Solitary Waves in Chains of High-Index Dielectric ... - ACS Publications

Sep 9, 2016 - solitary waves and effective generation of the new frequencies. ... KEYWORDS: silicon nanoparticles, discrete waveguides, solitary waves...
0 downloads 0 Views 4MB Size
Download PDF
Article pubs.acs.org/journal/apchd5

Solitary Waves in Chains of High-Index Dielectric Nanoparticles Roman S. Savelev,*,† Alexey V. Yulin,† Alexander E. Krasnok,† and Yuri S. Kivshar†,‡ †

Laboratory of Nanophotonics and Metamaterials, ITMO University, St. Petersburg 197101, Russia Nonlinear Physics Centre, Research School of Physics and Engineering, Australian National University, Canberra, ACT 2601, Australia

‡

S Supporting Information *

ABSTRACT: We study both linear and nonlinear propagation of pulses in waveguides composed of high-index dielectric nanoparticles supporting both electric and magnetic resonances. We reveal that short pulses (∼100 fs) broaden significantly in the linear regime after propagating only several tens of micrometers, due to a strong waveguide dispersion. In the nonlinear regime, the pulses propagating in the chains of spherical nanoparticles broaden even more strongly than in the linear regime due to defocusing nonlinearity. However, for the chains of nanodisks the pulse broadening can be compensated by the nonlinear effect, due to the interplay of the electric and magnetic resonances that can change the sign of the group-velocity dispersion for some frequencies, making possible the formation and propagation of solitary waves and effective generation of the new frequencies. Our results demonstrate that considered systems can serve as a promising platform for nonlinear and ultrafast nanophotonics, allowing the observation of strong nonlinear effects at the micrometer scales. KEYWORDS: silicon nanoparticles, discrete waveguides, solitary waves, Mie-type resonance, Kerr nonlinearity

T

generation.12,27−31 When considering a nonlinear regime of propagation in a waveguide, one expects other nonlinear effects, such as formation of optical solitons, i.e., optical pulses that propagate through the waveguide retaining their shape.32 Solitons in optical fibers were theoretically predicted33 and observed in experiment34 several decades ago and found practical applications in long-distance communication systems and in new coherent sources of light.32,35 Optical solitary waves are the phenomenon known not only in optical fibers but also in other optical systems, and they also have their spatial and spatiotemporal counterparts.32 The photonic systems supporting different types of optical solitons include photonic nanowires, 36−38 plasmonic nanostructures,39,40 semiconductors,41 photonic crystals,42−44 optical waveguide arrays,45 discrete waveguide arrays,46−49 etc. In the waveguide geometry, the length scale of the structure required for the soliton formation depends on several main parameters, including effective cross-section area, pulse duration, and nonlinear refractive index of the material. Thereby silicon nanowires with a very small cross-section area and very high nonlinear refractive index serve as one of the smallest platforms for observation of optical solitons and other related nonlinear effects. However, due to the weak waveguide dispersion, group velocity dispersion (GVD) of Si nanowires is of the order of ps2/m, and a typical dispersion length for 100 fs

he modern trends to replace the components of electronic integrated circuits with miniaturized optical devices require short interconnections and subwavelength waveguides.1−3 One of the promising realizations of such waveguides is a chain of high-index dielectric nanoparticles, such as silicon (Si) ones.4−7 The progress of optical technologies resulted in fabrication of the nanoparticles where both magnetic dipole (MD) and electric dipole (ED) resonances in the visible wavelength range were experimentally observed.8,9 To date, fabrication methods of different nanostructures composed of such nanoparticles were also reported.10−14 The unique optical properties of such nanostructures attracted much interest, and they have been extensively studied in the last several years.15 In particular, the high refractive index of silicon makes it possible to fabricate nanoparticle discrete waveguides having subwavelength transverse dimensions and demonstrating considerably lower attenuation rates compared with plasmonic waveguides. The presence of both MD and ED responses in high-index dielectric nanoparticles provides us with additional control over the waveguide dispersion and light scattering. Because of the aforementioned properties, silicon nanoparticles were also suggested as building blocks for nanoantennas,16,17 metasurfaces,13,18−22 and metamaterials23−26 with unique properties and functionalities. Linear properties of light scattering by nanostructures composed of Si nanoparticles are well studied.15 However, the study of their nonlinear properties has been initiated only in the last couple of years, mainly for the study of nonlinear scattering of light by Si nanopartices and third-harmonic © 2016 American Chemical Society

Received: June 8, 2016 Published: September 9, 2016 1869

DOI: 10.1021/acsphotonics.6b00384 ACS Photonics 2016, 3, 1869−1876

ACS Photonics

Article

pulses in such waveguides is about several millimeters.36−38 Photonic crystal waveguides are charaterized by much higher GVD (∼ps2/mm) due to their periodic structure, and therefore they allow the observation of solitons on a millimeter scale for longer picosecond pulses.43,44 Observation of temporal light− matter solitons on the length of 600 μm was reported in semiconductor waveguides coupled to quantum wells.41 In this paper, we show that the formation of optical solitons as well as other nonlinear temporal effects in the waveguides composed of Si nanoparticles (shown schematically in Figure 1) is possible on the length scale of several tens of micrometers.

