Optical Polarization Möbius Strips on All-Dielectric ... - ACS Publications

Subscriber access provided by HACETTEPE UNIVERSITESI KUTUPHANESI

Article

Optical polarization Möbius strips on all-dielectric optical scatterers Aitzol Garcia-Etxarri ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b00002 • Publication Date (Web): 11 Apr 2017 Downloaded from http://pubs.acs.org on April 16, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

ACS Photonics is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Photonics

Optical polarization M¨ obius strips on all-dielectric optical scatterers Aitzol Garcia-Etxarri∗ Donostia International Physics Center (DIPC), Donostia - San Sebastian 20018, Spain E-mail: [email protected]

Abstract In this article, we study the emergence of polarization singularities in the scattering of optical resonators excited by linearly polarized light. First, we prove analytically that spherical all-dielectric nanoparticles described by combinations of electric and magnetic isotropic polarizabilities can sustain L surfaces and C lines that propagate from the near-field to the far field. Based on these analytical results, we are able to derive anomalous scattering Kerker conditions using singular optics arguments. Next, through full-field calculations, we demonstrate that high refractive index spherical resonators present such topologically protected features. We calculate the polarization structure of light around the generated C lines, unveiling a Möbius strip structure in the main axis of the polarization ellipse when calculated on a closed path around the C line. These results prove that high-index nanoparticles are excellent candidates for the generation of polarization singularities and that they may lead to new platforms for the experimental study of the topology of light fields around optical antennas.

1

ACS Paragon Plus Environment

ACS Photonics

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Keywords Singular optics, Polarization singularities, High refractive index nanoparticles, Kerker conditions Mathematical singularities may be defined as points where a function is not defined or is not well behaved. In physics, singularities arise whenever a mathematical formalism fails at describing some particular physical phenomenon. Often, these singularities hint at interesting physical phenomena. Singularities in wave physics have been studied since 1830 in different kinds of waves (for a good overview, see 1 ). In wave optics, two general types of singularities have been identified and studied. 2 When the polarization of light is uniform throughout space, light can be mathematically described as a position dependent complex scalar wave multiplied by a constant polarization vector. In these situations (scalar optics), singularities arise whenever the amplitude of the wave is zero. In such cases, even though the amplitude of the wave is well defined, its phase cannot be unambiguously determined. These singularities are called phase singularities. Around these singularities, the phase of the scalar field varies gradually from 0 to 2πq, where q is an integer (positive or negative) named the topological charge of the singularity. For an extended discussion on the concept of topological charge of a phase singularity, please refer to the Supplementary Material. These phase singularities, also known as nodal lines or optical vortices, have been widely studied in the recent past due to their ability to carry orbital angular momentum. 3,4 In most situations, however, light presents a spatially varying polarization structure and the vectorial nature of electromagnetic fields cannot be disregarded. In these situations, optical vortices are rare since all of the three field components need to be exactly zero which is a condition much harder to achieve. Nonetheless, over a frequency cycle, the real part of the complex vector draws an ellipse (the polarization ellipse) and, as we proceed to detail, singularities arise as Polarization Singularities when light is either circularly polarized or lin2


Page 2 of 18

Page 3 of 18

ACS Photonics

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment

ACS Photonics

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

surfaces, called L surfaces. 2 On the other hand, when light is circularly polarized, the major and minor axes of polarization cannot be unambiguously determined, and both α and β vectors become identically zero. Since ϕ is a complex scalar function, its vanishing requires both its real and the imaginary parts to be zero. Thus, in 3D space, the points of circular polarization form lines (C lines). The phase of ϕ around any zero spans all the possible values from 0 to 2πl, l being an integer. The topological charge of the C line is defined as s = 2l . The division by 2 is justified because the orientation of the polarization ellipse (defined by its major or minor axis vectors) is indistinguishable under a rotation by π. For an extended discussion on the concept of topological charge of a polarization singularity, please refer to the Supplementary Material. Around any small closed curve surrounding a single C line, the major and minor axes of the polarization ellipse generate a multi twist ribbon with s twists.6,7 Thus, as it was recently experimentally demonstrated,8 for |s| = 21 , both α and β form a Möbius strip around any small closed path around a C line. Since the seminal contributions by Nye, 6–8 polarization singularities in optical fields have been studied extensively for their intrinsic theoretical interest. C lines and L surfaces have been identified in contexts as disparate as the skylight, 9 speckle fields, 10,11 tightly focused beams, 12,13 crystal optics, 14 photonic crystals 15,16 and plasmonic systems. 17,18 Nevertheless, experimental applications of such topological features remained elusive until very recently, when polarization singularities have proved to be useful for sensing, quantum information applications 19,20 and to identify points of purely transverse spin density. 21 In this paper, we explore the possibility of creating such polarization singularities in optical antennas illuminated by linearly polarized light. We prove that systems presenting a simultaneous isotropic electric and a magnetic polarizability (αe and αm respectively) are among the simplest nanostructures capable of sustaining C lines and L surfaces. Moreover, we prove the anomalous scattering Kerker conditions based on singular optics arguments alone. We verify our ideas by performing full field simulations on the optical response of a

