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Nov 27, 2017 - In this work, we investigate the contribution of longitudinal modes to the enhancement of third-harmonic generation process. Specifical...
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Enhancing Third-harmonic Generation with Spatial Nonlocality Hao Hu, Jingjing Zhang, Stefan A. Maier, and Yu Luo ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b01167 • Publication Date (Web): 27 Nov 2017 Downloaded from http://pubs.acs.org on November 29, 2017

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Enhancing Third-harmonic Generation with Spatial Nonlocality Hao Hu †, Jingjing Zhang†, Stefan A. Maier⊥ and Yu Luo*,† † School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, 639798, Singapore

⊥The Blackett Laboratory, Department of Physics, Imperial College London, London SW7 2AZ, U.K. Keyword: nanostructure, surface plasmons, nonlocal effect, longitudinal modes, third-harmonics Abstract – As the geometrical feature of a nanostructure approaches the Thomas-Fermi screening length, the electron-level interaction induced nonlocality leads to the longitudinal resonance modes above bulk plasmon resonance frequency. In this work, we investigate the contribution of longitudinal modes to the enhancement of third-harmonic generation process. Specifically, we study a single Ag nanowire and a Ag nanowire dimer coated with third-order nonlinear dielectrics. By implementing hydrodynamic and nonlinear models together, we find that the spectral overlap of the longitudinal resonance modes with 3rd order harmonics enables the improvement of the nonlinear conversion efficiency. The optimized results show that despite of reduced field enhancement for the fundamental resonance, the third-harmonic absorption intensities in nonlocal case can surpass the local calculation results by hundreds of times. Maximum third-harmonic scattering intensities can also be realized though appropriate design of both structures. In contrast to previous studies which mainly focus on the negative effects of nonlocality, our study indicates that the nonlocal effects may benefit our system with proper designs, opening a new door for quantum plasmonic research.

Surface plasmons allow light to be concentrated into nanoscale dimensions well below the diffraction limit, inducing strong enhancement of local optical fields.(1) These features of nanoplasmonics benefit a wide range of applications including enhanced nonlinear optics, surface-enhanced Raman scattering (SERS), efficiency-enhanced quantum dots and plasmon based absorber.(2-13) Recently, the tremendous development of nanofabrication and optical characterization techniques have enabled the reduction of nanostructure dimensions towards the atomic regime, e.g., gap system with size of 2.0 ± 0.6 Å has been achieved through top-down nanofabrication method.(14) In parallel to this exciting progress, growing attentions have been paid to the investigation of quantum properties of surface plasmons and the manipulation of light at subnanometric scale.(15-17) Metallic nanoparticles with dimension of tens of nanometers support localized surface plasmon (LSP) in the visible light regime, achieving high localized density of states (LDOS) around the surface and confined electric field hundreds of times larger than the incidence.(18-21) However, when the size of the nanostructures reduces to 10 nm or less, the smearing of electrons will have a noticeable impact on scattering features in far field as well as cause near-field saturation phenomenon.(22-24) In this regime, it is not sufficient to use classical description of metal response --- Drude model, where induced charge densities are perfectly localized at surface as impulse function, while ground-state charge densities are uniformly distributed in the bulk of the metal. In real metals, induced charge densities are not localized exactly at the surface but are slightly spread over a thickness around the boundary.(25-27) Therefore, the quantum description of permittivity must account for the spatial dispersion of charges. Full-quantum approaches, such as time-dependent density functional theory, can provide an electromagnetic description at an atomistic level, but the high demand for computational sources prevents their applicability to larger nanosystems.(26, 28) As a semiclassical theory, the hydrodynamic Drude model describes the spatial nonlocality of electrons in the form of the perturbation term of electron density from quantum repulsion.(25) Recently, Luo et al. has suggested that nonlocality can be represented by replacing the nonlocal metal with a composite material consisting of a thin dielectric layer located on top of a local metal. This simple model greatly simplifies the theoretical treatment of nonlocal systems, extending the domains of classical electrodynamics below the nanoscale.(29) Other quantum effect like quantum tunneling could affect gap system

