1. Introduction

The availability of ultrafast time-resolved spectroscopy methods of light and electrons enabled the investigation of dynamics on timescales down to attosecond duration [1–3] not only in the gas phase but also in solid samples. However the combination of ultrafast time resolution and nanometer spatial resolution is experimentally challenging and facing an exciting field of ultrafast science. The concepts of the attosecond nanoplasmonic field microscope [4] and the technique of attosecond photoscopy [5] demonstrate theoretically that ultrafast science at ultrasmall structures with high temporal and spatial resolution down to attosecond duration and nanometer spatial resolution is feasible. Nanoscale electric field imaging methods improved to map plasmonic fields directly [6–9]. Kubo et al. [10] and Marsell et al. [11, 12] imaged the plasmonic fields of a propagating plasmon on a silver grating and in silver nanowires, respectively, with interferometric time-resolved photoelectron emission micrographs (ITR-PEEM).

For a long time, temporal characteristics of plasmons have been investigated in the energy domain by analyzing the spectral linewidth. However the availability of femto- and attosecond pulses enable new excitation schemes [13] and analysis for the investigation of more complex nanostructures as coupled particles, resonators, or coupled antenna structures exhibiting both complex dynamic temporal and spatial modulations of the electric field on subcycle time scale [10, 11, 14]. The spatial characteristics of these structures under resonant excitation has been described by Prodan et al. in the model of plasmon hybridization [15]. Plasmonic coupling in nanostructures like nanoantennas or nanoshells builds up a symmetrical (bright) mode, where the charge density is unilateral modulated as a dipole or an antisymmetrical (dark) mode forming a multipole. The excitation of antisymmetrical modes is only possible from the interaction of bright modes, inhomogeneous far-field illumination or related excitation schemes [16].

In contrast to our studies presented here, Dmitriev et al. studied the hybridization in Au-SiO2-Au nanodisk sandwiches [17] in bulk media and compared extinction measurements to simulations. The sandwiches are composed of two gold disks (radius 55 nm and 44 nm, height 10 nm) that are separated by an insulating silica disk (diameter adapting top and bottom disk, height 10 nm. Hybridization in this system originates from two possibilities of dipole arrangements in the top and bottom gold disks. In the high energy state the dipole moments are aligned parallel, whereas in the low energy state anti-parallel. The aspect ratio of height to diameter of the smaller gold disk influences the appearance of the two states. In these studies an increased thickness changes this ratio and drives the system from anti-parallel dipole alignment to parallel alignment.

The temporal characteristics and in particular build-up time of the plasmonic fields in such hybridized resonators goes down to a few femtoseconds. So far, temporal characteristics of plasmons have been determined from the linewidth of resonant peaks. However, to directly observe the dynamics of plasmonic systems with a temporal resolution down to a few ten attoseconds, the technique of attosecond streaking can be adapted [5].

Studies based on nanodisk resonator structures with the thickness of the larger top disk decreased down to a few monolayers (1 nm formed by 2.5 monolayers) are presented. The plasmonic excitation in this type of nanoresonator is refined to the surface without any volume excitation observable at the surface layer. The key parameters for this behavior are analyzed and discussed. In particular the asymmetry of the diameter between top and bottom disk is identified as a key parameter for the predominant excitation of surface modes. The changes of absorption cross sections, electric near fields, and the temporal evolution of the electric near fields on a sub-femtosecond timescale are represented with respect to the variation of the key parameters.

2. Simulation setup

The simulations presented in this article were performed using the finite integration technique (FIT) [18] within the Computer Simulation Technology (CST) Studio Suite. This approach is similar to the finite difference time domain (FDTD) technique used for example in MEEP [19] or Lumerical [20]. With this approach the Maxwells equations in the time domain either in integral form (FIT) or differential form (FDTD) are solved to study the temporal evolution of the local distribution of electromagnetic fields. The target geometry is divided into small subvolumes with optimized shape and the electric and magnetic field is numerically retrieved for each time step.

