Modeling and design of all-dielectriccylindrical nanoantennas

Inder DeviReena DalalYogita KalraRavindra Kumar Sinha

Inder Devi, Reena Dalal, Yogita Kalra, Ravindra Kumar Sinha, “Modeling and design of all-dielectriccylindrical nanoantennas,” J. Nanophoton. 10(4), 046011 (2016), doi: 10.1117/1.JNP.10.046011.

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Modeling and design of all-dielectric cylindricalnanoantennas

Inder Devi,a Reena Dalal,a Yogita Kalra,a and Ravindra Kumar Sinhaa,b,*aDelhi Technological University, Technology Information, Forecasting and Assessment

Council-Center of Relevance and Excellence in Fiber Optics and Optical Communication,Department of Applied Physics, Bawana Road, Delhi 110042, India

bCSIR-Central Scientific Instruments Organization, Sector 30 C, Chandigarh 160-030, India

Abstract. We theoretically demonstrate ultradirectional, azimuthally symmetric forward scat-tering by dielectric cylindrical nanoantennas for futuristic nanophotonic applications in visibleand near-infrared regions. Electric and magnetic dipoles have been optically induced in the nano-cylinders at the resonant wavelengths. The cylindrical dielectric nanoparticles exhibit completesuppression of backward scattering and improved forward scattering at first generalized Kerker’scondition. The influence of gap between nanocylinder elements on the scattering pattern of thehomodimers has been demonstrated. Further, for highly directive applications, a linear chain ofultradirectional cylindrical nanoantenna array has been proposed. © 2016 Society of Photo-OpticalInstrumentation Engineers (SPIE) [DOI: 10.1117/1.JNP.10.046011]

Keywords: electric resonance; magnetic resonance; dielectric nanoantenna; Kerker’s condition;directional scattering.

Paper 16062 received Apr. 13, 2016; accepted for publication Oct. 7, 2016; published onlineNov. 16, 2016.

1 Introduction

Optical nanoantennas have been a topic of great interest in many applications from near-fieldmicroscopy to molecular and biomedical sensors, optical communication, solar cells, and opticaltweezers.1–7 Plasmonic nanoantennas are one of the promising candidates for nanophotonicapplications. However, the biggest drawback of plasmonic nanoantennas is metallic losses inthe visible and near-infrared regions, which limit their performances at a nanoscale level.8

The unavoidable problems of metallic structures motivated the study of high-dielectric nano-antenna structures. Interestingly, dielectric nanoantennas present sharp resonances and havelow dissipative losses in visible and near-infrared regions. High all-dielectric nanoparticlescan support both electric and magnetic resonances, which can be controlled independently.9

It has been observed in previous studies that an efficient response can be achieved when nano-particles possess both electric and magnetic resonances.10 These interesting advantages overmetallic counterparts make dielectric nanoantennas popular choices for directional scatteringin visible and near-infrared regions.

It is evident from past studies that the effective engineering of scattering radiation depends onthe electric and magnetic responses to the incident electromagnetic wave in dielectricnanoparticles.10,11 Krasnok et al.11 have shown the direct comparison of metallic and dielectricnanoantennas. In this paper, it has been reported that high permittivity all-dielectric nanoanten-nas are better than the metallic nanoantennas. The unusual electromagnetic scattering effects ofmagnetodielectric particles were first theoretically proposed by Kerker et al.12 The scatteringradiation of magnetodielectric particles exhibiting both electric and magnetic resonances canbe controlled under certain conditions for the values of electric permittivity ε and magnetic per-meability μ, as given by Kerker et al. According to Kerker, due to the interference of magnetic

*Address all correspondence to: Ravindra Kumar Sinha, E-mail: dr_rk_sinha@yahoo.com

Delhi Technological University was formerly Delhi College of Engineering, University of Delhi.

1934-2608/2016/$25.00 © 2016 SPIE

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and electric dipolar resonances, complete forward or backward scattering is possible. Theprediction by Kerker et al. has given new directions in the field of optical nanoantennas.At a zero-backward intensity condition or Kerker’s first condition, the scattered field is mainlyin the direction of the incoming wave with zero backward scattering which leads to interestingnanoantenna applications.

