Abstract
Nanophotonics is a rapidly developing field of research with many suggestions for a design of nanoantennas, sensors and miniature metadevices. Despite many proposals for passive nanophotonic devices, the efficient coupling of light to nanoscale optical structures remains a major challenge. In this article, we propose a nanoscale laser based on a tightly confined anapole mode. By harnessing the nonradiating nature of the anapole state, we show how to engineer nanolasers based on InGaAs nanodisks as onchip sources with unique optical properties. Leveraging on the nearfield character of anapole modes, we demonstrate a spontaneously polarized nanolaser able to couple light into waveguide channels with four orders of magnitude intensity than classical nanolasers, as well as the generation of ultrafast (of 100 fs) pulses via spontaneous mode locking of several anapoles. Anapole nanolasers offer an attractive platform for monolithically integrated, silicon photonics sources for advanced and efficient nanoscale circuitry.
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Introduction
In recent years, significant efforts have been devoted to realize nanoscale sources of coherent light aiming at filling the gap between photonics and electronics for both classical and quantum applications^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17}. Fundamental advances have been recently reported in semiconductor–dielectric heterojunctions, where nanolasers from direct bandgap materials have been demonstrated^{18,19,20}. Further progress with these materials requires to address many fundamental challenges, originating when the miniaturization of optical circuitry is pushed to the nanoscale. A first problem comes from the diffraction limit of light that introduces significant radiative losses that are detrimental and severely limit the performance of nanoscale devices^{21}. A second issue is represented by the efficient coupling of light from nanophotonics sources, whose emission is strongly nondirectional^{22}. All these problems originate from the fact that conventional optical sources couple energy to classical radiating states that are all diffraction limited and difficult to control at the nanoscale.
An intriguing idea is to introduce a nanolaser design concept based on states of matter with unconventional radiation properties that can overcome the aforementioned limitations. States with nonconventional emission of energy have been recently demonstrated in specific types of alldielectric nanoparticles, realized in both silicon and germanium semiconductors^{23,24,25}. Under specific conditions, the superposition of internal modes of these nanostructures can generate radiationless states, called anapoles, that are scatteringfree and invisible to the propagation of the electromagnetic fields^{26,27}.
A fundamental question is whether it is possible to employ a nonradiating state of matter, such as an anapole, to engineer a coherent light source based on stimulated light amplification. The possibility to achieve nonlinear amplification of an anapole state is counterintuitive, due to the lack of any optical feedback from a scatteringfree state that is totally invisible to a farfield observer. However, if such interaction could be successfully triggered, we would have at disposal a new type of source that is not limited by diffraction or any of the aforementioned problems.
By using rigorous firstprinciple simulations on the quantum electrodynamics of photons, here we show that the light–matter interaction in a nanolaser emitting at the anapole frequency gives rise to a surprisingly stable steady state, where light energy is strongly collected within the anapole and evanescently transmitted in a subwalength area outside the nanostructure. By leveraging on tunnelling effects and mutual synchronization, we illustrate that such ‘anapole nanolaser’ opens a large manifold of applications, ranging from efficient and polarization controlled energy coupling, to ultrafast pulse generation without the need of external design elements. Anapole nanolasers hold the promise to be an ideal energy source for nanoscale optics in siliconcompatible platforms.
Results
Design of an anapolebased nanolaser
To illustrate the concept of anapolebased nanolaser (see Fig. 1), we consider an alldielectric resonator composed of a semiconductor nanodisk that is optically pumped in the same scheme of a classical laser^{28,29,30,31}, with energy decaying in one or more indirect transitions to a final energy state (see Fig. 1b). Unlike a conventional laser, however, in our design the stimulated emission (see Fig. 1b red arrow) amplifies an anapole state, which can be identified by a dip in the scattering cross section of the structure (Fig 1c, blue line). The anapole mode is generated by the superposition of electric and toroidal dipoles^{24}, which produces an electromagnetic mode that is confined in the interior of the nanodisk and does not radiate in the farfield (see Fig. 1d). In the scheme illustrated in Fig. 1, the energy downconverted by stimulated emission at the anapole frequency excites electric and toroidal dipoles that interfere destructively in the farfield. Contrary to lasers based on dark modes, this scheme is not based on resonant states, nor does it require nonradiative energy transitions in order to allow the generation of radiationless states in purely dielectric materials^{28,29}. In contrast to bound states in the continuum (BIC) lasers recently proposed^{30,31}, the anapole laser does not require extended structures and the radiationless state is excited with no additional radiating components.