in what follows we neglect longitudinal modes and consider only transverse modes with fixed polarization. We also take the constant value of permittivity ε = 14 (silicon permittivity at the wavelength ∼800 nm) and therefore neglect the material dispersion and losses. It may be justified, since as it will be shown further, these factors make a quite small contribution to overall dispersion and waveguide properties of the considered structures. In the case of spherical nanoparticles with radius R = 90 nm and period a = 200 nm we obtained the dispersion for the transversely polarized modes shown in Figure 2a. Blue dashed curves are obtained with analytical calculation within the framework of the dipole model, and black solid curves with direct numerical calculation. Waveguide modes with β > ω/c are located under the light line β = k. The broadening of the pulse is governed by GVD parameter β2 = d2β/dω2, which is shown in Figure 2b as a function of normalized frequency. The simulations revealed that β2 is very high, ≳50 ps2/m, in most of the transmission band, where β is detuned from the light line and the radiation leakage is negligible. This means that very short (e.g., 100 fs) pulses will be significantly broadened after propagation for only tens of micrometers. Such values of dispersion length are very small compared to the conventional optical fibers32 and silicon nanowires36 and can be compared to that of photonic crsytals and fiber Bragg gratings.43,44,53 The change in β2 that would be caused by material dispersion of silicon is on the order of 1 ps2/ m in the near-infrared range, which is negligible, compared to the calculated β2 with the constant ε. Nonlinear properties of a nanoparticle waveguide depend on the nonlinear properties of the material. Here we consider the nanoparticles made of silicon, and this material exhibits a selffocusing-type Kerr-like nonlinear response like other high index dielectric materials.2 Calculation of the dependence of ω″ on normalized frequency for the chain of silicon nanospheres, shown in Figure 2c, reveals that it is negative in the whole transmission band, which means that at high intensities the nonlinearity accelerates the broadening of the pulse. This situation is always realized when MD and ED resonances are well-separated in frequency (i.e., the case of transversely polarized modes of spherical nanoparticles) or do not interact with each other (i.e., the case of longitudinally polarized modes). For the materials with self-focusing nonlinearity the dispersion of the nanoparticle waveguide ω″ has to be positive. It can be realized by exploiting the properties of nonspherical silicon nanoparticles. For the chain of spherical particles due to the weak interaction between MDs and EDs the corresponding dispersion branches are almost independent of each other. When resonance frequencies get closer, transversely polarized MD and ED modes start to hybridize,54 and because of this, the modes belonging to the second branch of the dispersion characteristics can have positive ω″. Let us also mention that the group velocity of the mode can be negative for some frequencies. Two examples of this situation are shown in Figure 2d−f and g−i, respectively. In the first case we simulated a chain of flatwise-located nanorods with radius R = 90 nm and height h = 180 nm, and in the second a chain of sidewiselocated nanodisks with radius R = 130 nm and height h = 120 nm. In both cases the period of the chain is a = 200 nm. To describe the response of nonspherical nanoparticles analytically, we employ approximate Lorentzian-shape expressions for the magnetic and electric polarizabilities (see Supporting Informa-

Figure 1. Schematic of a solitary wave propagating in a chain waveguide made of Si nanodisks. Each nanodisk is characterized by magnetic m and electric p dipole moments oriented perpendicular to the propagation direction.

We study theoretically both linear and nonlinear properties of such waveguides in the context of pulse propagation and reveal that, despite a low quality factor of a single Si nanoparticle and large GVD of nanoparticle waveguides, the pulse broadening due to the effect of the Kerr nonlinearity can be fully compensated by the negative dispersion in the waveguides created by chains of nanodisks with closely spaced electric and magnetic dipole resonances.

■

DISPERSION OF SI NANOPARTICLE WAVEGUIDES In order to achieve the full compensation of GVD by nonlinearity, specific linear properties of subwavelength waveguides are required. In our case, we need the nonlinear refractive index of the material and the dispersion parameter ω″ = d2ω/dβ2 (ω is the frequency and β is the wavenumber of the waveguide mode) to be either both positive or both negative. Therefore, we start with the calculation of the linear dispersion properties of the infinite chain of high-index dielectric (e.g., silicon) nanoparticles analytically and numerically. For an analytical description we employ a coupled MD and ED model6,50−52 (see Supporting Information for details), while numerically chain eigenmodes are calculated in CST Microwave Studio. Such dielectric nanoparticle waveguides support longitudinally polarized MD and ED modes and two (degenerate in the case of spherical particles) transversely polarized hybrid MD and ED modes.6 However, as it will be shown further, in the nonlinear regime we are interested only in the hybrid modes formed by interacting MD and ED resonances of the single particle at close frequencies. Therefore, 1870