According to Mie theory, the first and second lowest resonances of dielectric spherical nano-particles indicate the magnetic and electric dipole terms, respectively.13 For high permittivityparticles, the quality factor of Mie resonances increases resulting in high scattering efficiency.In literature, theoretical and experimental demonstrations of strong electric and magnetic dipolarresonances of high-dielectric spherical particles such as silicon (Si) or germanium (Ge) inthe visible and near-infrared regions have been reported based on Mie theory.14–16 It hasbeen shown that it is possible to tune electric and magnetic resonance at a particular frequencyfor Si nanoparticles by changing their shape and size.17 It has been observed from past studiesthat cylindrical-based structures have advantages over other geometries as they are easy tofabricate and can be tuned by changing the dimensions of the cylinder.18–20

Here, we propose the design of cylindrical nanoantennas using scattering properties of nano-cylinders by applying generalized Kerker’s (GK) condition for unidirectional forward scattering.We theoretically investigate and numerically demonstrate the scattering properties of dielectricnanocylinders. The ability of dielectric nanocylinders to control light scattering in the desireddirection efficiently makes them promising candidates for low loss, tunable, and ultradirectionalnanoantenna applications. The scattering by Si nanocylinder homodimers has been demon-strated. Next, an array of Si nanocylinders for highly directive nanoantenna applications hasbeen demonstrated. The effect of the number of array elements on the directivity of nanoantennashas been also studied. All numerical results are obtained using the finite element method.

2 Design Parameters and Scattering Profile of a Silicon Nanocylinder

A dielectric Si nanocylinder of radius r ¼ 70 nm and height h ¼ 150 nm has been considered asshown in Fig. 1(a). We have chosen Si as a scattering material due to its large refractive index(n ∼ 3.5) and low attenuation coefficient in the 600- to 1000-nm wavelength range as shownin Fig. 1(b).21 The nanocylinder dimensions are chosen such that the operating wavelengthfalls in the visible region, where the extinction coefficient is low for Si nanoparticles. The oper-ating wavelength can be tuned at the desired wavelength by changing the dimensions of thenanocylinder.

The nonmagnetic nanoparticle is characterized by the electric current density J and polari-zation P. The electric current density and polarization induced by external electromagnetic fieldsare related by J ¼ −iωP, where ω is the angular frequency.22,23 The relative magnetic permeabil-ity μ of the particle in a nonabsorbent nonmagnetic medium is given by μ ¼ 1. The time averageextinction power Pext (Pext ¼ Psca) for electromagnetic scattering in nonabsorbing nonmagneticmedium is given as

Fig. 1 (a) Schematic of a Si nanocylinder and (b) the plot showing optical properties of Si materialwith respect to wavelength.

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EQ-TARGET;temp:intralink-;e001;116;735Pext ¼ω

2Im

ZE�0ðrÞ:PðrÞdV 0; (1)

where E�0ðrÞ denotes the electric field of the incident electromagnetic waves, V 0 is the volume of

the scattering object, r denotes the position coordinate, and PðrÞ is the polarization vector, whichis defined as

EQ-TARGET;temp:intralink-;e002;116;665PðrÞ ¼ ε0ðεr − 1ÞEðrÞ; (2)

where ε0, εr denote the vacuum permittivity and relative permittivity of the nanoparticle, respec-tively, and EðrÞ is the total electric field, which is equal to the sum of the incident electric fieldand the scattered electric field.

From multipole decomposition of the scattering power, we get

EQ-TARGET;temp:intralink-;e003;116;586psca ≈ ppsca þ pm

sca þ pQsca þ pM

sca þ : : : ; (3)

where ppsca is the scattering power for an electric dipole, pm

sca is the scattering power for a mag-netic dipole, and pQ

sca, pMsca are the scattering powers for electric and magnetic quadrupoles,

respectively.The total scattering power can be written as

EQ-TARGET;temp:intralink-;e004;116;506psca ¼c2k40Z0

12πjpj2 þ c2k40Z0

12πjmj2 þ c2k60Z0

40π

XjQα 0β 0 j2 þ c2k60Z0

160π

XjMα 0β 0 j2 þ : : : ; (4)

where p is the electric dipole moment, m is the magnetic dipole moment, Q is the electric quad-rupole tensor, and M is the magnetic quadrupole tensor. Since the scattering powers at higherorder modes are insignificant, these higher modes can be neglected in calculations