4


Page 4 of 18

Page 5 of 18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Photonics

high index dielectric nanosphere iluminated by a linearly polarized planewave. We track the C lines generated by a Si nanoparticle over all the radiation regions and unveil the Möbius strip structure of the main axis of the polarization ellipse around them.

Results and discussion Dielectric nanoantennas made of materials with a high index of refraction have been intensely studied in the recent past due to their ability to sustain lossless electric and magnetic resonances in the visible and IR parts of the spectrum. 22 Their lossless character has proved instrumental in the fabrication of metamaterials and metasurfaces not limited by ohmic losses. 22–24 Furthermore, compared to plasmonic dipolar antennas, their additional magnetic dipolar character adds an additional degree of freedom in the manipulation of light that permits, for instance, to enhance chiral spectroscopy techniques, 25 and to shape the radiation characteristics in unprecedented ways. 26,27 In this article, we focus in the polarization aspect of the fields scattered by such resonators illuminated by linearly polarized light. Our first aim is to determine the necessary and sufficient conditions for the emergence of topologically protected polarization singularities in optical resonators. Let us start by considering the far field radiation characteristics of an electric dipole described by an isotropic electric polarizability. Excited by linearly polarized light (Einc = E0 x ˆ, Hinc = H0 y ˆ), the induced electric dipolar moment can be expressed as p = ε0 αe Einc = E 0 ε0 α e x ˆ, where ε0 is the vacuum permittivity and αe the electric dipolar polarizability. The far-field scattering of such a dipolar moment is given by:

EED scat

k2 k2 eikr = Ge p = [(n × p) × n] ε0 ε0 4πr

(4)

where Ge is the electric Green’s tensor in the far field approximation, k is the wavenumber of light in vacuum, r = |r − r0 | is the distance between the position of the dipole (r0 ) and the observation point r, and n = nx x ˆ + ny y ˆ + nz ˆ z is the unit vector in the direction of r − r0 . 5


ACS Photonics


Page 6 of 18

Page 7 of 18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Photonics

both the electric and magnetic dipole:

ED+MD Escat =

k2 Ge p + iZ0 k 2 Gm m. ε0

(6)

MD EED scat and Escat are not necessarily in phase nor are they parallel. Thus, the polarization

state in these systems (Figure 2c) becomes spatially inhomogeneous. Since the scattered fields are now generally elliptically polarized, it is logical to expect that these systems may sustain polarization singularities. ~ by substituting Eq.6 into We will begin by searching for L surfaces. Calculating |N| Eq.8, one obtains: ~ N =

k4 ℑ (αe∗ αm ) nx ny (4πr)2

(7)

~ = 0) in the far-field region. Thus, both the xz ˆ plane and the yz ˆ plane are L surfaces (|N| The analytical determination of C lines is more intricate. Deriving the polarization scalar ϕ from Eq.6 yields:

ϕ = k4

e2ikr 2 2 2 2 2 2 2 α n + α n + (α + α )n + 2α α n e m z e y m x e m z (4πr)2

(8)

(For a detailed derivation of equations 7 and 8, see the supplementary material). Light will be circularly polarized if ϕ = 0. Both real and imaginary parts of ϕ need to be zero in order to fulfill this condition. Thus,

2 2 2 )n2z + 2αe αm nz = 0 ℜ αm nx + αe2 n2y + (αe2 + αm

(9)

2 2 2 )n2z + 2αe αm nz = 0 ℑ αm nx + αe2 n2y + (αe2 + αm

(10)

.

