Metalens is a planar device for light focusing. In this work, we design and optimize c-Si nanosphere metalenses working on the magnetic dipole and quadrupole resonances of the c-Si nanoparticle. Resonant optical response of c-Si nanostructures is simulated by the multipole decomposition method along with the zero-order Born approximation. Limitations of this approach are investigated. The obtained results of optimization are verified by simulation via the T-matrix method

Baryshnikova, K.V.

Evlyukhin, A.B.

Kornovan, D.

Melnikov, R.

Ulyantsev, V.

Ustimenko, N.

English

Institutionelles Repositorium der Leibniz Universität Hannover

Journal of Physics: Conference SeriesPAPER • OPEN ACCESSLight focusing by silicon nanosphere structuresunder conditions of magnetic dipole andquadrupole resonancesTo cite this article: N Ustimenko et al 2021 J. Phys.: Conf. Ser. 2015 012160 View the article online for updates and enhancements.You may also likeA modelling & Simulation via CloudSim forLive Migration in Virtual MachinesAmbika Gupta-Combustion simulation via latticeBoltzmann and reduced chemical kineticsEliodoro Chiavazzo, Iliya V Karlin,Alexander N Gorban et al.-Simulation Study on KindergartenEvacuation Based on the ExperimentalDataYishu Yao and Wei Lu-This content was downloaded from IP address 89.245.22.218 on 31/07/2023 at 09:34Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd6th International Conference on Metamaterials and Nanophotonics METANANO 2021Journal of Physics: Conference Series 2015 (2021) 012160IOP Publishingdoi:10.1088/1742-6596/2015/1/0121601Light focusing by silicon nanosphere structures underconditions of magnetic dipole and quadrupoleresonancesN Ustimenko1, K V Baryshnikova1, R Melnikov1, D Kornovan1, VUlyantsev1, and A B Evlyukhin2,11ITMO University, St. Petersburg 197101, Russia2Leibniz Universitat Hannover, Hannover 30167, GermanyE-mail: nikita.ustimenko@metalab.ifmo.ruAbstract. Metalens is a planar device for light focusing. In this work, we design and optimizec-Si nanosphere metalenses working on the magnetic dipole and quadrupole resonances of the c-Si nanoparticle. Resonant optical response of c-Si nanostructures is simulated by the multipoledecomposition method along with the zero-order Born approximation. Limitations of thisapproach are investigated. The obtained results of optimization are verified by simulation viathe T-matrix method.1. IntroductionThe structures of many subwavelength particles for light focusing, namely metalenses, areconsidered by researchers as the perspective elements for future optical devices [1, 2]. However,to design widely-functional metalenses with the desired characteristics, advanced optimizationtechniques are needed [2, 3]. In this case, speed and efficiency of optimization are determinedby a physical model describing a collective response of metalens elements, typically arrangedin rings. In this work, we consider metalenses composed of c-Si nanosphere rings. Wesuggest the model when the optical response of the nanoparticle is associated with severalmultipoles while the collective response of nanoparticles structure is described in the zero-orderBorn approximation [4]. We find the limitations of our approach and apply an evolutionaryalgorithm [5] for optimization of ultra-thin Si nanosphere metalenses working at the magneticdipole and quadrupole resonances of the nanoparticle [6, 7]. The accuracy of the results obtainedin the Born approximation is verified by comparison with the rigorous T-matrix method [8].2. Results and DiscussionWe consider the structures consisting of N c-Si nanospheres with a diameter of 200 nm.According to Fig. 1(a), the optical response of single particle can be associated with the dipolesand quadrupoles of electric and magnetic types. The interaction of multipoles corresponding tothe different nanoparticles can be taken into account under the coupled multipole model (seeEq. (25) in [11]):Y = Y0 + V̂Y, (1)6th International Conference on Metamaterials and Nanophotonics METANANO 2021Journal of Physics: Conference Series 2015 (2021) 012160IOP Publishingdoi:10.1088/1742-6596/2015/1/0121602x y z Wavelength, λ (nm)σscaMDMQ200 nmSiλED600 700 800(a) (b)500 900 Figure 1. (a) Scattering spectrum of a silicon nanosphere having a diameter of 200 nmcalculated from the Mie-theory [9]. Red, orange, blue, and green lines correspond to the MD,ED, MQ, and EQ contributions, respectively. Dispersion of c-Si is taken into account [10].