EQ-TARGET;temp:intralink-;e005;116;427p ¼ ε0ð1 − εrÞZvEðr 0ÞdV 0; (5)

EQ-TARGET;temp:intralink-;e006;116;381m ¼ iωε0ð1 − εrÞ2c

Zv½r × Eðr 0Þ�dV 0; (6)

where c is the speed of light in vacuum, k0 ¼ ðω∕cÞ represents the free space wave number,z0 ¼ ð1∕ε0cÞ is the free space impedance, and α 0; β 0 ¼ x; y; z denote Cartesian components.24

From Eqs. (4)–(6), the scattering cross section (σsca) associated with each multipole can becalculated by simply normalizing each scattered power by the incident power density (I0), givenas

EQ-TARGET;temp:intralink-;e007;116;282σpsca ¼ ðppscaÞ∕I0; (7)

EQ-TARGET;temp:intralink-;e008;116;249σmsca ¼ ðppscaÞ∕I0: (8)

The scattering spectrum of an Si nanocylinder has been obtained as shown in Fig. 2(a). Thescattering spectrum peaks denote the modes at corresponding wavelengths, i.e., the electricresonance at λ ¼ 538 nm and magnetic resonance at λ ¼ 635 nm.

Figure 2(b) shows the two-dimensional (2-D) normalized polarization distribution at mag-netic and electric dipole resonances. At electric and magnetic dipolar resonance peaks, the scat-tering radiation pattern is symmetrically distributed in both the forward and backward directions.However, the unidirectional scattering can be obtained in the presence of electric and magneticinterferences. The direction of the scattering radiation is determined according to the polarity ofRe (αeα�m), whether positive or negative. The positive value of Re (αeα�m) indicates forwardscattering, where αe and αm are the electric and magnetic complex scalar polarizability ofthe nanoparticle, respectively, which are given as12,14

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EQ-TARGET;temp:intralink-;e009;116;532αe ≡px

ðε0jE0jÞ; (9)

EQ-TARGET;temp:intralink-;e010;116;487αm ≡my

ðjH0jÞ; (10)

where E0 andH0 are the electric and magnetic field of the incident plane wave, respectively, andpx and my are the electric dipole moment and magnetic dipole moment along the “x-axis” and“y-axis,” respectively.

Unidirectional forward scattering with complete suppression of backward scattering can beachieved when the interference between the electric and magnetic dipoles satisfies the first GKcondition. According to first GK condition

EQ-TARGET;temp:intralink-;e011;116;388

αeεs

¼ μsαm; (11)

which results in

EQ-TARGET;temp:intralink-;e012;116;335

dσscadΩ

ð180 degÞ ¼ 0; (12)

i.e., zero scattering in the backward direction,14 where εs and μs are the relative permittivity andpermeability of the surrounding medium, respectively.

The resultant of the interferences between the electric and magnetic dipole consists of thecoherent sum of two dipoles and the phase difference between dipoles is zero. Hence, zerobackward scattering is obtained when the electric and magnetic dipoles oscillate in phase.

In the case of dielectric nanoparticles, as the size changes the wavelength at which themagnetic resonance and electric resonance takes place shifts, resulting in the change of wave-length at which the first GK condition is satisfied. It has been observed as the radius of thenanocylinder is increased, the wavelengths corresponding to the electric and magnetic resonan-ces shift toward the higher side. Hence, the wavelength at which the first GK condition is sat-isfied also shifts toward the higher side. However, as the radius of the nanocylinder is decreased,the wavelength corresponding to the electric and magnetic resonance shifts toward the lowerside.

Further, the variation of electric and magnetic polarizabilities with respect to wavelength hasbeen obtained as shown in Fig. 3. At λ ¼ 676 nm, the first GK condition is satisfied, i.e.,αe ¼ αm, indicated by the point of intersection of electric and magnetic polarizability inFig. 3(a). Three-dimensional (3-D) and 2-D far-field patterns shown in Fig. 3(b) exhibit thetotal forward scattering with complete suppression of backward scattering at λ ¼ 676 nm.The observed scattering pattern exhibits complete azimuthal symmetry. Thus, high permittivitynanocylinders can be used for unidirectional nanoantenna applications.