7


ACS Photonics

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 18

Since n is a unit vector, a third condition also holds,

n2x + n2y + n2z = 1

(11)

Note that none of these equations depends on r. Thus, for some particular αe and αm , nx , ny and nz are the only unknowns in this set of 3 equations and their solutions specify the directions on which the C lines propagate on the far field. It is interesting to note that geometrically, Eq.9 and Eq.10 describe two hyperbolic surfaces, and that Eq. 11 describes a unitary sphere. So, for particular values of the complex electric and magnetic polarizability, the direction of the C lines will be determined by the intersection of these three surfaces. An analytic solution to this equations does exist, but unfortunately, it is too cumbersome to provide any intuitive interpretation. Nevertheless some interesting conclusions can be extracted under some particular assumptions. In the case where αe = αm , equations 9-11 predict that a single C line exists in the direction of n = −ˆz. According to Eq.7, however, the zˆ axis belongs to a L surface. Therefore, according to this formalism, light must be both linearly and circularly polarized in the backscattering trajectory. This is not feasible, unless the amplitude of the fields null. Therefore, no fields are radiated in this direction. This phenomenon is the well known Kerker condition for zero back-scattering, 26–29 but, to the best of our knowledge, it has never been approached before based on singular optics arguments. For αe = −αm , the second Kerker condition for zero forward-scattering can also be easily derived following the same procedure. To explore the existence of C lines in a realistic system, we consider the optical response of a 150 nm radius silicon sphere illuminated by linearly polarized light. We use tabulated bulk dielectric functions to characterize the response of silicon. 30 Figure 3a shows the geometrically normalized extinction cross section of this system (blue line) calculated using Mie theory. Yellow and red lines plot the contribution of the dipolar electric and magnetic terms in the Mie series to the total extinction spectrum. As expected, the lowest energy

8


Page 9 of 18

ACS Photonics


ACS Photonics

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

been chosen. Four intersection points can be identified (blue dots). These points indicate the directions on which C lines generated by the silicon nanoparticle will propagate in the far field. Note that the coefficients of Equations 9-11 are defined by the electric and magnetic polarizabilities of the sphere. If those coefficients would be perturbed by some reason, the surfaces in Fig. 3c would be distorted adiabatically and their intersections would be slightly displaced but they would continue existing. As a consequence, polarization singularities are stable upon perturbation, i.e. topologically protected. To verify our analytical predictions, we solve Maxwell’s equations for such a system using the Boundary Element Method. 31–33 In particular, we calculate ϕ at a distance 20 µm away from the Si nanoantenna. Figure 3c and Figure 3d plots the phase of ϕ and its absolute value. It easy to see that at the predicted positions, the amplitude of ϕ is zero and its phase becomes singular. Along these lines, the scattered fields are exactly circularly polarized. ~ which is positive for right handed fields Their handedness is determined by the sign of n · N and negative left handed fields. For an in depth discussion on the handedness of C lines in paraxial and non-paraxial fields please see Refs. 2,34,35 On the other hand, it is interesting to note that the plane of polarization (defined by the surface normal of the polarization ˜ may change along the C line. In the far-field, light propagates radially and the ellipse N) polarization plane is perpendicular to the radial direction. In the near-field instead, the fields are result of a superposition of plane waves and the plane of polarization becomes position dependent. If one is interested in studying the evolution of the polarization plane ˜ provides a simple and on the fields scattered by an optical antenna or any other object, N comfortable way to study such evolution. Finally, we calculate the polarization structure on a 3D volume encompassing all the radiation regions around the 150 nm Si nanoparticle excited by a 1110 nm linearly polarized plane wave. We track the C lines from the near-field to the far-field by the procedure described in the Methods section. This results are compiled in Figure 4. The gray sphere represents the Si nanoparticle while the red and green lines emerging from the sphere are

10


Page 10 of 18

Page 11 of 18

ACS Photonics


ACS Photonics

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

types: True Möbius polarization strips associated with topologically protected C lines and Möbius-like polarization structures associated with non-topologically protected points of pure transverse spin angular momentum. The Mobius strips presented in Figure 4, are associated to topologically protected C lines, and around them, the major axis of the polarization ellipse is never tangential to the trace around which the polarization structure. Consequently, the polarization structures presented in our contribution are true, topologically protected, Möbius polarization strips.

Conclusions In conclusion, we have analytically and numerically proved that high index nanoparticles support polarization singularities that propagate from the near-field to the far-field. In particular, C lines arise from the superposition of the fields scattered by electric and magnetic dipolar excitations on spherical silicon nanospheres. The study of polarization singularities is a rapidly growing field of research, due to both the intrinsic theoretical interest in the topology of light fields and because of the emergent applications on quantum information systems. This theoretical contribution proves that combinations of electric and magnetic dipoles supported by high index nanoparticles create such topologically protected features and may facilitate new experimental platforms to study polarization singularities.

Methods Identification and tracking of C lines around optical antennas In order to identify and track the C lines emerging from an optical antenna, we first calculate the vectorial electric fields on a meshed surface covering the nanoparticle at a short distance form its periphery (E0 (r)). In particular, to compute the results presented in Fig. 4, we choose to use the Metallic Nanoparticle Boundary Element Method MATLAB toolbox. 33

12


Page 12 of 18

Page 13 of 18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Photonics

Starting from E0 (r), the polarization scalar can be readily computed as:

ϕ0 (r) = u(r) + iv(r) = E0 (r) · E0 (r) = Ex (r)Ex (r) + Ey (r)Ey (r) + Ez (r)Ez (r).

