 to 4 nm . This large red shift, however, occurs only for gap sizes below 20 nm and can – with some limitations – be well approximated by exponential decay fits, known as the universal plasmon ruler equation , ,  Δλ/λ0=a·exp(−dgap/l) with a being a constant and dgap the gap size. The exponential decay fits are shown in Figure 2B by the red (Ga and Ga/He lithography) and blue (SaP) lines with decay length of LC=6.8 nm and LS=4.7 nm, respectively.
In comparison to the experimental data, the resonance positions extracted from scattering spectra simulated using a Maxwell equations solver (FDTD solutions) are shown by black circles with an exponential decay length of LSim=9.4 nm (black solid line). These decay lengths are similar to those simulated for a metallic nanoparticle dimer on a mirror configuration  but significantly shorter than the decay lengths of approximately 20 to 30 nm that have been reported for ellipsoid nanoparticles  and predicted for chemically synthesized nanorods . As it was pointed out by Dolinnyi , large deviations from the plasmon ruler equation are expected to occur for few-nanometer interparticle distances, and hence a biexponential may be more suitable. The observed increasing red shift for decreasing gap sizes is a clear signature of a near-field induced coupling of the two dimer arms , , , , resulting in mode splitting of the uncoupled LSPR mode of the individual rods into a dark, i.e. far-field invisible, slightly blue-shifted (anti-bonding) and a bright, (bonding) red-shifted mode , , . This mode splitting is indicated by the energy-level diagram in Figure 2C, where the formation of a bright, bonding mode and a dark, antibonding mode from the LSP mode of uncoupled nanorods (red rectangles) is sketched for two dimers with different gap sizes (green and blue dashed lines).
Next, we compare the quality factors Q of the bright, bonding mode of nanorod dimer antennas produced by the three fabrication processes. Here, the quality factor is defined as the ratio of the resonance frequency ω0 and the twice the damping constant γ, i.e. Q=ω0/2γ, or Q=λ0/ΔλFWHM. This quality factor is a measure for the dephasing time T2=1/γ of the electric field emitted by the antenna, i.e. the decay of the coherent polarisation within the gap region of the nanorod dimer antenna due to radiative and Ohmic damping , . The dephasing time thus determines the number of oscillation cycles of the free-induction decay of the SP oscillations  and is directly proportional to the field enhancement factor, i.e. the ratio between local field at the sample surface and incident field . This is of special interest when studying the coupling between the LSP mode and a quantum emitter placed in the spot of the gap antenna. When the emitter is weakly coupled to the LSP mode, a high Q and a small mode volume V in between the dimer antenna arms both increase the Purcell factor FP ~ Q/V, giving the enhancement of the spontaneous emission of the emitter beyond its vacuum value , . Such high Q factors are of particular interest as they result in a small damping γLSP of the antenna mode while enhancing vacuum field fluctuations in the gap. This is of key importance for reaching the strong-coupling regime, ΩR2>γLSP2/2+ γQE2/2, in which the strength of the emitter-plasmon coupling ΩR exceeds the sum of the dephasing rates of the QE and LSP mode , , . This regime can result in a periodic exchange of energy between LSP mode and emitter on a femtosecond time scale .
In Figure 3, the quality factor Q (A) and the dephasing time T2 (B) of the LSPR resonance of dimer antennas fabricated by the three techniques are displayed as a function of the gap distance. Again, open green and red squares represent antennas fabricated by Ga and Ga/He lithography, respectively, and open blue triangles show the SaP antennas. The black circles denote Q factors and T2 times of the simulated resonances. Solid lines in the respective colours are given as guides to the eye. Now, the Q factors for all Ga and Ga/He-fabricated structures are between 2.5 and 7 with an average value of 4.4. The Q factors of the SaP dimer antennas in contrast are significantly larger, ranging from about 6 to a maximum Q factor of 14.4. The average value for the SaP antennas is 10.1, even slightly larger than the average Q factor of 9.2 obtained from FDTD simulations. The quality factors for the SaP-produced dimer antennas are also larger than those reported for dimer antennas of similar size that were chemically synthesized , , fabricated using EBL , ,  and Ga-FIB , and also higher than those reported earlier for SaP-produced circular dimers . Even more, the quality factor approaches values that so far have been reached only for single-crystalline chemically synthesized single linear rods, for which, however, the field localisation is significantly smaller . For some of our SaP dimer antennas, we reach Q factors that are similar to those reported for encapsulated and annealed single nanorods . This again demonstrates the high fabrication quality of the SaP process that results in dephasing times longer than 10 fs, as can be seen in Figure 3B.