We design an integrated anapole source starting from standard In_{x}Ga_{1−x}As semiconductor that is obtained by combining InAs and GaAs and is a direct bandgap semiconductor that is silicon compatible for monolithic integration^{19}. Following the electrical or optical excitation of carriers, the materials sustain stimulated emission of radiation. Both the optical properties and the emission frequency can be tuned by varying on the molar concentration x of InAs in the compound^{32}. Figure 2a illustrates the bandgap energy of In_{x}Ga_{1−x}As as a function of the Indium molar fraction x. As shown in the figure, the emission of the material can be tuned in a large bandwidth from 873 nm in the near infrared to 3.6 μm in the infrared. In our analysis, we consider a molar fraction x=0.15, corresponding to an emission wavelength λ_{0}=948 nm. The value x=0.15 is a standard concentration of InAs that allows monolithic integration directly on silicon, as reported in ref. 18. In addition to that, In_{0.15}Ga_{0.85}As is characterized by low losses in the proximity of the emission frequency λ_{0}, as illustrated in Supplementary Fig. 1.
Figure 2b illustrates the scattering crosssection C_{sca}^{33} of a nanodisk of In_{0.15}Ga_{0.85}As with height 100 nm and different values of diameter d. The figure shows a region where the scattering of the nanodisk is suppressed in all directions (see Fig. 2b dashed line). Such special region, which extends in the visible and in the near infrared for In_{0.15}Ga_{0.85}As nanodisks, is sustained by the presence of nonradiating anapole states that do not possess farfield emission. This is further confirmed by the multipole decomposition of the electromagnetic fields as a function of the incident wavelength for a nanodisk with diameter d=440 nm and height h=100 nm (Supplementary Figs 2–5).
Figure 2d presents the scattering crosssection spectrum for a nanodisk with diameter d=440 nm and height h=100 nm, whose anapole wavelength coincides with the emission wavelength of the In_{0.15}Ga_{0.85}As semiconductor at 948 nm (red arrow). This configuration constitutes our design geometry as illustrated in Fig. 1c. To maximize the energy trapped inside the nanodisk for the pump beam, the pump wavelength can be placed on a scattering maximum of the nanodisk (see Fig. 2c, green arrow). In the proximity of the anapole frequency, the scattering of the structure is strongly suppressed within a bandwidth of Δλ≈50 nm. This is larger than the amplification bandwidth of the In_{0.15}Ga_{0.85}As semiconductor that is ∼20 nm (see Fig. 2d, red solid area) as experimentally measured in refs 18, 34. The anapole state is stable against geometrical changes in the structures, as we verified by calculating the scattering crosssection C_{sca} of the structure for varying nanodisk diameter d and height h (see Supplementary Fig. 6). The formation of anapole states dominate at λ_{0}=950 nm in a wide range of geometrical parameters corresponding to 85 nm≤h≤105 nm and 435 nm≤d≤475 nm that are well within the precision of optical nano fabrication technologies such as electron beam lithography.
In the case of lattice mismatch between the anapole nanodisk and the substrate, we do not expect in general any perturbation in the optical properties of the system, as long as the effect is limited to few lattice sites. However, to reduce latticemismatch nonidealities, the InGaAs layer can be growth by using III–V semiconductors on a latticematched substrate, and then transferred on a lowrefractive index substrate that is compatible with Si technology. At the proposed x=0.15 concentration of Indium, the InGaAs is lattice matched to both InP and GaAs that could be used as ultrathin epitaxial growth substrates. By introducing a sacrificial layer (for example, AlAs), the InGaAs/GaAs or InGaAs/InP substrate is easily transferred to a lowrefractive substrate. Such technique is widely used in the fabrication of III–V semiconductor nanolasers on silicon platforms, as discussed in refs 35, 36.
Our choice of InGaAs for the anapole laser also opens the possibility to design electrically pumped systems. Although this topic clearly falls beyond the scope of this article, some points can be highlighted. To realize an electrically pumped anapole laser, the main point is to design a semiconductor heterostructure that ensures threedimensional (3D) confinement of the anapole mode. A possible solution is to consider a core–shell geometry. As discussed in ref. 26, anapole states can be excited in core–shell structures by acting on the multilayer thicknesses. By employing this strategy, an electrically pumped anapole laser can be composed of InGaAs/GaAs or InGaAs/InP. This structure, after adding top and bottom electrodes for electric carriers injection, can provide an initial setup for an electrically pumped anapole laser, where the anapole state is generated in the intrinsic layer InGaAs. Another important point is the optical footprint of the electrical circuitry. To this extent, dielectric conductors such as indium tin oxide are preferred to metal contacts. The use of dielectric contacts helps in minimizing optical losses. An intriguing possibility is the introduction of graphenebased contact layers or nanowires, such as the ones proposed in ref. 37, that is a promising design for a roomtemperature electrically pumped anapole laser.