DOI: 10.1021/acsphotonics.6b00384 ACS Photonics 2016, 3, 1869−1876

ACS Photonics

Article

Figure 2. (a, d, g) Dispersion curves for the transversely polarized modes of an infinite chain of silicon (a) nanospheres with radius R = 90 nm, (d) nanodisks with radius R = 90 nm and height h = 180 nm, and (g) nanodisks with radius R = 130 nm and height h = 120 nm; period is a = 200 nm in all cases. βa/π is the normalized Bloch wavenumber and ka/π is the normalized frequency. (b, e, h) GVD parameter β2 and (c, f, i) dispersion parameter ω″ (only numerical calculations are shown) for the modes of an infinite chain of nanospheres, nanorods, and nanodisks, respectively. Solid black curves are obtained via direct numerical calculations and dashed blue curves via an analytical dipole model.

tion). Comparison of the analytical results (solid curves) and exact solutions (dashed curves) in Figure 2d,e,g,h shows that the dipole model can adequately describe the dispersion properties of a nonspherical-nanoparticle chain. In the case of nanorods with approximately the same diameter and height the resonance frequencies of MD oriented along the diameter of the cylinder and ED oriented along the cylinder axis are closer to each other than in the case of spherical particles.11,55 From Figure 2d−f we observe that it results in the change of the signs of group velocity and ω″ of

the upper branch (Figure 2e,f). In the case of nanodisks with radius R = 130 nm and height h = 120 nm resonance frequencies of MD and ED oriented along the diameter almost coincide,56,57 and the change in the upper branch is more pronounced (Figure 2g−i).

■

TEMPORAL EVOLUTION AND MAIN PARAMETERS In order to describe the evolution of the induced magnetic and electric moments in time and take into account the resonance frequencies shifts due to the nonlinear response, we transform 1871

DOI: 10.1021/acsphotonics.6b00384 ACS Photonics 2016, 3, 1869−1876

ACS Photonics

Article

Figure 3. Dynamics of electric dipole moments in a chain of Si (a, b) nanospheres, (c, d) flatwise-oriented nanorods, and (f, e) sidewise-oriented nanodisks, excited by illuminating the first particle with a Gaussian 100 fs pulse with normalized center frequency of 0.551, 0.594, and 0.553, respectively. (a), (c), and (f) correspond to the linear regime and (b), (d), and (e) to the nonlinear regime with intensities I = 3 GW/cm2, I = 2.2 GW/cm2, and I = 2.5 GW/cm2, respectively.

γe and γm but also by the imaginary parts of the coupling constants. The radiative losses are finite for a single particle, but a Bloch mode can propagate in a chain of particles without radiative losses. This happens because of the destructive interference of the individual emissions from the particles. It is obvious that the nearest neighbors approximation does not allow accounting for this effect and long-range interactions become important. Let us note here that for the realistic parameters the losses are very small even for quite short pulses. The main difficulty in experimental studies of the nonlinear light propagation in discussed waveguides is that the pulses must be short enough to be well accommodated in the waveguides and that their intensity must be high enough to compensate for the dispersion of pulses that short. Let us estimate the pulse intensity needed to observe the nonlinear effect in the suggested silicon nanoparticle waveguiding systems. The values of n2 for silicon reported in the literature vary in the range n2 = (2−10) × 10−18 m2/W.2,59−61 We choose here n2 = 8 × 10−18 m2/W. This value should be multiplied by ∼10, which is approximately the value of enhancement of the electric field intensity inside the particles averaged over the nanoparticle volume at both MD and ED resonances. Therefore, for realistic intensities I = 5 GW/cm2, reported in experiments with silicon nanoparticles,27,28 the relative refractive index and consequently the relative resonance frequency can be shifted for Δn/n = Δωr/ωr ≈ 0.4% (see Supporting Information). In our calculations we used values of intensity up to 7 GW/cm2, which allows us to neglect other types of nonlinear response in Si, due to free carrier generation and thermal nonlinearity. We also note that low-loss silicon nanoparticles have a very high damage threshold, over 100 GW/cm2, according to the known data from the literature.28,62

frequency domain equations into the time domain (see Supporting Information). To simplify the model, we neglect the dispersion of coupling constants. We assume that the interaction between the dipoles is instantaneous, which is a quite good approximation, despite that we take into account the interaction between all dipoles in the chain. This is so because the interaction strength rapidly decreases with the distance and the retardation time between a few neighboring dipoles is much smaller than the propagating pulse duration. In addition, in the modeling of the nanoparticle nonlinear response we suggested that the spatial distribution of the electric field inside the particles does not change significantly even at the maximum possible intensity. That is, the increase of intensity only shifts the resonance frequency of the particle, but the field distributions in eigenmodes remain practically unchanged. We emphasize that this is a good approximation since the nonlinear response of a single nanoparticle is relatively weak; that is, the shift of the relative resonance frequency due to nonlinearity is very small (on the order of 0.1%). Because of the weak nonlinear response and the small overlap of the field distributions of the modes corresponding to the MD and ED resonances in the single particle, we can also assume that these modes also do not interact in the nonlinear regime. We also neglected the nonlinear absorption coefficient (two-photon absorption), which is assumed to be insignificant compared to the nonlinear refractive index n2.58 It is worth noting that taking into account long-range coupling is essential especially in the case of transverse polarization. This is so, first, because the parameter ωr/c (r is the distance between the two interacting dipoles) is of the order of 1 for the neighboring dipoles, and, second, because the effective radiative losses are governed not only by the constants 1872