Fig. 2 Scattering analysis for Si nanocylinder: (a) scattering cross-section of the electric (ED),magnetic dipoles (MD), and total scattering cross-section. (b) Normalized electric field distributionin V∕m for MD at λ ¼ 635 nm and ED at λ ¼ 538 nm.

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3 Scattering Pattern of Cylindrical Nanoantenna Homodimers

Two identical Si nanocylinders of radius r ¼ 70 nm and height h ¼ 150 nm with spacing d ¼250 nm have been considered as shown in Fig. 4(a). The far-field pattern of homodimers hasbeen observed at λ ¼ 676 nm as shown in Fig. 4(b).

The far-field pattern obtained for homodimers with d ¼ 250 nm gets enhanced as comparedto the far-field pattern for a single nanocylinder, which results from the constructive interferencesof far-fields of two individual nanocylinders. Complete forward scattering in directionθ ¼ 0 deg has been observed. It has been observed that cylindrical homodimers offer betterdirectionality over individual cylindrical nanoantennas. Further, the far-field patterns have beenobtained by varying the spacing between nanocylinder homodimers as shown in Figs. 5(a)–5(f).The effect of d on the scattering pattern has been demonstrated by comparing three cases:d < λ∕2, d > λ∕2, and d > λ.

Fig. 3 Variation of electric and magnetic polarizabilities with respect to wavelength: (a) spectrashowing variation of polarizabilities. At λ ¼ 676 nm, the first forward scattering condition is satis-fied with zero backscattering, (b) 3-D and 2-D electric (red line) and magnetic (green line) far-fieldpatterns obtained at λ ¼ 676 nm.

Fig. 4 (a) Schematic of Ge nanocylinder homodimers of radius r ¼ 70 nm and h ¼ 150 nm.(b) 3-D and 2-D electric (red curve) and magnetic (green curve) far-field patterns at λ ¼ 676 nm.

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In the first case, where d has been kept as d < λ∕2 (d ¼ 250 nm), one major is observed asshown in Figs. 5(a) and 5(d). In the second case, where d > λ∕2 (d ¼ 400 nm), the directivity isfurther increased as compared to the directivity of homodimers in the case of d less than λ∕2. Asd increases, the phase delay between cylindrical homodimers leads to collective grating diffrac-tions. The phase delay between homodimers is given by Δφ ¼ kdðcos θ − 1Þ ¼ 0, wherek ¼ 2π∕λ. The directivity of nanocylinder dimers depends on the d∕λ ratio. As d exceeds ahalf wavelength, one major lobe with two minor side lobes has been observed at first-orderdiffraction angles as shown in Figs. 5(b) and 5(e). The first-order diffraction is given byβ ¼ arccosð1 − λ∕dÞ.25 For d ¼ 400 nm, the first-order diffraction is observed at 134.4 degand 225.57 deg.

Next, for d > λ (d ¼ 750 nm,), directivity increases but the scattering strength furtherdecreases. One major lobe with four side lobes at 81.4 deg, 134.4 deg, 225.6 deg, and 278.6 deghas been observed as shown in Figs. 5(c) and 5(f).

For higher order diffractions, the diffraction angles can be calculated at βm ¼arccosð1 −mλ∕dÞ, where m is the order of the diffraction which is same as obtained fromthe above far-field pattern. It has been observed that the directivity of the homodimers increaseswith an increase in d, but as d increases, the diffraction grating effect becomes dominant leadingto the increase in the number of side lobes. The diffraction grating effect results in scattering inundesired directions, which further reduces the scattering strength of the main lobe.

4 Linear Array of Cylindrical Nanoantenna

As has been observed, the directionality is enhanced by using a pair of cylindrical nanoantennas.To further enhance the directivity, an array of Si cylindrical nanoparticles has been deployed. Anarray of Si nanocylinders of radius ¼ 70 nm and height ¼ 150 nm with spacing d ¼ 300 nm

(d < λ∕2) so as to avoid the diffraction grating effect has been considered for forward scatteringat wavelength 676 nm. It has been observed that with a variation of 20 nm in the wavelength, i.e.,Δλ ¼ �20 nm, directivity decreases on either side by 0.37 dBi.