(12)

The objective is to find the exact zeros of ϕ on the surface surrounding the nanoparticle. In order to do so, we first find the local minima of |ϕ| on the calculated mesh, and then run a minimization routine (fminsearch.m in Matlab) on |ϕ(r)| using the identified minima as seeds for the minimization routine. After verifying that these minima are actual zeros of ϕ(r), we take them as the starting points of the C lines (r0 ). The direction of the C line can be calculated though the following expression: 2 1 d(r0 ) = ∇ϕ∗ (r0 ) × ∇ϕ(r0 ) = ∇u(r0 ) × ∇v(r0 ) 2

(13)

Having calculated the origin of the C line and its direction, it is easy to calculate the seed of the next point on the C line as

rs1 = r0 + ld(r0 ),

(14)

with l = l0

d(r0 ) · ˆ r |d(r0 ) · ˆ r|

(15)

being the target distance to the next point on the line, l0 , corrected for the fact that d may point towards the nanostructure. Minimizing ϕ around this new position, it is easy to calculate the exact position of the next point on the C line. One can compute the entire trajectory of the C line by following this procedure recursively.

13


ACS Photonics

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 18

Associated content Supporting information Additional details on the concepts of topological charge of phase and polarization singularities and the analytical calculation of the polarization scalar ϕ the polarization ellipse surface ~ (Eqs. 7-11). normal N

Acknowledgement We thank J. J Saenz for insightful discussions and XX for the careful reading of the manuscript. A. G.-E. received funding from the Fellows Gipuzkoa fellowship of the Gipuzkoako Foru Aldundia through FEDER ”Una Manera de hacer Europa”

References (1) Berry, M. Making waves in physics. Nature 2000, 403, 21. (2) Dennis, M. R. Topological singularities in wave fields. University of Bristol (PhD Thesis) 2001, (3) Soskin, M. S.; Gorshkov, V. N.; Vasnetsov, M. V.; Malos, J. T. Topological charge and angular momentum of light beams carrying optical vortices. Phys. Rev. A 1997, (4) Molina-Terriza, G.; Torres, J. P.; Torner, L. Twisted photons. Nat. Phys. 2007, 3, 305–310. (5) Berry, M. V. Index formulae for singular lines of polarization. J. Opt. A: Pure Appl. Opt. 2004, 6, 675. (6) Nye, J. F. Lines of Circular Polarization in Electromagnetic Wave Fields. Proc. R Soc. A 1983, 389, 279–290. 14


Page 15 of 18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Photonics

(7) Nye, J. F. Polarization Effects in the Diffraction of Electromagnetic Waves: The Role of Disclinations. Proc. R Soc. A 1983, 387, 105–132. (8) Nye, J. F. Natural Focusing and Fine Structure of Light; IOP Publishing Ltd: Bristol, 1999. (9) Berry, M. V.; Dennis, M. R.; Lee, R. L., Jr Polarization singularities in the clear sky. New J. Phys. 2004, 6, 162. (10) Flossmann, F.; Kevin, O. H.; Dennis, M. R.; Padgett, M. J. Polarization singularities in 2D and 3D speckle fields. Phys. Rev. Lett. 2008, 100, 203902. (11) Egorov, R. I.; Soskin, M. S.; Kessler, D. A.; Freund, I. Experimental measurements of topological singularity screening in random paraxial scalar and vector optical fields. Phys. Rev. Lett. 2008, (12) Lindfors, K.; Priimagi, A.; Setälä, T.; Shevchenko, A.; Friberg, A. T.; Kaivola, M. Local polarization of tightly focused unpolarized light. Nat. Photonics 2007, 1, 228–231. (13) Schoonover, R. W.; Visser, T. D. Polarization singularities of focused, radially polarized fields. Opt. Express 2006, 14, 5733–5745. (14) Flossmann, F.; Schwarz, U. T.; Maier, M.; Dennis, M. R. Polarization singularities from unfolding an optical vortex through a birefringent crystal. Phys. Rev. Lett. 2005, 95, 253901. (15) Burresi, M.; Engelen, R.; Opheij, A.; Van Oosten, D. Observation of polarization singularities at the nanoscale. Phys. Rev. Lett. 2009, 102, 033902. (16) Rotenberg, N.; le Feber, B.; Visser, T. D.; Kuipers, L. Tracking nanoscale electric and magnetic singularities through three-dimensional space. Optica 2015, 2, 540. (17) de Hoogh, A.; Rotenberg, N.; Kuipers, L. Optical singularities in plasmonic fields near single subwavelength holes. J. Opt. (Bristol, U. K.) 2014, 16, 114004. 15