DOI: 10.1021/acsphotonics.6b00384 ACS Photonics 2016, 3, 1869−1876

ACS Photonics

■

Article

PULSE PROPAGATION IN SI NANOPARTICLE WAVEGUIDES We perform calculations of MD and ED dynamics by solving the system of evolution equations (see Supporting Informa-

dynamics of pulse propagation are shown in Figure 3c,d for the chain of nanorods and in Figure 3e,f for the chain of nanodisks. GVD parameters β2 are approximately the same as in Figure 3a; therefore dynamics in the linear cases (Figure 3a,c,e) are similar to each other. However, the nonlinear regime of the pulse propagation is very different from the one shown in Figure 3b. It is seen that in the chain of nonspherical particles the pulse broadening is completely eliminated at a certain value of intensity. In Figure 4 the width of the propagating pulse, calculated at the 75th, 25th, and first particles (for the case of sidewiseoriented nanodisks) is shown as a function of the normalized intensity. The intensity required for full compensation of GVD is ∼2.5 GW/cm2, while at higher intensities we observe further pulse compression. We emphasize that the predicted selftrapping of the propagating pulse in such nanoparticle waveguides requires the existence of both magnetic and electric dipole responses at close frequencies, what cannot be achieved in the spherical dielectric nanoparticles. We also model the collision of two pulses with normalized frequencies 0.549 and 0.553, launched in a chain of 1000 sidewise-oriented nanodisks. Delay times for the first and second pulses are τd = τp and τd = 12τp, respectively. The second pulse corresponds to the one in Figure 3e, and the intensity required for the soliton formation is I2 = 2.5 GW/cm2 in this case. The first pulse is characterized by lower group velocity and higher GVD, and therefore the required intensity is higher, I1 = 7 GW/cm2. In Figure 5a we demonstrate the typical case of soliton interaction, when pulses remain practically unchanged after the collision. In Figure 5b intensities are 2 times lower (I1 = 3.5 GW/cm2, I2 = 1.3 GW/cm2) and the nonlinear interaction is accompanied by the energy redistributions between pulses. As an example of other temporal nonlinear effects we consider the generation of dispersive waves spectrally shifted from the soliton peak frequency. We launch the pulse with I = 2.5 GW/cm2 intensity in the chain of 600 nanodisks (Figure 2g−i) at the normalized frequency ka/π = 0.556, which is close to zero GVD frequency ka/π = 0.56. Spectra of the soliton calculated at 50th, 250th, and 450th particles are shown in Figure 6a with black, blue, and red curves, respectively. We observe that after propagating about 100 particles additional

Figure 4. Width of the pulse with 0.553 normalized center frequency propagating through the chain of sidewise-oriented nanodisks (Figure 2g−i) as a function of the incident pulse intensity calculated at the 75th particle (solid curve), 25th particle (dashed curve), and first particle (dashed-dotted curve). Spectral width of the incident pulse is 100 fs.

tion) with the Runge−Kutta method of fourth-order. We assume that the first particle is illuminated by a Gaussian pulse with duration τp = 100 fs and the maximum amplitude of magnetic field H͠ 0 ; that is, the envelope of the magnetic field is H(t) = H0 exp(−2 ln 2(t − τd)2/τp2), where we choose the delay time τd = τp. First, we consider the chain of nanospheres with negative ω″ and launch the pulse at the normalized frequency ka/π = 0.551. The dynamics of a 100 fs pulse propagation is shown in Figure 3a for the linear case and in Figure 3b for the nonlinear case. Clearly the pulse is broadening very fast and the increase of intensity accelerates the broadening. Then we consider two cases with positive ω″: we excite the chain of flatwise-oriented nanorods at the normalized frequency ka/π = 0.594 and the chain of sidewise-oriented nanodisks at the normalized frequency ka/π = 0.553, respectively. The

Figure 5. Dynamics of the two pulses propagating in a chain of sidewise-oriented silicon nanodisks (only central parts of the chains are shown). Normalized pulse center frequencies are 0.549 and 0.553, respectively. Intensities of the external field are (a) I1 = 7 GW/cm2, I2 = 2.5 GW/cm2, and (b) I1 = 3.5 GW/cm2, I2 = 1.3 GW/cm2. Pulses time delays are τd1 = τp and τd2 = 12τp. 1873