As both, the fractional red shift as an indicator for the field enhancement, and the quality factor Q as a measure for the storage time of electric field inside the dimer antennas are the relevant quantities for the antenna design, the two parameters are plotted in two-dimensional distribution in Figure 3C and are compared to literature values. As can be seen, the fabricated dimer antennas using the SaP approach are located on the upper right corner, indicating their superior ability to concentrate electromagnetic fields to nanometer dimensions for significantly long times.
To confirm that we indeed probe the longitudinal SPR of the dimer antenna alone, we have performed detailed polarisation-sensitive measurements of isolated SaP-produced nanorod dimer antennas with gap sizes of 5 and 25 nm, shown in Figure 4A. Here, the scattering spectra are displayed for a small-gap (solid lines) and a large-gap (dashed line) dimer for varying polarisation angles of θ of the linearly polarized incident light. The polarisation angle is defined as the angle between electric field vector and long axis of the nanorod dimer. The angle-dependent, spectrally-integrated intensity for the dimer with 5-nm gap size is shown in (B) as a polar plot as green squares together with a cos2(θ) fit (black line) to the experimental data. The spectra indicate that for the light being polarized along the connection axis of the two nanorods antennas (θ=0°), mainly the longitudinal, red-shifted resonance of the dimer antenna with a small gap is excited. This resonance shifts to smaller wavelengths for increasing gap sizes as can be seen for the large gap size, indicated by the dashed lines. The suppression ratio, i.e. the ratio of spectrally-integrated intensity at the single resonance with the light being polarized along the long axis compared to light polarized perpendicular to the long axis, is around 90:1 for the 5-nm gap and 85:1 for the 25-nm gap. This again underlines the high structural quality of our SaP antennas.
Our experimental findings indirectly evidence the strong coupling and hence high field localisation in the gap region of the two antenna arms. To further support this assertion, we compared our experimental results to three-dimensional FDTD simulations of the dimer antenna spectra. In Figure 5A, the simulated linear scattering spectra (dashed lines, left y-axis) are compared to normalized experimental spectra (right y-axis) of SaP dimers with varying gap sizes. Both resonance position and the spectral line shape are fairly well reproduced in the simulation, even though the red shift for the 5-nm gap sizes is still underestimated in the simulations. We ascribe this to an imperfect, since unknown matching of the simulated dimer antenna geometries to those in the experiment. In Chapter S4 of the Supporting Information, we further compare the scattering intensity P=ʃI(λ)d(λ) of the simulated and the experimental nanorod dimer antennas. This shows that for decreasing gap size the measured scattered signals increases slightly, just as it is also visible in the increase of the scattering cross section in the simulation.
To illustrate the coupling of the two antenna arms in more detail, Figure 5B shows the colour-coded field enhancement factor Lres(x,y)=<|Ex(x,y,z=20nm)|>/<|E0|> at nearly half height for two different gap distances of 5 nm (left) and 25 nm (right) at the resonance wavelengths of 825 and 735 nm, respectively. In Figure 5C, cross-cuts of Fres along the x-axis at the white dashed line in (B) are shown for the two respective gap sizes. As can be seen, the electric field for the 25-nm gap is concentrated at the corners of the structure, reaching Fres=28. However, the field decays already in the gap region due to the larger gap of 25 nm. For the 5-nm gap, the field is concentrated exclusively in the gap region, where a spatially homogenous, capacitor-like electric field distribution is created, reaching a maximum field enhancement factor F of 81. This spatially homogeneous electric near field in the gap is excellently suited for coupling the plasmon field to that of a quantum emitter sample positioned inside the gap.
The combination of Ga+- and He+-ion beam lithography with a novel SaP technique provides a high, few-nanometer precision for the fabrication of isolated, nanometer-sized plasmonic particles. We have used this technique to fabricate nanorod dimer antennas of 2×90-nm length from a 30-nm-thick gold substrate with varying gap sizes down to 5 nm. By using linear, polarisation-resolved dark-field spectro-microscopy, the ability of these structures to confine electromagnetic fields in their gap region was probed. We took the linear response, i.e. the red shift of the optically bright bonding mode and the quality factor of the nanorod dimer antennas as a measure for the field localisation in the gap region. The SaP-produced dimers were compared to dimer antennas fabricated using a standard Ga-FIB and He-FIB lithography approach. We find an enormous red shift for the SaP-produced dimers of more than δλ/λ0=0.31, larger than those reported for chemically synthesized nanorods  and those fabricated using EBL . The average quality factor Q=λ0/Δλ of the SaP-produced dimers exceeded 10, more than a factor of two larger compared to the conventionally fabricated rods using EBL or Ga-FIB. Finite-difference time-domain simulations of the linear scattering spectrum and the electric field confined in the gap region support our conclusion that by using the presented fabrication technique nanorod dimer antennas with superior fabrication quality and the ability to concentrate light into nanometer dimensions could be produced. Our results suggest that the improvement in fabrication quality will not only affect the near- and far-field radiation pattern of the nanorod dimers but may also lead to a significant enhancement of their nonlinear optical efficiency. Experimental studies of these properties are currently underway. We anticipate that the presented technique can be a cornerstone for the fabrication of complex plasmonic nanostructures, where the interaction of localized plasmon and propagating surface plasmon polariton waves with excitonic materials is probed on nanometer dimensions .
