Source: https://knepublishing.com/index.php/KnE-Energy/article/view/2029/4592
Timestamp: 2019-04-25 15:51:19+00:00

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Copyright © 2018 V. S. Pereskokov and I. V. Dzedolik.
We present the results of simulation of interference of surface plasmon-polaritons (SPPs) which are falling and reflecting from the curvilinear boundary of inhomogeneity area in the metal layer. The plasmon vortices with a screw phase dislocation appear in the singular points of the field as a result of the SPP interference after reflection from the boundary of inhomogeneity in the dovetail form. The position of the plasmon vortices on the surface of metal layer can be controlled by means of the external electrostatic field. Negative charges localized at the control probes cause the change of the boundary curvature of the permittivity of inhomogeneity area on the metal layer, which leads to displacement of the vortex localization points. When the vortex is localized under the readout nanowire probe with angular thread, the maximum or minimum of the signal takes place in the probe depending on the helicity of the thread and the topological charge of the vortex.
In the last decade surface plasmon polaritons (SPPs) attract attention of researchers in connection with controlling of the electromagnetic fields of optical frequencies by the SPP-devices and creating logic gates for optical processors, spasers, other devices and elements of plasmon technology [1-10]. The SPPs with the dependence of field components in the form ∼exp-αx+iβz-ωt can be excited on the interface of metal with a negative real part of the permittivity ReεM<0 , and a dielectric medium with the permittivity Reε0>0 . When ReεM<0 the SPPs propagation takes place on the surface of the metal; but with positive values ReεM>0 the SPPs can not propagate because the boundary conditions εMα0=-ε0αM are not satisfied, where α0>0 and αM>0 are the decrements along the normal axis x to the metal surface (Figure 1).
The curvature of the wavefront and the direction of propagation of the SPPs change while reflecting from the curvilinear boundary of the inhomogeneity in the metal layer. Scattering of SPPs on the inhomogeneities of various configurations at the boundary between the dielectric and metal leads to enrichment of the mode composition of the SPPs, as well as to the interconnection of modes in microwave guides and microcavities, and to radiation from the metal and dielectric interface of bulk electromagnetic waves. In this field of research a large volume of theoretical and experimental works are devoted to scattering of the SPPs on various inhomogeneities in the metal layers [11-14], to focusing and controlling the SPPs by electromagnetic fields, and to various dielectric and metallic nanostructures on a chips and plasmon lenses [15-26].
Generation of the SPPs in the metal layer according to the Kretchmann scheme, and reflection of the SPPs from the boundary of permittivity inhomogeneity in the layer: φ is the angle between the tangent to the boundary of the inhomogeneity and the transverse axis y .
It is known, that the optical vortices may appear when the waves with wavefronts of different configurations formed under reflection, refraction or diffraction of waves have interference [27,28]. Optical vortices are still actively investigated in connection with a wide field of their application . Interference of the SPPs can also lead to the formation of plasmon-polariton vortices under certain conditions. The plasmon-polariton vortices are excited when the SPPs pass through plasmon lenses which are curved slits, or cavities and protrusions in a metal layer [30-33], as well as under normal incidence of an optical vortex beam on the metal surface . The singular points with a screw phase dislocation arise on the surface of the metal layer, in which plasmon-polariton vortices are formed at the interference of incident and reflected SPPs from the inhomogeneity of permittivity with curvilinear boundary [35-37]. The plasmon vortices do not arise in case of superposition of the modes reflected from a straight line boundary of the inhomogeneity.
The boundary of inhomogeneity in the form of dovetail (Figure 1) can be formed in the metallic layer as a result of the action of an electrostatic field of the negative charges that localized on the control probes which are located above the metal layer. In this case, the negatively charged probes can change the permittivity of metal in such a way that it acquires positive values εM'>0 in the optical frequency range. Then, in the area of the electrostatic field the boundary propagation conditions are violated, and the SPPs are scattered at the boundary of the artificially created inhomogeneity of the metal permittivity. It is possible to control the position of the SPP vortices by changing the radius of the circles of the dovetail varying the intensity of the electrostatic field.
The purpose of our work is to find the conditions for the appearance of SPP vortices formed on the interface of the metal and dielectric when the SPPs are reflected from the inhomogeneity of permittivity in the metal layer in the form of dovetail. By varying the values of the negative charges on the control probes it is possible to change the form of the inhomogeneity area of the metal permittivity. This change of the boundary curvature leads to the change of the topology of the interference field of the incident and reflected SPPs. Thus, it is possible to control the configuration of the vortex lattice of the SPPs by means of the external electrostatic field arising between the probes with negative charges and the metal layer.