(b) Schematic view of a single nanosphere ring with the shown incident wave and Cartesiancoordinates system.where Y is the vector of self-consisted multipole moments:Y =[p1x, ..., pNz ,m1x, ...,mNz , Q1xx, ..., QNzz,M1xx, ...,MNzz]T24N, (2)Y0 is the vector of moments excited only by the incident wave:Y0 =[p10,x, ..., pN0,z,m10,x, ...,mN0,z, Q10,xx, ..., QN0,zz,M10,xx, ...,MN0,zz]T24N, (3)V̂ is the 24N × 24N interaction matrix composed of Green’s tensors blocks.Solving Eq. (1), we can find the self-consisted multipole moments of nanoparticles:Y = (Û− V̂)−1Y0, (4)if det(Û − V̂) 6= 0, where U is the 24N × 24N unity matrix. If the electromagnetic couplingbetween particles is weak enough, it can be neglected, which corresponds to the zero-order Bornapproximation (ZBA). The ZBA multipole moments of nanoparticlesY = Y0 (5)connect with the incident fields E0 and H0 through the Mie-polarizabilities [11]:pj0 = αpE0 (rj) , (6)mj0 = αmH0 (rj) , (7)Q̂j0 = (αQ/2) [∇E0 (rj) + E0 (rj)∇] , (8)M̂ j0 = (αM/2) [∇H0 (rj) + H0 (rj)∇] , (9)Knowing the multipole moments, we can calculate the total electric E and magnetic H fieldsin the system (using Eqs. (13)-(16) from [11]), and the electromagnetic intensity I(r)/I0 =(|E(r)|2 + Z2|H(r)|2)/2E20 where r is the observation point, Z is the free-space impedance, E0is the amplitude of electric field in the incident plane wave.Using the ZBA, we study light focusing by a single ring of Si nanospheres shown in Fig. 1(b).Due to the axial symmetry, the ring has a focus on its axis (here z-axis). Hence, ring focal6th International Conference on Metamaterials and Nanophotonics METANANO 2021Journal of Physics: Conference Series 2015 (2021) 012160IOP Publishingdoi:10.1088/1742-6596/2015/1/012160311 12 13 14 15 16 17Ring Radius, R/λ0.811.21.41.61.82Distance, D/λ1415161718192021222324z/λFocal Length, f/λMagnetic Quadrupole resonance(a)10 12 14 16106 8Normalized Intensity(b)R/λ=10R/λ=9Figure 2. Features of light focusing at the MQ resonance λ = 574 nm. (a) Focal length of singlenanosphere ring as a function of the ring radius and center-center distance between particles.(b) Normalized (on the maximum) electromagnetic intensity distribution along the z-axis for thering radius R/λ = 9 (blue dashed line) and R/λ = 10 (red solid line). The number of particlesin both rings N = 32. The arrows indicate the focal lengths of rings. The results are obtainedin the ZBA.length is defined as the distance between the ring plane and the position of the highest intensitypeak on the ring axis. Dependence of the ring focal length on the ring parameters for the MQresonance is shown in Fig. 2(a). The ring focal length is almost completely determined by itssize. Moreover, this dependence is non-monotonic due to the interference origin of the intensitypeaks. As a result of constructive interference between the incident wave and the scatteredwaves, the bigger ring may have a shorter focal length [see focal lengths in Fig. 2(b)]. Similarbehavior of the focal length is observed at the MD resonance.The focal length and focal intensity are the basic parameters of metalens. The error ofcalculating these quantities in the ZBA was studied for the single ring at the MD and MQresonances. This error was defined as follows: ∆(V ) = 100% × |V CMM − V ZBA|/V CMM, whereV CMM and V ZBA are values of V calculated using moments (4) and (5), respectively. Wecalculate errors of the ring focal length ∆(f) and intensity ∆(I) as functions of ring radius 6 µm≤ R ≤ 10 µm and distance between centers of particles 200 nm ≤ D ≤ 1200 nm. We obtainthat the focal length error ∆(f) ≤ 2.5% at the both resonances for all ring parameters; thefocal intensity error ∆(I) . 