The sandwich is composed of three individual disks with a gold bottom disk, a silica center disk and a gold top disk as depicted in Fig. 1. The initial diameters of the bottom and top disks are 88 nm and 108 nm, respectively. The center disk shape is conical and adapts to the disks below and above. The disk thickness of the bottom and center disks are 10 nm and of the top disk 1 nm. We excite the plasmonic oscillation with an ultrashort laser pulse with a Fourier limited pulse duration of below 5 fs covering a spectral range from 550 to 1500 nm in normal incidence (see Fig. 8). The carrier envelope phase (CEP) of the electric field is π/2 corresponding to a sine-like electric field.

 

Fig. 1 Simulation setup: a gold-silica-gold nanodisk sandwich is illuminated from above with an ultra short laser pulse (λ = 550to1500 nm, τ ≤ 5 fs) or monochromatically at the resonance wavelength. The sandwich has a total height of 21 nm and a radius of 54 nm of the top thin disk and 44 nm of the bottom thick disk.

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The simulations cover the variation of several geometrical parameters:

In the simulation presented in this paper, the dielectric function for gold is taken from [21]. For ultrathin films, the dielectric properties change [22] also with respect to physical properties like the formation of islands and the coalescence during the growth process on the experimental side and approaching the quantum limit of atomically thin films in particular. Wang et al. [23] demonstrated experimentally that gold film thicknesses from two to ten nanometers could be carefully treated in the Drude circle regime and identified the cluster size in film growth as the major effect for permittivity changes. A theoretical investigation of the permitivity changes in comparison to the Lorentz-Drude model by Campbell et al. [24] supports this finding for the visible wavelength range. Furthermore, the plasmon in the simulations presented here is excited with an electric field direction in the surface plane representing the direction of extended material where the influence of the film thickness to the permittivity is less compared to the out-of-plane component [25]. Therefore, we restrict the presented results on the data for the dielectric function based on a Lorentz-Drude model. As a systematic simulation within a Lorentz-Drude model we trust our results within the range of the film thickness presented according to a careful evaluation of the optical properties in the given thickness range. The decreasing film thickness of the top layer will have a substantial impact on the permittivity in the out-of-plane axis approaching a few-layer system below 6 layers which needs a more careful evaluation and an adequate adaptation for the simulation framework which is not included in the simulations presented here.

3. Simulation results

3.1. Influence of the top disk thickness

The formation of surface plasmon modes in the nanodisk system is investigated in this section. Starting with a thickness of 1 nm (10-10-1) of the thin gold disk, we increase the thickness up to 3 nm (10-10-3) in 0.5 nm steps. Other geometrical parameters are constant. We evaluate the absorption cross sections for each thickness. The cross sections are presented as false color plot in Fig. 2(a) and two individual spectra for the (10-10-3) and (10-10-1) sandwiches are plotted in Figs. 2(b) and 2(c).

 

Fig. 2 False color plot of the absorption cross section spectra for increasing thickness of the thin gold disk (a) and the spectra for 3 nm and 1 nm thickness (b) and (c), respectively. The corresponding electric near field component perpendicular to the sandwich surface is monitored after monochromatic excitation at the different resonance wavelengths and shown next to the corresponding peak.

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The false color plot reveals four resonances that undergo a shift and are merging with increasing thickness of the thin gold disk. The resonance wavelengths in the (10-10-1) case are located at

The electric near field at the thin disk surface of the sandwich is different for both thicknesses exhibiting an amplitude distribution similar to the spherical transverse electro-magnetic modes (TEM) that depends on the excitation wavelength. To investigate these modes the sandwich is illuminated with monochromatic light at the corresponding resonance wavelength. The resulting electric near fields are shown next to the corresponding peaks. The near fields are phase averaged. For the (10-10-1) sandwich we find modes from TEM 11 to TEM 41 for the resonances λ₄ to λ₁, respectively. Increasing the disk thickness, the amplitudes of the higher TEM modes 31 and 41 decrease and vanish. The electric near field on the (10-10-3) sandwich that is excited with ${\lambda }_1^{*}$ has changed to a pure dipole structure. Whereas, the near field excited with ${\lambda }_2^{*}$ still reveals the TEM 11 pattern. Since the ${\lambda }_1^{*}$ resonance shows no surface related structure, this mode is identified as a volume mode.