The permittivity of a metal at optical frequencies is a complex quantity with a negative real part εM=-εM'+iεM'' . Therefore, the propagation constants of the SPPs β=β'+iβ'' are also complex quantities. Their imaginary parts characterize the attenuation of the SPPs along the axis z , i.e. they determine the propagation length of the SPPs along the lower and upper surfaces of the metal layer L=1/2β'' . Generally, the decrements α0=β2-c-2ω2ε0 1/2 and αM=β2-c-2ω2εM 1/2 are complex quantities as well as the SPP propagation constant. Thus, when the SPP propagates along the metal surface, the components of its field oscillate at the frequency ω attenuating both along the longitudinal axis z and along the normal axis x to the media interface.
The TM-mode of the SPPs with the field components Ex,Ez,By is formed at the homogeneous interface of the non-magnetic metal and dielectric . The TM-mode propagates along the homogeneous metal surface with the mode propagation constant β that is parallel to the axis z ; the electric vector of the TM-mode rotates in the plane x,z , and the wavefront of the surface wave is flat. But the surface plasmon-polariton wave is reflected from the inhomogeneity boundary if there is an inhomogeneity in the metal layer, for example, if the metal layer is broken off or the boundary conditions -α0εM=αMε0 are violated. The real part of the permittivity ReεM=-εM'<0 of the metal at optical frequencies is negative. However, under the influence of the external electrostatic field of the negative charge located at the control probe above the metal layer, the area with positive permittivity ReεM=εM'>0 can be formed in the metal. The SPPs will be scattered on such area of inhomogeneity of the metal permittivity.
There are the evanescent waves directed from the boundary to the inhomogeneity area in the direction of the axis z , and the SPPs directed back from the boundary of inhomogeneity against the axis z . In this case, the mode composition of the reflected SPP wave is enriched; the modes with field components Ex,Ey,Ez,By,Bz are formed [35-37]. However, because of the boundary conditions, the normal component of the magnetic vector Bx does not arise from the inhomogeneity reflection of the TM mode, and the rotation plane of the electric vector turns about the axis x . The singular points localized in the field minima may arise in result of the interference of the incident TM-mode and reflected modes of the SPPs. At these points, the corresponding components of the SPP field are zero, the phase of the interference field is not determined, and the interference fringes of the SPP field are split. The Poynting vector S=c/4π E×B of the SPPs at such singular points has three components and it precesses around an axis normal to the plane of the metal layer, i.e. plasmon-polariton vortices arise.
(1) where A=const , ϕT=-α0x+iβz . The dispersion equations of the SPPs have the form c-2ω2ε0+α02-β2=0 in the air, and c-2ω2εM+αM2-β2=0 in the metal. From the condition of continuity of the tangential components of the electric field Ez on the surface of the metal layer, the boundary condition αMε0=-α0εM holds for the TM-mode of the SPPs at ReεM<0 .
where ϕR=-α0x+iβysin2φ-zcos2φ , and the normal to the surface component of the magnetic field Bx=0 remains equal to zero [35-37]. The number of modes of reflected SPPs depends on the shape of the inhomogeneity boundary, i.e. the angles of reflection of the SPPs are 2φ . The same modes are formed in the metal layer near the upper surface, but in this case the decrement in the expressions for the field components must be replaced -α0→αM , and the permittivity is also replaced ε0→εM . The boundary conditions for the reflected SPPs are not violated, and the transverse component of the SPP wavevector κ is added to the dispersion equations at the boundary with the air c-2ω2ε0+α02-κ2-β˜2=0 , and in the metal layer c-2ω2εM+αM2-κ2-β˜2=0 , where κ=βsin2φ , β˜=βcos2φ , i.e. κ2+β˜2=β2 . The propagation constant at the boundary of the metal layer and the air has the real value β=k0εM/1+εM 1/2=k0εM'2+εM''2/1-εM'2+εM''2 1/2 , where k0=ω/c .
The distribution of the field on the metal surface x=0 during the SPP mode interference depends on the angle of reflection 2φ of the TM-mode from the boundary of the inhomogeneity in the metal, i.e. from the curvature of the boundary. The maxima and minima of the components of the electric and magnetic vectors of the SPPs arise in the area of the existence of the SPPs as a result of the mode interference; accordingly, the Poynting vector Sj=Sjaexpiϕj of the SPP has the maxima and minima, where Sja=ReSj2+ImSj2 1/2 is the amplitude, ϕj=arctan ImSj/ReSj is the phase of the interference field, j=x,y,z . At least three plasmon-polariton waves arrive to the zero points of the SPP interference field: the incident wave and two reflected waves at different angles 2φ from the curvilinear boundary of the inhomogeneity, then the screw dislocation takes place in the phase of interference field. At these points the SPP interference fringes are split, and plasmon-polariton vortices arise. Modern methods of apertureless near-field microscopy with the resolution of units of nanometers [38-40] are based on the detection of the normal component Ex of the electric vector; therefore we will analyze hereinafter the distribution of Ex .