ACS Photonics

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(18) de Hoogh, A.; Kuipers, L.; Visser, T.; Rotenberg, N. Creating and Controlling Polarization Singularities in Plasmonic Fields. Photonics 2015, 2, 553–567. (19) Le Feber, B.; Rotenberg, N.; Kuipers, L. Nanophotonic control of circular dipole emission. Nat. Commun. 2015, 6 . (20) Young, A. B.; Thijssen, A.; Beggs, D. M.; Kuipers, L.; Rarity, J. G.; Oulton, R. Polarization engineering in photonic crystal waveguides for spin-photon entanglers. Phys. Rev. Lett. 2015, 115, 153901. (21) Bauer, T.; Neugebauer, M.; Leuchs, G.; Banzer, P. Optical Polarization Möbius Strips and Points of Purely Transverse Spin Density. Phys. Rev. Lett. 2016, 117, 013601. (22) Garcia-Etxarri, A.; Gómez-Medina, R.; Froufe-Perez, L. S.; López, C.; Chantada, L.; Scheffold, F.; Aizpurua, J.; Nieto-Vesperinas, M.; Sáenz, J. J. Strong magnetic response of submicron Silicon particles in the infrared. Opt. Express 2011, 19, 4815–4826. (23) 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. (24) 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. (25) Garcia-Etxarri, A.; Dionne, J. A. Surface enhanced circular dichroism spectroscopy mediated by non-chiral nanoantennas. Phys. Rev. B 2013, 87, 235409. (26) Geffrin, J. M.; Garc´ıa-Cámara, B.; Gómez-Medina, R.; Albella, P.; Froufe-Pérez, L. S.; Eyraud, C.; Litman, A.; Vaillon, R.; González, F.; Nieto-Vesperinas, M.; Sáenz, J. J.; Moreno, F. Magnetic and electric coherence in forward- and back-scattered electro-

16


Page 16 of 18

Page 17 of 18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Photonics

magnetic waves by a single dielectric subwavelength sphere. Nat. Commun. 2012, 3, 1171. (27) Person, S.; Jain, M.; Lapin, Z.; Sáenz, J. J.; Wicks, G. Demonstration of zero optical backscattering from single nanoparticles. Nano Lett. 2013, 13, 1806–1809. (28) Kerker, M.; Wang, D. S.; Giles, C. L. Electromagnetic scattering by magnetic spheres. J. Opt. Soc. Am. 1983, 73, 765–767. (29) Zambrana-Puyalto, X.; Fernandez-Corbaton, I.; Juan, M. L.; Vidal, X.; MolinaTerriza, G. Duality symmetry and Kerker conditions. Opt. Lett. 2013, 38, 1857. (30) Palik, E. D. Handbook of optical constants of solids; Academic press, 1998. (31) Garc´ıa de Abajo, F.; Howie, A. Retarded field calculation of electron energy loss in inhomogeneous dielectrics. Phys. Rev. B 2002, 65, 115418. (32) Garc´ıa de Abajo, F.; Howie, A. Relativistic Electron Energy Loss and Electron-Induced Photon Emission in Inhomogeneous Dielectrics. Phys. Rev. Lett. 1998, 80, 5180–5183. (33) Hohenester, U.; Tr¨ ugler, A. MNPBEM–A Matlab toolbox for the simulation of plasmonic nanoparticles. Comput. Phys. Commun. 2012, (34) Freund, I. Observer-dependent sign inversions of polarization singularities. Opt. Lett. 2014, 39, 5873–5876. (35) Freund, I. Topological events on the lines of circular polarization in nonparaxial vector optical fields. Opt. Lett. 2017, 42, 611–614. (36) Freund, I. Hidden order in optical ellipse fields: I. Ordinary ellipses. Opt. Commun. 2005, 256, 220–241. (37) Freund, I. Cones, spirals, and Möbius strips, in elliptically polarized light. Opt. Commun. 2005, 249, 7–22. 17


ACS Photonics

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(38) Bauer, T.; Banzer, P.; Karimi, E.; Orlov, S.; Rubano, A.; Marrucci, L.; Santamato, E.; Boyd, R. W.; Leuchs, G. Observation of optical polarization Mobius strips. Science 2015, 347, 964–966.

18


Page 18 of 18

Optical Polarization Möbius Strips on All-Dielectric ... - ACS Publications

Recommend Documents