3.2. Influence of the bottom disk diameter

In this section we study the impact of asymmetry of the top and bottom gold disk radii for the (10-10-1) and (10-10-3) sample geometry on the absorption cross section. For both sandwiches the bottom gold disk radius is increased in 1 nm steps up to equalizing the top disk radius. The false color plots in Figs. 3(a) and 3(b) show the calculated absorption cross sections for the (10-10-1) and (10-10-3) geometries, respectively.

 

Fig. 3 False color plots of the absorption cross section spectra for increasing radius of the thick bottom gold disk for a thickness of 1 nm and 3 nm of the thin gold disk (a) and (b), respectively.

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Both sandwiches differ in their number of resonant modes, which was already found above (four resonances for the (10-10-1) and two resonances for the (10-10-3) sandwich). However, if the radius in the (10-10-1) sandwich is increased up to 54 nm, the intensity of the long wavelengths resonances at λ₂, λ₃ and λ₄ decreases and vanishes. These resonance wavelengths are almost not shifted. The short wavelength resonance at λ₁ is the remaining mode and is only slightly shifted to longer wavelengths. The (10-10-3) sandwich obtains two well separated resonances at 44 nm radius of the bottom disk. Increasing the radius the short wavelength resonance becomes the prominent resonance, whereas the long wavelength resonance is suppressed. In both systems the remaining mode is located around 700 nm.

Whereas in the variation of the top disk thickness the remaining mode was originating from the low order TEM 11, in the variation of the bottom disk radius the remaining mode is originating from the high order TEM 41 in the (10-10-1) case or more general the short wavelength mode.

From these results we conclude that the formation of surface modes on the top surface of the nano disk sandwich needs two requirements to be fulfilled:

3.3. Temporal evolution of the electric near field

To study the temporal evolution of the superposition of the excited modes in the disk sandwich, the time dependent electric near field is calculated. The modes are excited by a coherent ultrashort Fourier-limited pulse quasi-simultaneously with a pulse duration much shorter than the expected plasmon lifetime. However, the temporal appearance of the individual mode oscillations is not synchronized and the total electric near field is supposed to oscillate with a complex structure. The near field is calculated in 0.2 fs steps and ends when the amplitude drops below 1/e² of its maximum value. We monitor the electric field along a line perpendicular to the electric field and in the surface plane at three positions: on top of the thin gold disk, in the center of the spacing layer and below the bottom gold disk. We investigate the asymmetric sandwich with 44 nm bottom disk radius and the symmetric one with 54 nm radius. The results are presented as false color plots in Fig. 4.

 

Fig. 4 Temporal evolution of the out-of-plane component of the electric field evaluated along a line parallel to the incident electric field crossing the center of the sandwich for the asymmetric (a) – (c) and symmetric sandwich (d) – (f) monitored at three different positions. The surface position monitoring is shown in (a) and (d), the center in (b) and (e) and at the bottom in (c) and (f). The electric field is normalized to the maximum value. The electric near field evolution on the surface of the asymmetric and symmetric sandwich are shown in Visualization 1 and Visualization 2 as video, respectively.

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The top, center and bottom monitors of the asymmetric sandwich are shown in Figs. 4(a)–4(c), and of the symmetric sandwich in Figs. 4(d) – 4(f), respectively. One observes positively and negatively charged areas that are represented by the red and blue colors. A pure dipole mode is represented by a half negatively and half positively charged disk resulting in a red and blue oscillation separated by a center line. A splitting of the charged area indicates that the superposition of the excited modes forms a more complex pattern of two or more amplitude modulations along the line of observation.