DOI: 10.1021/acsphotonics.6b00384 ACS Photonics 2016, 3, 1869−1876

ACS Photonics

Article

lower frequencies of ka/π ≈ 0.552. To understand the nature of this peak in Figure 6b, we plot the waveguide dispersion β̃(k) in the reference frame of a moving soliton for the initial frequency ka/π = 0.556 and shifted frequency ka/π = 0.552. Here β̃ = (β(vs − 1) + ks)/vs, ks = ωs/c, ωs is the soliton frequency, and vs is the group velocity of the soliton in units of c. The crossing points of the soliton linear dispersion (horizontal line) and the waveguide dispersion define the frequencies of the dispersive waves (marked points), which are excited through resonance energy transfer from the propagating soliton. The shaded area between these two points indicates the range of possible frequencies of excited dispersive waves. Since the modes under the light line are radiative, propagating dispersive waves are excited only in the approximate frequency range from 0.57 to 0.573, which well corresponds with Figure 6a. In order to charaterize the propagating signal in both the spectral and time domain, we employ the cross-correlation frequency resolved optical gating (XFROG) method.63 That is, we plot the convolution of the pulse spectrum calculated at 450th particle with the spectrum of the reference Gaussian pulse with duration τ = 250 and maximum at different points in time τref. We note that measurement of such a map can be done experimentally by exploiting a nonlinear material with quadractic nonlinearity. In Figure 6c we observe that a resonantly excited dispersive wave that has a sligthly higher group velocity than that of the soliton detaches from the initial pulse in both time and frequency domains after propagating several tens of micrometers. We mention that the subwavelength size of the nanoparticles makes it difficult to couple light in one nanoparticle, and for this a nanoantenna design may be required.7 However, the solution to this problem could decrease the minimum intensity required for a dispersion compensation. Another issue we would like to note is that in our simulations we did not take into account the substrate holding the nanodisks. The dispersion properties of the chain strongly depend on the permittivity of the substrate, and their modification is most pronounced for the substrate made of material with high refractive index, while for materials with a refractive index much lower than that of silicon, e.g., silica, the influence on the dispersion is expected to be quite small, especially for certain geometries.64

■

Figure 6. (a) Spectra of the pulse propagating along the chain of 600 particles, measured at the nth particle; n = 50, black solid curve; n = 250, blue long-dashed curve; n = 450, red short-dashed curve; straight lines ka/π = 0.556 and ka/π = 0.56 correspond to the center frequency of the external pump and the zero GVD frequency, respectively. (b) Black and blue curves show the upper dispersion branch (including radiative modes with complex frequencies) of the chain of nanodisks (see Figure 2g) in the moving frame of reference at two different normalized frequencies, ka/π = 0.556 and ka/π = 0.552; horizontal lines are the linear soliton dispersion k = vsβ, where vs is the group velocity at the corresponding frequency; dashed oblique lines are the light lines. Shaded area indicates the approximate frequency range of generated dispersive waves. (c) Characterization of the pulse at the 450th particle with cross-correlation frequency resolved optical gating (XFROG) in logarithmic scale.

CONCLUSIONS

We have shown that femtosecond pulses propagating in the waveguides made of Si nanospheres broaden significantly in the linear regime after propagating only tens of micrometers. In the waveguides formed by nonspherical Si nanoparticles, the positive dispersion ω″ can be achieved for closely separated magnetic and electric dipole resonances, and this dispersion can compensate for the effective Kerr nonlinearity, making possible the formation and propagation of solitary waves. We have suggested two possible designs of such all-dielectric waveguides made of Si nanodisks when the soliton propagation can be realized due to an interplay of electric and magnetic resonances. We have shown that a resonant transfer of energy from the propagating soliton to the frequency-shifted dispersive waves is possible on the micrometer scale.

spectral components start to appear, while the center frequency of the soliton shifts from the initial frequency ka/π = 0.556 to 1874