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with atomic length-scale by accumulation of charges with opposite sign on both sides of the gap, reducing the maximum enhancement of electric field predicted by classical theory.(14, 30, 31) Various efforts have been made to probe the limits of plasmonic enhancement caused by the quantum effects. Ciraci et al studied a gap system containing a film-coupled Au nanosphere above a Au film, and found that when the gap size decreases below 1 nm, nonlocality causes the reduction of field enhancement up to 2 orders of magnitudes and blueshift of the resonant frequency compared to the local theoretical prediction.(25) Crozier et al numerically and experimentally investigated the plasmonic enhancement of metallic dimer, and observed the negative impact of electron tunneling, which is significant especially when the distance of the gap is under 3 Å.(14) While a host of theoretical and experimental studies have been performed to explore how the quantum effects will limit the performance of plasmonic systems, e.g. setting a bound for the field enhancement property, little attention has been paid to the benefits that quantum plasmonic features may bring to us. Here, we study the longitudinal plasmon resonances in nonlocal model and their prospects for enhancing high harmonic generation.(32 – 34) We consider plasmonic nanostructures coated by nonlinear materials, and find that when the frequencies of longitudinal modes overlap with the multiples of fundamental resonance frequency, the electric fields of the high order modes will be enhanced, thereby enhancing the high harmonic generation. Our study indicates that instead of suppressing nonlocal effects, we may in turn make use of them with judiciously designed plasmonic systems. This may pave a new path for quantum plasmonic research, and bring new perspectives for exciting applications such as ultraviolet source, ultraviolet detect, etc.

HYDRODYANMIC AND NONLINEAR IMPLE MENTATION The schematic diagrams of the composite we consider are shown in Figure 1, where infinitely long nanocylinders are embedded in a nonlinear dielectric medium with permittivity ԑd. The many-body electron dynamic of plasmonic material is described by the hydrodynamic model, which takes electron-level interaction into account. The hydrodynamic current density J and electric field E inside a nanoparticle are described by ∇ × ∇ ×  = i −i ∞   +   , ⑴ ⑵

β ∇∇ ∙  +  + iγ = i ω ε ,

where ω is the operating frequency of the incidence, ωp is the plasmon frequency, γ is the collision frequency, ԑ(∞) is the relative permittivity, ԑ0 is the vacuum permittivity, μr is the relative permeability and μ0 is the vacuum permeability. Parameter β is proportional to the Fermi velocity vF as β = 3/5 vF.(33) The inclusion of the pressure term β gives rise to the spatially dependent permittivity for longitudinal response, whereas the transverse response remains unchanged.(35, 36) Here, we focus our discussion on third-harmonic generation (THG) process, because it is available for inversion symmetrical crystals. We applied undepleted pump approximation method to simplify the calculation for THG.(3743) This method is valid only if the nonlinear polarizability is far less than the linear polarizability. Under this assumption, two steps are involved to solve for the electric field of THG. First, nonlinear polarization field for isotropic material is estimated by ⑶



!

"

= ε χ& ' ∙ ' ' #

,

where E1 is the electric field at the fundamental frequency inside the nonlinear dielectric under the linear polarized excitation. Secondly, the nonlinear polarization serves as an external source to excite the system, where the electric field E3 at the third-harmonic frequency can be obtained. The interaction between the nonlinear polarized field and the metallic cores stimulates the longitudinal plasmon resonance, thus enhancing the scattering and absorption of E3 by cores at discrete frequencies. To investigate the contribution of nonlocality to the enhancement of THG, we implement coupled nonlinear hydrodynamic equations and calculate the normalized absorption cross section (ACS) and scattering cross section (SCS) of the core-shell structure under the undepleted pump approximation. All the numerical analysis is carried out with commercially available software COMSOL Multiphysics. Note that our 2D model is not restricted to the specific structures we consider but can be used to investigate the nonlocality enhanced high-harmonic generation for arbitrary nanowire geometries.