The complex superposition can be seen best on the top and center monitor in Fig. 4(a) and 4(b) for the asymmetric and 4(d) and 4(e) for the symmetric sandwich. In the asymmetric sandwich shown in Fig. 4(a) a dipole mode is excited until around 18 fs a higher mode dominates the superposition leading to a split up of the charged area into two regions. At 20 fs the superposition of the modes is dominated by an even higher mode leading to another splitting. Locally the superposition pattern is not stable but moves towards the disk center in the beginning. The propagation direction is represented by the gradient of the particular minima/maxima along the time axis. Around 32 fs the propagation direction changes and the pattern moves towards the disk edge. This process is denoted with phase variation in the following. These phase variations repeat at 40 fs and 48 fs. The position of the localization of the minima/maxima varies over time, shown by the angle of the separating white lines. The symmetric sandwich shown in Fig. 4(d) uncovers a locally more stable superposition of higher modes. The white lines are almost straight. The first and second phase variations appear 3 fs later than in the asymmetric sandwich, and the third one almost 6 fs later. The monitoring in the center of the spacing layer in Figs. 4(b) and 4(e) shows the phase variation processes more clearly. This position is experimentally not accessible. If one compares the superposition of the modes on the surface in Figs. 4(a) and 4(d) with the position in the dielectric in Figs. 4(b) and 4(e), respectively, an increased field strength in the asymmetric sandwich is observed whereas in the symmetric sandwich the field strength is decreased (see Appendix Figs. 9(a)–9(f)). The superposition of the excited modes appears different. The width of the center part is modulated significantly stronger. The maximum number of separated minima/maxima is reduced in the asymmetric sandwich. Moreover, in the region of the phase variation in the center there is an extended plateau (±10 nm) with a very weak modulation. This denotes a strongly localized change of the polarization of the free electron density. The electric near-field on the bottom of the sandwich shows only a dipolar volume structure and does not seem to be influenced by the superposition of the modes on the top surface. The dipolar mode decays much faster than the surface modes. Comparing the asymmetric to the symmetric sandwich a significantly longer lifetime of the volume mode in the symmetric sandwich > 30 fs than in the asymmetric sandwich < 30 fs is observed.

Resolving dynamics in the few femtosecond down to the attosecond regime directly in the time domain, the technique of attosecond streaking [1] is widely used and has been transfered to solid samples [3] a few years ago. With respect to plasmonic samples this pump-probe technique utilizes a femtosecond near infrared laser pulse to excite a plasmonic field followed by a subsequent attosecond extreme ultraviolett pulse with a well defined time delay probing the electronic excitation by XUV photoemission. This general idea was developed a few years ago and we call it attosecond photoscopy [5]. The initial kinetic energy of the released photoelectrons is modified by the interaction with the vector potential of a present electric field, i.e. the plasmonic nearfield [1]. The kinetic energy of the photoelectrons is analyzed by a time of flight detector which is sensitive to the electric field component pointing towards the detector entrance. Delaying both pulses with respect to each other, the dynamics of the electric field is sampled in the energy spectrum of the collected photoelectrons. According to the geometry, plasmonic interaction can be distinguished from laser interaction in the photoemission process [5]. In an attosecond streaking experiment the lateral resolution is limited by the focus size of the XUV pulse to approximately 10 μm. This is much bigger than the structure itself resulting in a photoelectron wavepacket that is spread by the local field strengths at the place of birth and along the trajectories of the individual electrons getting an additional vector momentum from the local plasmonic nearfield. Having an array of resonant nanodisk sandwiches one has to ensure to avoid coupling between the individual disks. The coupling in a nanostructured array is controlled by the distance between the structures and can be avoided by using a large distance (several times the particle dimensions) between the individual objects.

The electric field over each half disk on top of the surface of the asymmetric and symmetric sandwich was integrated to analyze the total dipolar behavior of the half disks with respect to each other. The results are plotted in Figs. 5(a) and 5(b) for the asymmetric and symmetric case, respectively. In the asymmetric and symmetric sandwich a high frequency oscillation of the plasmonic field and a plasmon build-up time of approximately 6 fs is observed but the dephasing and damping is different. The dephasing of the plasmonic field in the asymmetric sandwich is superposed with a low frequency oscillation resulting from the different dominating modes analyzed above. The dephasing in the symmetric sandwich is not modulated because of the locally stable and ordered structure of the superposition showing a comparable behavior to classical plasmon excitation and damping with comparable timescales according to the disk size. Thus we attribute the complex surface plasmon excitation predominantly to the asymmetric disk geometry as a key parameter for our findings.