DOI: 10.1021/acsphotonics.6b00384 ACS Photonics 2016, 3, 1869−1876

ACS Photonics

■

Article

A. E.; Brener, I.; Fedyanin, A. A.; et al. Nonlinear Interference and Tailorable Third-Harmonic Generation from Dielectric Oligomers. ACS Photonics 2015, 2, 578−582. (13) Decker, M.; Staude, I.; Falkner, M.; Dominguez, J.; Neshev, D. N.; Brener, I.; Pertsch, T.; Kivshar, Y. S. High-Efficiency Dielectric Huygens’ Surfaces. Adv. Opt. Mater. 2015, 3, 813−820. (14) Dmitriev, P. A.; Makarov, S. V.; Milichko, V. A.; Mukhin, I. S.; Gudovskikh, A. S.; Sitnikova, A. A.; Samusev, A. K.; Krasnok, A. E.; Belov, P. A. Laser fabrication of crystalline silicon nanoresonators from an amorphous film for low-loss all-dielectric nanophotonics. Nanoscale 2016, 8, 5043−5048. (15) Krasnok, A.; Makarov, S.; Petrov, M.; Savelev, R.; Belov, P.; Kivshar, Yu. Proc. SPIE 2015, 9502, 950203. (16) Krasnok, A. E.; Miroshnichenko, A. E.; Belov, P. A.; Kivshar, Y. S. All-dielectric optical nanoantennas. Opt. Express 2012, 20, 20599− 20604. (17) Rolly, B.; Stout, B.; Bonod, N. Boosting the directivity of optical antennas with magnetic and electric dipolar resonant particles. Opt. Express 2012, 20, 20376−20386. (18) Staude, I.; Miroshnichenko, A. E.; Decker, M.; Fofang, N. T.; Liu, S.; Gonzales, E.; Dominguez, J.; Luk, T. S.; Neshev, D. N.; Brener, I.; et al. Tailoring Directional Scattering through Magnetic and Electric Resonances in Subwavelength Silicon Nanodisks. ACS Nano 2013, 7, 7824−7832. (19) Lin, D.; Fan, P.; Hasman, E.; Brongersma, M. L. Science 2014, 345, 298−302. (20) Khorasaninejad, M.; Aieta, F.; Kanhaiya, P.; Kats, M. A.; Genevet, P.; Rousso, D.; Capasso, F. Achromatic Metasurface Lens at Telecommunication Wavelengths. Nano Lett. 2015, 15, 5358−5362. (21) Arbabi, A.; Horie, Y.; Bagheri, M.; Faraon, A. Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission. Nat. Nanotechnol. 2015, 10, 937−943. (22) Shalaev, M. I.; Sun, J.; Tsukernik, A.; Pandey, A.; Nikolskiy, K.; Litchinitser, N. M. High-Efficiency All-Dielectric Metasurfaces for Ultracompact Beam Manipulation in Transmission Mode. Nano Lett. 2015, 15, 6261−6266. (23) Vynck, K.; Felbacq, D.; Centeno, E.; Căbuz, A. I.; Cassagne, D.; Guizal, B. All-Dielectric Rod-Type Metamaterials at Optical Frequencies. Phys. Rev. Lett. 2009, 102, 133901. (24) Huang, X.; Lai, Y.; Hang, Z.; Zheng, H.; Chan, C. Dirac cones induced by accidental degeneracy in photonic crystals and zerorefractive-index materials. Nat. Mater. 2011, 10, 582−586. (25) Moitra, P.; Yang, Y.; Anderson, Z.; Kravchenko, I. I.; Briggs, D. P.; Valentine, J. Realization of an all-dielectric zero-index optical metamaterial. Nat. Photonics 2013, 7, 791−795. (26) Jahani, S.; Jacob, Z. All-dielectric metamaterials. Nat. Nanotechnol. 2016, 11, 23−36. (27) Shcherbakov, M. R.; Neshev, D. N.; Hopkins, B.; Shorokhov, A. S.; Staude, I.; Melik-Gaykazyan, E. V.; Decker, M.; Ezhov, A. A.; Miroshnichenko, A. E.; Brener, I.; et al. Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response. Nano Lett. 2014, 14, 6488−6492. (28) Makarov, S.; Kudryashov, S.; Mukhin, I.; Mozharov, A.; Milichko, V.; Krasnok, A.; Belov, P. Tuning of Magnetic Optical Response in a Dielectric Nanoparticle by Ultrafast Photoexcitation of Dense Electron−Hole Plasma. Nano Lett. 2015, 15, 6187−6192. (29) Shcherbakov, M. R.; Vabishchevich, P. P.; Shorokhov, A. S.; Chong, K. E.; Choi, D.-Y.; Staude, I.; Miroshnichenko, A. E.; Neshev, D. N.; Fedyanin, A. A.; Kivshar, Y. S. Ultrafast All-Optical Switching with Magnetic Resonances in Nonlinear Dielectric Nanostructures. Nano Lett. 2015, 15, 6985−6990. (30) Smirnova, D. A.; Khanikaev, A. B.; Smirnov, L. A.; Kivshar, Y. S. Multipolar third-harmonic generation driven by optically-induced magnetic resonances. ACS Photonics 2016, 3, 1468−1476. (31) Baranov, D. G.; Makarov, S. V.; Milichko, V. A.; Kudryashov, S. I.; Krasnok, A. E.; Belov, P. A. Nonlinear Transient Dynamics of Photoexcited Silicon Nanostructures. ACS Photonics 2016, DOI: 10.1021/acsphotonics.6b00358.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsphotonics.6b00384. Coupled-dipole model, evolution equations, and nonlinear parameters estimation (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The reported study was funded by RFBR, according to the research projects No. 16-37-60092 mol_a_dk, No. 16-37-60076 mol_a_dk, No. 15-57-45141 and the Government of the Russian Federation (projects GZ 2014/190, GZ 3.561.2014/ K). A.Y. acknowledges support from the Government of the Russian Federation (Grant 074-U01) through the ITMO University early career fellowship. Y.K. acknowledges support from the Australian Research Council.