RESULT AND DISCUSSION

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Here, we consider a super-thin silver (Ag) nanowire coated by third-order nonlinear dielectrics with radius R of 20 nm. The optical parameters of Ag are taken from (44): the plasmon frequency ћωp=9.0 eV, the damping rate ħγ =0.018 eV and the dielectric constant of metal due to the background electron screening in high frequency ԑ(∞)=5. Meanwhile, the relative permittivity and the third-order susceptibility of nonlinear dielectrics are 2.2 and 4.4 × 1020 m2/V2, respectively. We compare the optical responses (e.g. field enhancements and intensities of high-harmonic signals) of the composite structures under the local and nonlocal treatments. The absorption and scattering spectra of THG is calculated by sweeping the geometrical parameters such as the radius of the single nanowire a and the gap width d of the nanowire dimer. The optimized geometries which maximize the contribution of longitudinal resonances to the intensities of THG can then be determined. The nonlocal results are then compared with the calculations of the local model. Ag single nanowire We first examine the nonlocal response of the single nanowire case. The absorption spectra for radius dimensions from 0.3 nm to 0.8 nm are calculated using both the local and nonlocal models (Figure 2). for small metallic particles (size below 20nm), the local plamonic resonance is determined by equation ԑm = - ԑd, independent  of the radius, where the dielectric function of Ag ε( = ∞ − ),  + *+ . The resonances at both the fundamental and high-harmonic frequencies can enhance THG. However, in the local case, all the surface plasmon modes happen below the bulk plasma frequency ) . Hence, only the field enhancement at the fundamental frequency contributes to the THG process. According to the classical Drude model, all the local plasmon resonance modes degenerate at the surface plasmon frequency ) = 3.35eV, as shown in Figure 2(a). Correspondingly, the third harmonic signal is enhanced at = 10.05eV and its intensity remains almost unchanged for different particle sizes (see Figure 2(c)). This observation is valid for structure dimensions larger than 5 nm where only local responses should be considered. However, the situation becomes quite different for the nonlocal case, where the third harmonic intensities vary nonmonotonically when the particle size decreases (see Figure 2(d)). This nonmonotonic behavior results from the nonlocal electron-electron interaction, which modifies the distribution and the intensity of electric fields, resulting in resonance blueshift and broadening of the transverse surface modes as well as the emergence of longitudinal plasmon modes above ) (see Figure 2(b)).(45 – 47) Different from transverse surface modes which evanescently decay into the metallic particle, the longitudinal modes strongly resonate along the radial direction,(33) as shown in Figure 2(e). For particular particle sizes, these resonances overlap with the third harmonic frequency (the white dash line in Figure 2(b)), and thus provide additional enhancement to the third harmonic intensity, giving rise to two local maxima (i.e. hotspots) in Figure 2(d). In other words, with appropriate design, the nonlocal longitudinal plasmon resonances could be exploited to increase the efficiency of THG. To further investigate the contribution of nonlocality in the THG process, we plot the field enhancements, scattering and absorption spectrum for different nanowire sizes. The top and bottom panels correspond to nanowire radii of 0.6nm and 0.76nm, respectively. For both nanowire sizes, the nonlocal effects result in approximately three-time reduction of the field enhancement as compared to the local calculations, as shown in Figure 3(a) and (d). Accordingly, the absorption and scattering cross sections of 0.6-nm nanowire calculated with nonlocal model are dramatically suppressed (below 40%) compared to that predicted by local model (see Figure 3(b) and (c)). In contrast, as the radius of nanowire is tuned to 0.76nm, we can observe significant enhancement of absorption exactly at the frequency of 3rd harmonics (see Figure 3(e)). This is because for 0.76nm nanowire, the third-harmonic frequency overlaps the frequency of 3rd longitudinal mode while for 0.6nm nanowire this condition is not satisfied (see Figure 2(b)). Moreover, compared with case of Figure 3(c), the scattering intensity also experiences an intensive enhancement by 3rd longitudinal mode, as shown in Figure 3(f). Even though the longitudinal mode cannot offer a complete compensation for the field attenuation induced by nonlocality, the THG efficiency still could be maximized by controlling the particle sizes. Ag nanowire dimer The optical response of Ag nanowire dimer has also been investigated under the quasi-static approximation. Compared to the single nanowire, the dimer structure might be worth more interests in the study of nonlocal problems, because experimentally realizing atomic length scale gap is more feasible than fabricating a nanowire in the same scale. In addition, the nanogap creates plasmonic hotspot with considerable confinement of light, effectively harvesting the electromagnetic energy.(14, 48 – 53) The amplified electric field leads to the intensive field-particle interaction. Hence, strong THG intensity can be triggered by enhanced nonlocal effect. We study the

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cases where the separation of the dimer d>0.4 nm, so that the ignorable quantum tunneling effect can be assumed.(22, 54) Here, we fix the radius of each nanowire at 1.08 nm, and vary the gap size of the dimer d from 0.9nm to 0.4nm. Two peaks can be observed from the local study of absorption spectrum, corresponding to the dipolar mode and the overlapping high order bright modes (see Figure 4(a)), which all show redshift.(55-59) Note that both dipolar mode and high order modes can enhance third harmonics. However, the high order modes are weaker than the dipolar mode by one order of the magnitude. Thus, their contribution to the third harmonics is insignificant here (see Figure 4(c)). Moreover, the decrease of the gap size strengthens the interaction between the nanowires and enhances the electric field at the gap centre. Therefore, a monotonic increase of the absorption in both fundamental and third harmonic frequencies can be observed (see Figure 4(a) and (c)).(60) On the other hand, this monotonic feature breaks down when considering the of nonlocal effects, which results in the saturation of mode redshift as well as the field enhancement (see Figure 4(b)). Therefore, the third harmonic frequency (the white dashed line in Figure 4 (b)) overlaps with the longitudinal mode for a wide range of gap size d, as depicted by Figure 4(d). The saturation of resonance shifts due to nonlocality makes the enhancement of third harmonic generation much less sensitive to the geometrical parameter (gap size here) than in the single nanowire case (radius of nanowire). Thus, this dimer structure is robust against geometrical perturbations, and greatly relaxes the requirement for the fabrication precision in related experiments. Figure 4(e) indicates that the longitudinal modes inside particles dominates the transverse modes localized in the gap, so that the third-harmonic intensity is further promoted.