In the present work, we have discussed the red shift of the dimer SP resonance in terms of the hybridisation model introduced in Ref. . This classical coupled-dipole model predicts three distinct signatures of a strong coupling between the two rods of a dimer: (A) the appearance of two energetically split, hybrid SP resonances of the dimer: a symmetric, optically bright mode and an antisymmetric, optically dark mode; (B) an increase in radiative damping of the bright mode and a decrease in radiative damping of the dark mode (as opposed to that of the individual rods); (C) an increased (decreased) scattering cross section (effective dipole moment) of the bright (dark) mode in comparison to that of the individual rods . The strong coupling regime is reached when the energetic splitting between the coupled resonances exceeds the linewidth of the uncoupled resonances , , . We have taken the observed red shift of the dimer resonance of up to 450 meV (Figure 2B), much larger than the FWHM of the dimer resonance for large gap distances (170 meV, Figure 2A) as the distinct signature of strong coupling in our antennas. Two more features in the far-field optical spectra of such dimers may be taken as additional indications for strong coupling: (A) the change in radiative linewidth and (B) the change in the ratio between scattering and absorption cross section of the dimer resonance with coupling strength, i.e. with decreasing gap size. Both quantities are more difficult to analyse than the spectral shift of the dimer resonance as also nonradiative damping contributes significantly to the lineshape. A more in-depth analysis of the effect of the gap distance on the linewidth and on the ratio between absorption and scattering cross sections  might reveal additional insight into coupled dimer antennas. In principle, additional information on the coupling between the antenna arms may also be deduced by analysing the optical properties of antisymmetric dark mode , either by asymmetric far-field optical excitation schemes , electron energy loss spectroscopy, or near-field optical spectroscopy. In our far-field experiments, we did not find clear signatures of this dark mode, implying that the optical properties of two coupled rods are too similar to result in a significant oscillator strength of the dark mode.
An alternative signature of strong coupling in such nanodimer antennas is the periodic, oscillatory exchange of energy between the two coupled rods, mediated by their near-field coupling , , . In order to probe these dynamic couplings, it is necessary to locally excite one of the rods with an optical pulse that is shorter in duration than the Rabi period, i.e. the time it takes for the energy to oscillate between the two arms of the antenna. This oscillatory energy transfer is a distinct manifestation of strong coupling in coupled nanosystems. Due to the short time and small length scales involved, such experiments are challenging and have so far reported only very few selected hybrid systems , , . Importantly, the signatures of such strong couplings in the linear optical regime are identical in classical coupled dipole models, systems of two coupled quantum-mechanical harmonic oscillators, and for two-level systems coupled to plasmonic harmonic oscillator modes (Jaynes–Cummings model). Nonlinear experiments are therefore needed to reveal the quantum signatures of strong coupling.
Financial support by the Deutsche Forschungsgemeinschaft (SPP 1840 and GRK 1885) (SPP 1839, Funder Id: http://dx.doi.org/10.13039/501100001659 and SPP1840) and the Korea Foundation for International Cooperation of Science and Technology (Global Research Laboratory project, K20815000003). M.S. wishes to thank the BMBF for a personal research grant “Photonic Transistors” in the NanoMatFutur program, Funder Name: Bundesministerium für Bildung und Forschung, Funder Id: http://dx.doi.org/10.13039/501100002347, Grant Number: NanoMatFutur FKZ: 13N13637. The He ion beam patterning was performed in the Corelab Correlative Microscopy and Spectroscopy at Helmholtz-Zentrum Berlin.
Conflicts of interest: There are no conflicts to declare.
Author information: M.G. built up the optical setup and performed the optical experiments, together with H.K. The samples were produced by M.G., H.K., and K.H. The data were analysed by M.G. and M.S. C.L. and M.S. initiated the experiment. FDTD simulations were performed by V.S. and M.G. together with M.S. The manuscript was written by M.S. with contributions from all authors. All authors have given approval to the final version of the manuscript.
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