We observe that the anapole state is not a resonance of the system, therefore it is not possible to associate intuitive quantities such as quality factors which are employed in the description of resonant laser cavities. Resonant states are in fact observed in the spectral response as Lorentzian energy peaks with full width half maximum equal to ω_{0}/Q, with ω_{0} being the resonant frequency. An anapole state does not possess a Lorentzian shape and is conversely observed in points of the spectrum where the scattering crosssection vanishes.
Anapole amplification by stimulated emission of radiation
We investigated the process of amplification of anapoles through stimulated emission by the Maxwell–Bloch finitedifference timedomain (MBFDTD) (see Methods). This technique has proven to furnish extremely realistic results that have been verified against many experiments^{38,39,40,41}. In our simulations, we modelled the exact dispersion curve of the In_{0.15}Ga_{0.85}As (see Supplementary Fig. 1), and we assumed a Lorentzianshaped gain profile with a dephasing time of τ=40 ps, whose value has been experimentally measured in refs 18, 34.
Figure 3 summarizes our 3D results for a pumping beam linearly polarized along the x axis and perpendicular to the nanodisk axis. In these simulations we considered a nanodisk of In_{0.15}Ga_{0.85}As with diameter 440 nm and height 100 nm, exposed to different pumping rates that are measured through the density of carriers ρ_{0}. Figure 3a,b report the intensity and the linewidth of the electromagnetic field of the anapole state amplified inside the nanodisk. The behaviour of the anapole amplitude versus pumping rate (see Fig. 3a,b) is that of a standard laser, with a linear relationship between pumping rate and amplified intensity (see Fig. 3a). As in plasmonic and dielectric nanoparticle lasers^{18,42}, the amplification of anapole states directly follows from stimulated amplification as a balance between field enhancement in the gain material and the system losses, including absorption and nearfield energy leakage. The anapole laser also shows an equivalent Schawlow–Townes linewidth (see Fig. 3b) that quickly reaches a stable value of ∼2 nm. This value is in accordance with experiments on lasing emission from InGaAs nanostructures^{18}. The carrier threshold density to lasing, observed in Fig. 3a, depends on the pumping rate ρ_{0} that is proportional to the population inversion density, or equivalently, to the input pumping rate of the system. By increasing the latter, we increase ρ_{0} and control the laser threshold condition.
A characteristic time evolution of the E_{x} field is displayed in Fig. 3a (inset) that shows the reaching of a stable stationary emission state after an initial transient. The corresponding intensity spectrum, reported in Fig. 3b (inset), further confirms that the emission corresponds to the amplification of the anapole state at λ_{0}=948 nm.
Figure 3c–e illustrates the spatial distribution of the electromagnetic energy of the system at steady state. Here we observe a radically different scenario if compared with a classical nanolaser. The emitted field is in fact totally confined within the nanodisk resonator, with electromagnetic energy decaying exponentially outside the nanodisk region. The lack of farfield emission provides a characteristic signature of the anapole state that is being amplified through stimulate emission, and is the hallmark of a new type of nearfield nanolaser that does not radiate in the far field. Inside the nanodisk, we observe that the electromagnetic energy distribution preserves the dipolar shape of the linear anapole state (see Supplementary Fig. 7) that is perpendicular to the polarization of the pumping field E_{x}. We evaluated the correlation between the energy distribution of the anapole state amplified in Fig. 3c–e and that of an ideal anapole in a lossless InGaAs structure, obtaining a correlation amplitude of 96.1%. This shows that the anapole laser amplifies through stimulated emission an almost ideal radiationless anapole state. Supplementary Note 1 and Supplementary Figs. 2–5 provide an indepth analysis of the role of losses in the system.