■

REFERENCES

(1) Maier, S. A.; Brongersma, M. L.; Kik, P. G.; Meltzer, S.; Requicha, A. A. G.; Atwater, H. A. Plasmonics−A Route to Nanoscale Optical Devices. Adv. Mater. 2001, 13, 1501−1505. (2) Leuthold, J.; Koos, C.; Freude, W. Nonlinear silicon photonics. Nat. Photonics 2010, 4, 535−544. (3) Halir, R.; Bock, P. J.; Cheben, P.; Ortega-Moñux, A.; AlonsoRamos, C.; Schmid, J. H.; Lapointe, J.; Xu, D.-X.; Wanguemert-Pérez, J. G.; Molina-Fernández, I.; et al. Waveguide sub-wavelength structures: a review of principles and applications. Laser Photonics Rev. 2015, 9, 25−49. (4) Evlyukhin, A. B.; Reinhardt, C.; Seidel, A.; Luk’yanchuk, B. S.; Chichkov, B. N. Optical response features of Si-nanoparticle arrays. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 82, 045404. (5) Du, J.; Liu, S.; Lin, Z.; Zi, J.; Chui, S. T. Dielectric-based extremely-low-loss subwavelength-light transport at the nanoscale: An alternative to surface-plasmon-mediated waveguiding. Phys. Rev. A: At., Mol., Opt. Phys. 2011, 83, 035803. (6) Savelev, R. S.; Slobozhanyuk, A. P.; Miroshnichenko, A. E.; Kivshar, Y. S.; Belov, P. A. Subwavelength waveguides composed of dielectric nanoparticles. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 89, 035435. (7) Bakker, R. M.; Yu, Y. F.; Paniagua-Domínguez, R.; Luk’yanchuk, B.; Kuznetsov, A. Silicon Nanoparticles for Waveguiding. Frontiers in Optics 2015 2015, FM1B.2. (8) Kuznetsov, A. I.; Miroshnichenko, A. E.; Fu, Y. H.; JingBo, Z.; Luk’yanchuk, B. S. Magnetic light. Sci. Rep. 2012, 2, 492. (9) Evlyukhin, A. B.; Novikov, S. M.; Zywietz, U.; Eriksen, R. L.; Reinhardt, C.; Bozhevolnyi, S. I.; Chichkov, B. N. Demonstration of Magnetic Dipole Resonances of Dielectric Nanospheres in the Visible Region. Nano Lett. 2012, 12, 3749−3755. (10) Chong, K. E.; Hopkins, B.; Staude, I.; Miroshnichenko, A. E.; Dominguez, J.; Decker, M.; Neshev, D. N.; Brener, I.; Kivshar, Y. S. Observation of Fano Resonances in All-Dielectric Nanoparticle Oligomers. Small 2014, 10, 1985−1990. (11) Bakker, R. M.; Permyakov, D.; Yu, Y. F.; Markovich, D.; Paniagua-Domínguez, R.; Gonzaga, L.; Samusev, A.; Kivshar, Y.; Luk’yanchuk, B.; Kuznetsov, A. I. Magnetic and Electric Hotspots with Silicon Nanodimers. Nano Lett. 2015, 15, 2137−2142. (12) Shcherbakov, M. R.; Shorokhov, A. S.; Neshev, D. N.; Hopkins, B.; Staude, I.; Melik-Gaykazyan, E. V.; Ezhov, A. A.; Miroshnichenko, 1875

DOI: 10.1021/acsphotonics.6b00384 ACS Photonics 2016, 3, 1869−1876

ACS Photonics

Article

(32) Kivshar, Y.; Agrawal, G. Optical Solitons: from Fibers to Photonics Crystals; Academic Press, 2003. (33) Hasegawa, A.; Tappert, F. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl. Phys. Lett. 1973, 23, 142−144. (34) Mollenauer, L. F.; Stolen, R. H.; Gordon, J. P. Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers. Phys. Rev. Lett. 1980, 45, 1095−1098. (35) Agrawal, G. Fiber-optic Communication Systems; John Wiley and Sons: New York, 2002. (36) Zhang, J.; Lin, Q.; Piredda, G.; Boyd, R. W.; Agrawal, G. P.; Fauchet, P. M. Optical solitons in a silicon waveguide. Opt. Express 2007, 15, 7682−7688. (37) Foster, M. A.; Turner, A. C.; Lipson, M.; Gaeta, A. L. Nonlinear optics in photonic nanowires. Opt. Express 2008, 16, 1300−1320. (38) Ding, W.; Gorbach, A. V.; Wadsworth, W. J.; Knight, J. C.; Skryabin, D. V.; Strain, M. J.; Sorel, M.; De La Rue, R. M. Time and frequency domain measurements of solitons in subwavelength silicon waveguides using a cross-correlation technique. Opt. Express 2010, 18, 26625−26630. (39) Noskov, R. E.; Belov, P. A.; Kivshar, Y. S. Subwavelength Modulational Instability and Plasmon Oscillons in Nanoparticle Arrays. Phys. Rev. Lett. 2012, 108, 093901. (40) Noskov, R. E.; Belov, P. A.; Kivshar, Y. S. Oscillons, solitons, and domain walls in arrays of nonlinear plasmonic nanoparticles. Sci. Rep. 2012, 2, 873. (41) Walker, P. M.; Tinkler, L.; Skryabin, D. V.; Yulin, A.; Royall, B.; Farrer, I.; Ritchie, D. A.; Skolnick, M. S.; Krizhanovskii, D. N. Ultralow-power hybrid light−matter solitons. Nat. Commun. 2015, 6, 8317. (42) Christodoulides, D. N.; Efremidis, N. K. Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals. Opt. Lett. 2002, 27, 568−570. (43) Colman, P.; Husko, C.; Combrie, S.; Sagnes, I.; Wong, C. W.; Rossi, A. D. Temporal solitons and pulse compression in photonic crystal waveguides. Nat. Photonics 2010, 4, 862−868. (44) Blanco-Redondo, A.; Husko, C.; Eades, D.; Zhang, Y.; Li, J.; Krauss, T.; Eggleton, B. Observation of soliton compression in silicon photonic crystals. Nat. Commun. 2010, 5, 3160. (45) Christodoulides, D.; Lederer, F.; Silberberg, Y. Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature 2003, 424, 817−823. (46) Christodoulides, D. N.; Eugenieva, E. D. Blocking and Routing Discrete Solitons in Two-Dimensional Networks of Nonlinear Waveguide Arrays. Phys. Rev. Lett. 2001, 87, 233901. (47) Sukhorukov, A. A.; Kivshar, Y. S.; Eisenberg, H. S.; Silberberg, Y. Spatial optical solitons in waveguide arrays. IEEE J. Quantum Electron. 2003, 39, 31−50. (48) Lederer, F.; Stegeman, G. I.; Christodoulides, D. N.; Assanto, G.; Segev, M.; Silberberg, Y. Discrete soliton in optics. Phys. Rep. 2008, 463, 1−128. (49) Keil, R.; Heinrich, M.; Dreisow, F.; Pertsch, T.; Tünnermann, A.; Nolte, S.; Christodoulides, D. N.; Szameit, A. All-optical routing and switching for three-dimensional photonic circuitry. Sci. Rep. 2011, 2, 94. (50) Mulholland, G. W.; Bohren, C. F.; Fuller, K. A. Light Scattering by Agglomerates: Coupled Electric and Magnetic Dipole Method. Langmuir 1994, 10, 2533−2546. (51) Merchiers, O.; Moreno, F.; González, F.; Saiz, J. M. Light scattering by an ensemble of interacting dipolar particles with both electric and magnetic polarizabilities. Phys. Rev. A: At., Mol., Opt. Phys. 2007, 76, 043834. (52) Shore, R. A.; Yaghjian, A. D. Complex waves on periodic arrays of lossy and lossless permeable spheres: 1. Theory. Radio Sci. 2012, 47, RS2014/1−16. (53) Mok, J. T.; De Sterke, C. M.; Litter, I. C. M.; Eggleton, B. J. Dispersionless slow light using gap solitons. Nat. Phys. 2006, 2, 775− 780.