The directivity and far-field pattern of the linear array of a cylindrical nanoantenna with avarying number of elements in the array has been obtained as shown in Fig. 6. It has beenobserved that the directivity increases with respect to the increase in the number of array ele-ments. From Fig. 6, it has been observed that the directivity increases linearly up to six

Fig. 5 Electric and magnetic far-field patterns showing the effect of spacing d on directionality ofnanocylinder homodimers. 2-D and 3-D far-field patterns when d < λ∕2 [(a) and (d)] havingdirectivity ¼ 4.90 dBi, d > λ∕2 [(b) and (e)] having directivity ¼ 5.13 dBi, and d > λ [(c) and (f)]having directivity ¼ 5.49 dBi.

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nanoparticles, after that directivity becomes stable having a constant value 9 dBi. Therefore, thelinear chain of the array should be as small as six nanoparticles and as large as up to ten nano-particles for better directivity and for compactness of the device.

5 Conclusion

In this paper, unidirectional scattering by Si nanocylinders has been demonstrated. The Kerker’stype, forward scattering in the visible region at wavelength 676 nm for Si nanocylinders has beendemonstrated. The scattering by Si nanocylinder homodimers has been analyzed by varying thespacing between nanocylinders. It has been observed that as the gap increases the directivityincreases, but as gap exceeds λ∕2 the diffraction grating effect becomes dominant, which resultsin scattering in undesired directions. Due to scattering in undesired directions, the strength ofscattering in the desired direction is reduced. Thus, for a nanocylinder array, the gap betweennanoantenna elements must not exceed λ∕2 in order to avoid diffraction grating effects. Next, forhighly directive forward scattering applications, a linear chain of nanoantennas array has beendemonstrated. It has been demonstrated that with an increase in the number of nanoantennaelements in an array the directionality increases, however, after a specific limit the increasein directivity is not significant.

Acknowledgments

The authors gratefully (i) acknowledge the initiatives and support from Technology Information,Forecasting and Assessment Council (TIFAC)-Center of Relevance and Excellence in FiberOptics and Optical Communication at Delhi College of Engineering, now DTU, and Delhiunder Mission REACH Program of Technology vision-2020, Government of India andSupport through (ii) DST-RMES (Indo-Russian) joint research project on “All dielectric, plas-monic and hybrid photonic nanostructures” and (iii) DST-RFBR (Indo-Russian) joint researchproject on “From plasmonics to dielectric and hybrid nanoantennas: novel approaches to controlelectromagnetic waves and light.”

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Inder Devi graduated in electronics and communication engineering from the UniversityInstitute of Technology, Punjab University, Chandigarh, in 2014. Recently, she has done herMTech in microwave and optical engineering from Delhi Technological University. Her areaof interest is the design of dielectric nanoantenna structures for nanophotonics applications.

Reena Dalal received her MSc degree from Guru Jambheshwar University of Science andTechnology, Hisar, in 2009. She is pursuing her PhD from Delhi Technological University,Delhi. Her area of interest is all-dielectric based-metamaterials and photonic crystal fibers.

Yogita Kalra received her PhD from the University of Delhi, India, in 2007. Currently, she is anassistant professor at the Department of Applied Physics and coordinator of Technology Infor-mation, Forecasting and Assessment Council (TIFAC)-Center of Relevance and Excellence inFiber Optics and Optical Communication, Delhi Technological University, India. She isa member of SPIE, Optical Society of America, and Optical Society of India.

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Ravindra Kumar Sinha received his PhD from the Indian Institute of Technology, Delhi, India,in 1990. He is a professor of applied physics and chief coordinator of (TIFAC)-Center ofRelevance and Excellence in Fiber Optics and Optical Communication, Delhi TechnologicalUniversity, India. Currently, he is a director of CSIR–Central Scientific InstrumentationOrganization, Chandigarh. He is a fellow of SPIE, Institution of Electronics and Telecommu-nication Engineers, and the Photonics Society of Institute of Electrical and ElectronicsEngineers.

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