Spontaneously polarized nanolasers and photonics couplers
The possibility to exploit nearfield emission in conjunction with the dipolar nature of the anapole mode opens new pathways to develop spontaneously polarized nanolasers and nanophotonic routers. Polarization control, which is crucial in many linear and nonlinear applications^{43,44,45,46}, is quite a challenging task for nanoscale sources and it is in general addressed with the introduction of metals and/or specific geometries^{47,48,49}. By employing anapole sources we can introduce a new design concept that can be easily integrated into twodimensional circuits. Figure 4 illustrates our idea, where we consider a nanodisk of In_{0.15}Ga_{0.85}As with 440 nm diameter and 100 nm height placed in proximity of silicon nanowires of width 150 nm and thickness 100 nm. The nanowires are distributed in cross configuration, with the anapole nanolaser placed at the intersection point of the circuit. When the incident beam impinges on the anapole with linear polarization (see Fig. 4a), it excites the corresponding anapole state with localized dipolar field aligned perpendicularly to the pump polarization. This depends on the fact that the energy distribution of the anapole, resulting from the nearfield interference between electric and toroidal dipoles, is strongly asymmetric in one spatial direction orthogonal to the input polarization (please see Supplementary Fig. 7). When the anapole is nonlinearly amplified by stimulated emission, the input pump polarization selectively excites the corresponding symmetry direction, leading to different energy distributions in the final anapole state (see Fig. 4a). The electromagnetic field distribution of the anapole couples only into the channels that overlap with the anapole mode, routing light according to the polarization of the input field. Figure 4b reports a volumetric mapping of the spatial distribution of the electromagnetic energy at steady state and illustrate this dynamics. The nearfield characteristics of the anapole allows electromagnetic energy to be efficiently coupled inside nanowire channels even of such a small size. To evaluate this functionality more quantitatively, we considered the same scenario with a standard nanolaser, characterized by a classical spherical nanoparticle with diameter d=448 nm and resonance at 948 nm (Fig. 4e, inset). Figure 4c illustrates the outcome. Due to the strongly radiating nature of the components amplified in the nanoparticle, light scatters almost everywhere without coupling in the nanowire channels. By measuring at the light energy in the nanowire channels, we observe that with an anapole nanolaser we have four orders of magnitude more energy inside the channel with respect to the standard nanoparticle laser (see Fig. 4d,e). Such remarkable result depends on the fact that an anapole nanolaser enables by design nearfield energy coupling, a mechanism that is extremely efficient in transferring energy in neighbouring structures. In Supplementary Fig. 8 we considered in detail the comparison with another integrated lasing structure, namely a nanodisk with the same size of the anapole laser but now set to emit at 1,125 nm. At this wavelength, the structure shows a strongly dipolar energy emission that well couples with the fundamental mode of the nanowire channels. In this case the energy couples inside all channels. The energy transferred in the waveguides is 15 times lower that the one observed with the anapole, showing the superior performances of the anapole laser.
The tunability of the formation of an anapole state in the In_{0.15}Ga_{0.85}As material (see Fig. 2) and the compact geometry of the anapole nearfield nanolaser allows a large design flexibility, where multiple anapoles can be optically pumped within hundreds of nm, feeding light energy into complex nanoscale circuits. In this scenario, even the use of a scanning nearfield optical microscope nanotip or a plasmonic nanoantenna is very challenging, as the macroscopic size of the coupling structure does not easily allow the excitation of selective channels nearby in space.
Anapole mode locking and onchip ultrafast pulse generation
The evanescent emission of energy from anapole nanolasers opens to new concepts for ultrafast pulse generation that results from a mechanism of mode locking that is based on the nearfield synchronization of different anapoles. In photonics, synchronization phenomena that are usually exploited for the generation of optical pulses require expensive setup with specific design elements, such as Qswitching components, saturable absorbers and optical modulators, that cannot be miniaturized to the nanoscale.
Recent experiments reported the observation of mutual synchronization in space of laser sources emitting at the same frequency^{50,51,52} that can reconfigure the spatial distribution of the emitted energy in the far field. These results open the question of whether it is possible to exploit the mutual synchronization in time of different anapole lasers, designing a new process of mode locking that generates pulses without the need of external design elements than the laser itself.
Figure 5a shows the design principle of our idea, where we consider a linear array of anapole nanolasers with slightly different radii. Due to the radius variation, each anapole possesses a different emission frequency ω_{n}=ω_{0}+n·Δω that lies in the 20 nm amplification bandwidth of the In_{0.15}Ga_{0.85}As semiconductor (see Fig. 2b). The collective emission from the anapole chain is collected by a tapered structure into a singlemode nanowire channel of width 150 nm and height 100 nm.