(54) Zywietz, U.; Schmidt, M. K.; Evlyukhin, A. B.; Reinhardt, C.; Aizpurua, J.; Chichkov, B. N. Electromagnetic Resonances of Silicon Nanoparticle Dimers in the Visible. ACS Photonics 2015, 2, 913−920. (55) Wang, C.; Jia, Z. Y.; Zhang, K.; Zhou, Y.; Fan, R. H.; Xiong, X.; Peng, R. W. Broadband optical scattering in coupled silicon nanocylinders. J. Appl. Phys. 2014, 115, 244312. (56) Evlyukhin, A. B.; Reinhardt, C.; Chichkov, B. N. Multipole light scattering by nonspherical nanoparticles in the discrete dipole approximation. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 235429. (57) Savelev, R. S.; Filonov, D. S.; Kapitanova, P. V.; Krasnok, A. E.; Miroshnichenko, A. E.; Belov, P. A.; Kivshar, Y. S. Bending of electromagnetic waves in all-dielectric particle array waveguides. Appl. Phys. Lett. 2014, 105. (58) Reitze, D. H.; Zhang, T. R.; Wood, W. M.; Downer, M. C. Twophoton spectroscopy of silicon using femtosecond pulses at above-gap frequencies. J. Opt. Soc. Am. B 1990, 7, 84−89. (59) Lin, Q.; Zhang, J.; Piredda, G.; Boyd, R. W.; Fauchet, P. M.; Agrawal, G. P. Dispersion of silicon nonlinearities in the near infrared region. Appl. Phys. Lett. 2007, 91, 021111. (60) Tsang, H. K.; Liu, Y. Nonlinear optical properties of silicon waveguides. Semicond. Sci. Technol. 2008, 23, 064007. (61) Vallaitis, T.; Bogatscher, S.; Alloatti, L.; Dumon, P.; Baets, R.; Scimeca, M. L.; Biaggio, I.; Diederich, F.; Koos, C.; Freude, W.; et al. Optical properties of highly nonlinear silicon-organic hybrid (SOH) waveguide geometries. Opt. Express 2009, 17, 17357−17368. (62) Yang, Y.; Wang, W.; Boulesbaa, A.; Kravchenko, I. I.; Briggs, D. P.; Puretzky, A.; Geohegan, D.; Valentine, J. Nonlinear Fano-Resonant Dielectric Metasurfaces. Nano Lett. 2015, 15, 7388−7393. (63) Linden, S.; Giessen, H.; Kuhl, J. XFROG  A New Method for Amplitude and Phase Characterization of Weak Ultrashort Pulses. Phys. Status Solidi B 1998, 206, 119−124. (64) Miroshnichenko, A. E.; Evlyukhin, A. B.; Kivshar, Yu. S.; Chichkov, B. N. Substrate-induced resonant magnetoelectric effects for dielectric nanoparticles. ACS Photonics 2015, 30, 2589−2598.

1876

DOI: 10.1021/acsphotonics.6b00384 ACS Photonics 2016, 3, 1869−1876