Similarly, the contribution of nonlocality in the THG process is quantified by analyzing two extreme cases, as shown in Figure 5. The field enhancement in the nonlocal case is one-ninth of that in local case, for both gap widths 0.52 nm and 0.72 nm. Similar to the single nanowire case, we find that the absorption at third harmonic frequency is dramatically enhanced only when the 3rd order harmonics overlap with the longitudinal mode (when d=0.72nm). The scattering peak for d=0.72nm in Figure 5(f) also shows a double growth, compared to the one in Figure 5(c) (d=0.52nm), where the longitudinal frequency does not overlap with the third harmonic frequency.

CONCLUSIONS In summary, we propose the idea of using the nonlocal longitudinal modes to enhance the third-harmonic generation (THG) process. To maximize the THG intensities, the design of single nanowire and nanowire dimer follows the same rule, i.e. the third-harmonic frequency should overlap with longitudinal resonance frequency. In this condition, the absorption intensities could surpass the local prediction by hundreds of times and the scattering intensities could reach their maxima for both structures. We also demonstrate that the absorption spectra yield nonmonotonic behavior due to discrete energies of longitudinal oscillation. Notably, the dimer system shows higher practical importance over the single nanowire structure, as its THG enhancement property is robust to the fabrication perturbation. Such nonlocality-enhanced nonlinear optical phenomenon could offer a possible route toward atomic-length-scale photonic devices at ultraviolet frequencies.(61-63)

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AUTHOR INFORMATION Corresponding Author *Yu Luo, Email: [email protected].

Notes All authors have given approval to the final version of the manuscript. The Authors declare no competing financial interest.

ACKNOWLEDGMENTS Y. L., H.H. and J.Z. acknowledge the support by Nanyang Technological University start-up grant, Singapore Ministry of Education (MOE) under grant no. MOE2015-T2-1-145, and NRF-CRP grant under grant no. NRF2015NRF-CRP002-008. S.A.M. acknowledges the EPSRC (EP/L 204926/1), the Royal Society, and the Lee-Lucas Chair in Physics.

ABBREVIATIONS SERS, surface-enhanced Raman scattering; LSP, localized surface plasmon; LDOS, localized density of states; THG, third-harmonic generation.

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53. Fernandez-Dominguez, A.I.; Zhang, P.; Luo, Y.; Maier, S.A.; Garcia-Vidal, F.J.; Pendry, J.B. Transformation-optics insight into nonlocal effects in separated nanowires. Phys Rev B 2012, 86(24), 241110. 54. Savage, K.J.; Hawkeye, M.M.; Esteban, R.; Borisov, A.G.; Aizpurua, J.; Baumberg, J.J. Revealing the quantum regime in tunnelling plasmonics. Nature 2012, 491(7425), 574-577. 55. Cherqui, C.; Wu, Y.Y.; Li, G.L.; Quillin, S.C.; Busche, J.A.; Thakkar, N. et al. STEM/EELS Imaging of Magnetic Hybridization in Symmetric and Symmetry-Broken Plasmon Oligomer Dimers and All-Magnetic Fano Interference. Nano Lett 2016, 16(10), 66686676. 56. Huang, Y.; Ma, L.W.; Hou, M.J.; Li, J.H.; Xie, Z.; Zhang, Z.J. Hybridized plasmon modes and near-field enhancement of metallic nanoparticle-dimer on a mirror. Sci Rep-Uk 2016, 6, 30011. 57. Li, G.C.; Zhang, Y.L.; Lei, D.Y. Hybrid plasmonic gap modes in metal film-coupled dimers and their physical origins revealed by polarization resolved dark field spectroscopy. Nanoscale 2016, 8(13), 7119-7126. 58. Lei, D.Y.; Aubry, A.; Maier, S.A.; Pendry, J.B. Broadband nano-focusing of light using kissing nanowires. New J Phys 2010, 12, 093030. 59. Fernandez-Dominguez, A.I.; Wiener, A.; Garcia-Vidal, F.J.; Maier, S.A.; Pendry, J.B. Transformation-Optics Description of Nonlocal Effects in Plasmonic Nanostructures. Phys Rev Lett 2012, 108(10), 106802. 60. Zhu, WQ.; Esteban, R.; Borisov, A.G.; Baumberg, J.J.; Nordlander, P.; Lezec, H.J. et al. Quantum mechanical effects in plasmonic structures with subnanometre gaps. Nat Commun 2016, 7, 11495. 61. Manjavacas, A.; Liu, J.G.; Kulkarni, V.; Nordlander, P. Plasmon-Induced Hot Carriers in Metallic Nanoparticles. Acs Nano 2014, 8(8), 7630-7638. 62. Vadai, M.; Selzer, Y. Plasmon-Induced Hot Carriers Transport in Metallic Ballistic Junctions. J Phys Chem C 2016, 120(37), 21063-21068. 63. Li, D.; Wang, Y.; Nakajima, M.; Hashida, M.; Wei, Y.; Miyamoto, S. Harmonics radiation of graphene surface plasmon polaritons in terahertz regime. Phys Lett A 2016, 380(25-26), 2181-2184.