The anapole chain, in this scheme, behaves as a system of weakly coupled, nonlinear oscillators. Nonlinearity is here provided by the presence of the amplifying optical semiconductor. Weak coupling, conversely, arises from the spatial overlapping of the anapole emitted fields (see Fig. 3). Each anapole behaves initially as an independent oscillator at the frequency ω_{n} and phase ϕ_{n}, as illustrated in Fig. 5b. During the time evolution of the system, the frequencies ω_{n} and phases ϕ_{n} interact are allowed to interact via nearestneighbour coupling. To understand the dynamical outcomes of these interactions, we developed a simplified model of light–matter interaction by generalizing the theory of coupled laser systems^{53} to the case of multiple anapole lasers emitting at different frequencies (see Supplementary Note 2 for a full derivation and for a detailed analysis of the model):
where is the complex electric field amplitude of the the nth anapole in the chain, t=t/τ_{p} a dimensionless time measured in units of the photon carrier lifetime τ_{p}, N_{n} the population inversion, α the linewidth enhancement factor, μ=τ_{p}/τ_{s} the ratio of photons τ_{p} to carrier τ_{s} lifetime, P the pumping rate and coupling coefficients, arising from spatial overlaps between weakly overlapping nearestneighbour anapoles.
For α=0, equation (1) reduce to the wellknown Kuramoto model, a universal model of synchronization dynamics that has been largely studied in the past^{54}. At steady state, for increasing pumping rate P and nonzero coupling between the anapoles, both amplitudes and phases tend to lock to a common value A_{n}(t)→A_{0}, ϕ_{n}(t)→ϕ_{0}. Supplementary Figure 9a,b shows an example of this dynamics. For a realistic anapole laser with nonzero α, the frequency of oscillation of each electric field becomes influenced by the carrier population N_{n}. On a timescale smaller than 1/μ, no effect is practically observed and the anapole network behaves as a classical Kuramoto model. Conversely, for times larger than 1/μ the synchronization conditions between the anapole oscillators change, with some anapoles getting detuned from the others. Supplementary Figure 9c,d illustrates an example, obtained for α=1 in an ensemble of 10 anapoles.
Synchronization events in the anapole chain open the interesting possibility to control the time dynamics of the laser at the ultrafast scale. When a synchronization event takes place, in fact, the total emission intensity I(t) acquires the temporal profile of a train of pulses, expressed by the superposition of the harmonics that becomes synchronized:
The period of the train is T=2π/δω and the individual pulse width is Δ=2π/Nδω. These quantities depend on the number N and frequency spacing δω of the anapoles that become synchronized. Following equation (1), the synchronization process is controlled by the pumping rate P and coupling coefficients C_{mn}, as shown in Supplementary Fig. 9.
To validate this theoretical picture for a realistic anapole laser, we performed firstprinciple simulations on the anapole chain. Figure 5d–g shows the lasing behaviour of the anapole array. Figure 5d shows a snapshot of a 3D volume mapping of the energy distribution of the system, showing the nearfield coupling of energy into the nanowire optical channel and the high degree of confinement ensured by the evanescent coupling of the anapoles radiation into the tapered structure. In the steady state, the anapoles mutually synchronize and the structure naturally generates train of pulses, as shown in Fig. 5e where we report the emitted intensity at the end of a 5 μm waveguide. Fig. 5f provides a zoomed view of a single pulse, characterized by an ultrashort duration Δt=95 fs. The spectrum of the electric field (see Fig. 5g) well illustrates the locking of the various emitted frequencies around the amplification band edge of the In_{0.15}Ga_{0.85}As semiconductor (see Fig. 5g dashed line). By extracting the mode spacing Δω=4 × 10^{12} rad s^{−1} and their spectral amplitudes, it is possible to compare the emitted intensity predicted by equation (2) that shows perfect agreement with the results from firstprinciple calculations (see Fig. 5e). Remarkably, the modelocking mechanism is a stable and repeatable feature of the system, as we verified by performing a statistical campaign of 50 distinct simulations. Due to the stochastic properties of quantum noise, each simulation corresponds to a different set of initial conditions for the phases of the nanolasers along the chain.
The dynamical model described by equation (1) of the anapole chain suggests that the number of anapole that becomes synchronized can be changed by acting on the coupling matrix elements C_{mn}. This observation is stimulated by the similitude of equation (1) and the dynamics of oscillatory neural networks^{55}. Simple calculations on equation (1) show that in the case where the connection matrix is suitably engineered to possess real eigenvalues ɛ with orthogonal eigenvectors, the populations converge to the same value N_{n}(t)=−ɛ and all oscillators tend to synchronize their phases. This case is numerically illustrated in Supplementary Fig. 9e,f.