10% when the inter-particle distance D ≥ 0.66λ and D ≥ 0.92λ atthe MQ and MD resonances, respectively. We will call these distance values limiting. Hence,to simulate Si nanosphere metalenses via the ZBA-based approach, the inter-particle distanceshould be higher than limiting.Based on the ZBA calculation scheme is directly comparable with the optimization algorithm.To optimize metalenses, we choose an evolutionary SEMO algorithm [5]. We use this algorithmto optimize metalenses at the MQ and MD resonances. The objective functions of optimizationare both desired focal length (here f = 5 µm) and high focal intensity. During the optimization,the algorithm varies the number of rings, their radii, and the number of particles in the ring.The fixed parameters: diameter of Si nanoparticle is 200 nm, metalens size to 20 µm, minimalinter-particle distance is equal to 1.11λMQ and 1.07λMD. The optimized metalenses with f = 5µm and their intensity profiles are shown in Fig. 3.We compare intensity profiles, obtained in ZBA, with the ones calculated by the T-matrixmethod [8] and obtain a good agreement. Both methods determine the same focal length equalled6th International Conference on Metamaterials and Nanophotonics METANANO 2021Journal of Physics: Conference Series 2015 (2021) 012160IOP Publishingdoi:10.1088/1742-6596/2015/1/0121604to 5 µm and 4.9 µm for MQ and MD resonances, while the error of focal intensity: 5.7% and3%, respectively.10403020-5 5x (μm)Intensity, I/I0246810z (μm)(b) λ = 770 nm10403020Intensity, I/I0-5 5x (μm)246810z (μm)(a) λ = 574 nmFigure 3. Normalized electromagnetic intensity distribution in the xz-plane for metalensesoptimized at the MQ (a) and MD (b) resonances (I0 is the intensity of incident wave). Bothmetalenses have the focal length f = 5 µm. In insets: distribution of nanoparticles in thexy-plane for optimized metalenses. The results are obtained in the ZBA.3. ConclusionWe studied light focusing by c-Si nanosphere rings. The optical response of such structures wasdescribed using the dipole-quadrupole model along with the zero-order Born approximation.We showed that this approach provides a low-error simulation of the resonant optical responseif the distance between particles & λ0. We optimized positions of Si nanospheres, using anevolutionary algorithm, and obtained effective metalenses with the desired focal lengths, workingat the magnetic dipole and quadrupole resonances of the single nanosphere. The results ofoptimization in ZBA demonstrate a good agreement with the T-matrix method simulation.4. AcknowledgmentsK.V.B., and D.K. acknowledge support from the Foundation for the Advancement of TheoreticalPhysics and Mathematics “BASIS”.References[1] Lalanne P and Chavel P 2017 Laser Photonics Rev. 11 1600295[2] Chen W T, Zhu A Y and Capasso F 2020 Nat. Rev. Mater. 5 604–620[3] Yao K, Unni R and Zheng Y 2019 Nanophotonics 8 339–366[4] Keller O, Xiao M and Bozhevolnyi S 1993 Surf. Sci. 280 217–230[5] Kalyanmoy D 2002 Multi-Objective Optimization using Evolutionary Algorithms (John Wiley & Sons, LTD,New York, USA)[6] Kuznetsov A I, Miroshnichenko A E, Brongersma M L, Kivshar Y S and Luk’yanchuk B 2016 Science 354aag2472[7] Evlyukhin A B, Novikov S M, Zywietz U, Eriksen R L, Reinhardt C, Bozhevolnyi S I and Chichkov B N2012 Nano Lett. 12 3749–3755[8] Egel A, Pattelli L, Mazzamuto G, Wiersma D S and Lemmer U 2017 J. Quant. Spectrosc. Radiat. Transf.199 103–110[9] Bohren C F and Huffman D R 1983 Absorption and Scattering of Light by Small Particles, (Wiley, NewYork)[10] Aspnes D E and Studna A A 1983 Phys. Rev. B 27 985[11] Babicheva V E and Evlyukhin A B 2019 Phys. Rev. B 99 195444

Light focusing by silicon nanosphere structures under conditions of magnetic dipole and quadrupole resonances

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Light focusing by silicon nanosphere structures under conditions of magnetic dipole and quadrupole resonances

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