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Title: Enhancing third-harmonic generation with spatial nonlocality Authors: Hao Hu, Jingjing Zhang, Stefan A Maier and Yu Luo Synopsis: The benefits of the nonlocal effects are systematically studied for the first time in this work. By coating the Ag nanostructures with third-order nonlinear dielectrics, the third harmonics of fundamental frequency can excite strong longitudinal mode resonance in Ag dimers with proper geometric design. Thereby the radiation of longitudinal modes can provide an enhancement for third-harmonic generation (THG) process in ultraviolent frequency.

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Figure 1. Schematics of a single nanowire (a) and a nanowire dimer (b) coated by nonlinear dielectric. The radii of the nanowire and the dielectric shell are a and R, respectively. The gap in the nanowire dimer is d. 92x50mm (300 x 300 DPI)

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Figure 2. The absorption cross section of single Ag nanowire, as the function of radius of the nanowire a and the frequency of incoming photons in eV unit. (a), (c) show the results calculated by the local model, and (b), (d) are obtained by the nonlocal model. Panels (a) and (b) plot the absorption cross-section at the fundamental frequency, while panels (c) and (d) correspond to the intensity of the third harmonic generation. The colorbars of (a) and (b) are prepared in the logarithm scale. The white dashed line denotes the triple frequency of the fundamental mode. Panel (e) shows the electric field distributions of the plasmonic system at the third-harmonic frequency for local and nonlocal cases. 230x273mm (300 x 300 DPI)

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Figure 3. Electric field enhancements (Panel (a)/(d)), normalized absorption cross-sections (Panel (b)/(e)) and scattering cross-sections (panel (c)/(f)) versus the incident photon energy. For panels (a), (b) and (c), the radius of the nanowire is 0.6 nm, where the frequencies of the longitudinal resonances deviate from the third harmonic one. For panels (d), (e) and (f), the radius of the nanowire is set 0.76 nm, and hence the third-order longitudinal mode occurs at the third-harmonic frequency. 91x49mm (300 x 300 DPI)

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Figure 4. The absorption cross section of Ag nanowire dimer, as the function of gap width of the nanowire d and the frequency of incoming photons in eV unit, and the radii of nanowires are 1.08 nm. (a), (c) show the results calculated by the local model, and (b), (d) are obtained by the nonlocal model. Panels (a) and (b) plot the absorption cross-section at the fundamental frequency, while panels (c) and (d) correspond to the intensity of the third harmonic generation. The colorbars of (a) and (b) are prepared in the logarithm scale. The white dashed line denotes the triple frequency of the fundamental mode. Panel (e) shows the electric field distributions of the plasmonic system at the third-harmonic frequency for local and nonlocal cases. 230x272mm (300 x 300 DPI)

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Figure 5. Electric field enhancements (Panel (a)/(d)), normalized absorption cross-sections (Panel (b)/(e)) and scattering cross-sections (panel (c)/(f)) versus the incident photon energy. For panels (a), (b) and (c), the gap width of the nanowires is 0.52 nm, where the frequencies of the longitudinal resonances deviate from the third harmonic one. For panels (d), (e) and (f), the gap width of the nanowires is set 0.72 nm, and hence the third-order longitudinal mode occurs at the third-harmonic frequency. The radii of the nanowires are 1.08 nm. 89x47mm (300 x 300 DPI)

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