In the anapole chain, coupling coefficients can be altered by acting on the geometric topology of the array. A thorough analysis on this topic goes beyond the scope of this article, and is deferred to a future specialized work. We will here consider a simple and illustrative case, where we shift by 6° the anapole array along the direction y (see Fig. 6a). In this configuration the emission dynamics exhibits the appearance of a larger number of synchronized frequencies in the spectrum (see Fig. 6e), and the stable emission of a pulse train of period 1.65 ps (see Fig. 6c) and pulse duration 145 fs (see Fig. 6d). Figure 6b shows a section along the z plane of the energy energy distribution in the structure, and well illustrates the mechanism of formation of the energy pulses inside the waveguide channel by the complex interactions among the different anapoles.
By comparing the emission spectra of Figs 5g and 6e we observe that the dynamics of mode locking in the anapole nanolasers involve the selective locking of different spectral frequencies that vary in number and position according to the spatial configuration considered (linear in Fig. 5 versus shifted array in Fig. 6). This result opens interesting possibilities to use complex anapole arrays as sources for future optical integrated neurocomputers^{55}.
The modelocking dynamics of Figs 5 and 6 is a characteristic feature of anapoles that is not observed in conventional nanoparticle lasers. Supplementary Fig. 10 reports the results for nanoparticles with diameter 448 nm, arranged in the same geometry of Fig. 5. In this case there is no pulse generation: the collective emission does not mutually synchronize and very little energy is coupled into the nanowire.
Methods
Scattering calculations
All scattering efficiencies for 3D structures have been computed by means of the freely available package DDSCAT^{33}. As an input condition, we considered a monochromatic plane wave illumination directed along the nanodisk axis and with polarization perpendicular to the nanodisk axis (see Supplementary Fig. 7).
MBFDTD simulations
We used a fully vectorial, 3D implementation of the MB equation, where the gain material is described by a quantummechanical fourlevel system^{56,57,58}. The interaction between the FDTD electromagnetic fields E, H and the atoms composing the amplifier is expressed by the polarization response of the material P=P_{lin}+P_{Nlin} that is decomposed into a linear P_{lin} and nonlinear contribution P_{Nlin}. The latter is related to the atomic transitions in the amplifier by the following set of MB equations:
where q_{0} is the quantum displacement length, e is the electric charge, N_{a} represents the density of polarizable atoms in the excited state and τ_{j} are the atomic decaying constant rates. In equation (3), S_{j} is the jth component of the Bloch coherence vector S that groups different complexvalued elements of the density matrix ρ_{mn} into a realvalued vector state. A detailed derivation of equation (3), including the expressions for the coupling Γ_{lm} and Hamiltonian , components can be found in refs 57, 58. As discussed in ref. 41, our MB implementation explicitly considers a realistic source of quantum noise, while the density of excited atoms N_{a} plays the role of an external pumping rate. The optical properties of InGaAs were taken from published experimental data^{32,59}. The dephasing time for the active levels of InGaAs is set as τ=40 fs, following experimental measurements available in the literature^{18,34}. Material dispersion was included in the FDTD algorithm by employing a representation based on a multipolar expansion that correctly represents the media response at visible wavelengths^{58}.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Additional information
How to cite this article: Gongora, J. S. T. et al. Anapole nanolasers for modelocking and ultrafast pulse 2 generation. Nat. Commun. 8, 15535 doi: 10.1038/ncomms15535 (2017).
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Acknowledgements
A.F. acknowledges funding support from KAUST (Award No. OSR2016CRG52995). For the computer time, we have used the resources of the KAUST Supercomputing Laboratory and the Redragon cluster of the PRIMALIGHT group.
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J.S.T.G. and A.E.M. carried out scattering calculations. J.S.T.G. performed firstprinciple parallel simulations. Y.S.K. and A.F. supervised the research. All authors contributed to the analysis of data. All authors contributed equally to manuscript preparation.
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Totero Gongora, J., Miroshnichenko, A., Kivshar, Y. et al. Anapole nanolasers for modelocking and ultrafast pulse generation. Nat Commun 8, 15535 (2017). https://doi.org/10.1038/ncomms15535
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DOI: https://doi.org/10.1038/ncomms15535
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