 

Fig. 5 Time dependent integrated ut-of-plane electric field for left and right half disk on the sandwich surface calculated for the asymmetric (a) and symmetric (b) (10-10-1) sandwich. The splitting of the sandwich is perpendicular to the electric field polarisation vector

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4. Conclusion

Dmitriev et al. showed the hybridization of nanodisk sandwiches with layer thicknesses of 10 nm, where either parallel or anti-parallel alignment of dipole moments inside the sandwich occurs. Tuning the layer thickness with the bigger radius or the spacing layer thickness one can distinctly influence the excitation of only one of the two modes. Here, we presented the hybridization of a disk sandwich structure by refining the plasmonic oscillation to the surface reducing the disk thickness of the bigger disk down to a few monolayers. To our current interpretation with in teh Lorentz-Drude model this behavior is strongly related to the increasing free electron relaxation time (Drude damping) and will be even more influenced by experimental conditions as film roughness and non-closed films increasing free electron scattering on bounderies much more then predicted. The predominant excitation of surface modes is possible by satisfying two criteria: very thin disk thickness with up to a few gold monolayers and the asymmetry of the disk diameters. The coherent excitation with a broadband ultrashort linearly polarized laserpulse launches a plasmonic electric field with complex spatio-temporal structure resulting from the temporal evolution of the superposition of the excited modes. With respect to the time-dependent dominant mode, the superposition of the spatial modes shows a varying number of minima and maxima that are comparable to TEM modes which are evolving on a sub-femtosecond timescale.

From these initial findings we conclude that the hybrid nano thindisk resonators presented in this article are prototype systems for experiments with subcycle time resolution. Tailored spectral phases and amplitudes according to the plasmonic resonances of the exciting laser pulse will lead to coherent control of the temporal evolution of the superposition of all excited modes. In both cases a time-resolved experiment will be the basic requirement to follow the time evolution of the plasmonic excitation in real time. The verification of the theoretical results requires experimentally the fabrication of large area, high quality few- to monolayer surfaces for the top disk which is technologically possible following current research of two-dimensional materials. This experimental approach defines a new class of hybrid nano-resonators combining bulk material and two-dimensional material on the nanoscale for attosecond time-resolved spectroscopic research providing complex surface dynamics following ultrafast broadband optical excitation.

A. Simulation robustness

By simulating the sandwich as described by Dmitriev et al. in [17] we test our simulation setup. The absorption cross sections for three top layer thicknesses and the monochromaticly excited electric fields in the (10-10-10) sample are shown in Figs. 6(a) and 6(b), respectively. The results correspond to the extinction spectra calculated and measurend by Dmitriev qualitatively. The phase is presented as a false-colour plot in Figs. 7(a) and 7(b). By reproducing the results the simulation environment is expected to be stable and to deliver reliable results.

 

Fig. 6 Simulation of (10-10-6) to (10-10-36) disk sandwiches with top disk radius of 44 nm and bottom disk radius of 54 nm: (a) spectra and (b) sectional view of the electric fields

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Fig. 7 Sectional view of the phase in the high (a) and low (b) energy resonance for the (10-10-10) sandwich

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B. Excitation pulse

The short pulse used as excitation signal is shown in Fig. 8. The spectrum of the laser pulse is shown in Fig. 8(a) and chosen such that the resulting pulse has a Fourier limit of approximately 4 − 5 f s. The spectral shape is Gaussian in the frequency domain. The electric field shown in Fig. 8(b) reveals a sine like structure. The electric field is normalized to a maximum value of 1.

 

Fig. 8 Broadband excitation pulse (a) spectrum and (b) the normalized temporal electric field. The e-field vector is parallel to the sandwich surface.

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C. Temporal evolution of electric near field - quantitative results

 

Fig. 9 Temporal evolution of the electric field evaluated along a line parallel to the incident electric field crossing the center of the sandwich for the asymmetric (a) – (c) and symmetric sandwich (d) – (f) monitored at three different positions. The surface position monitoring is shown in (a) and (d), the center in (b) and (e) and at the bottom in (c) and (f).

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Funding

Landesexzellenzcluster “Frontiers in Quantum Photon Science”; Joachim Herz Stiftung.

Acknowledgments

We are grateful to Walter Pfeiffer for fruitful discussions.

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