Negatively charged control probes (Figure 1) create the areas of inhomogeneity of the metal permittivity in the form of circles with radii r0 with the positive “mirror” charges in the metal that arise by the displacement of conduction electrons. We estimate the parameters of the experimental device assuming that a point charge -q is placed at the tip of the probe, creating a field strength E0=-q4πε0ehp2-1 , where ε0e=8.84×10-12F/m . On the boundary of the circle with the radius r0 , i.e. at a distance ξ=hp2+r02 1/2 from the charge, the intensity of the electrostatic field will decrease as a ratio Eξ/E0=hp2/ξ2 . The intensity of the electrostatic field decreases as the ratio Eξ/E0≈10-5 at the boundary of the inhomogeneity area of the permittivity at the radius r0=10 μm with the height of the probe hp=30 nm above the surface of the layer. We obtain the ratio Eξ/E0≈10-3 at the radius r0=1 μm , and the ratio Eξ/E0≈10-1 at the radius r0=100 nm . The voltage at the control probe placed at the height hp is U=E0hp . With the voltage at the probe U=1 mV the field intensity under the probe will be E0=33.3×103 V/m , and the magnitude of the positive “mirror” charge is equal to q=4πε0ehpU=0.111×10-12C . It is possible to increase the “mirror” charge at constant voltage, if dielectric medium with permittivity ε0→ε is placed between the control probe and the metal layer. It leads to the electrocapacity increasing in the space between the control probe and the metal layer, then the charge value is q=Cε U .
The scattering of the SPPs at the boundary of the inhomogeneity area is inelastic. However, in this case the reflected SPPs directed against the axis z are generated, and they interfere with the SPPs falling along the axis z . One can change the radius r0 of the area of permittivity inhomogeneity of the metal by varying the voltage U at the control probe U2/U1=q2/q1=r022+hp2 1/2r012+hp2-1/2 at the fixed probe height hp above the surface of the metal. Then the radius of the inhomogeneity area varies as r02=g2r012+g2-1hp2 1/2 , where g=U2/U1 , i.e. the radius r0 varies in proportion to the voltage between the control probe and the layer.
Figure 2 shows the normalized distribution of the normal component of the electric vector Ex=Exaexpiϕx , where Exa=ReEx2+ImEx2 1/2 is the amplitude, ϕx=arctan ImEx/ReEx is the phase of the SPP interference field at a certain time for the superposition of TM-modes at the air-metal interface: ε0=1 and εM=-εM'+iεM''=-12.64+i 1.40 (the layer of polycrystalline gold with the thickness 53 nm ) , the laser beam has the wavelength in the air λ0=630 nm . In the case under consideration the propagation constant of the SPP is equal β=1.08×105cm-1 , then the wavelength of the SPP is λ=2π/β=581 nm . The decrements of the SPP have values: α0=4.19×104cm-1 in the air, and αM=3.71×105cm-1 in the metal layer, that corresponds to the distances along the axis x normal to the surface: h0=238 nm , and hM=26.9 nm , where the SPP is attenuating.
Distribution of the normal component Ex of electric vector at the interference of the incident and reflected SPPs from the inhomogeneity boundary in the metal layer in the form of dovetail: (a) the interference fringes of the amplitude at the radius of curvature boundary r0=5 μm ; (b) the interference fringes of the amplitude at the radius of curvature boundary r0=10 μm , (c) the phase distribution, r0=5 μm ; (d) the phase distribution, r0=10 μm ; (e) the SPP vortices with topological charge ℓM=+1 (red arrow, anti-clockwise) and with topological charge ℓM=-1 (green arrow, clockwise) in the fence, r0=5 μm ; (f) the SPP vortices in the fence, r0=10 μm ; the values along the axes y,z are marked in micrometers.
As a result of SPP interference the plasmonic vortices arise at the points of splitting of the interference fringes of the field minima, when the SPPs are reflected from the curvilinear boundary of the inhomogeneity in the metal layer, (Figure 2 (a), dark lines). The change of the boundary curvature leads to a shift in the minima of the SPP field. The varying of the potential of the control probe over the metal surface leads to decreasing or increasing of the radius r0 of the permittivity inhomogeneity area in the metal, which causes to the vortices shifting from their original positions (Figure 2 (c) and (d)).
If the readout probe is placed above the point of localization of the SPP vortex on the metal surface, then surface plasmon modes can be excited in the nanowire of the probe [42-43]. To excite the SPP modes in the nanowire, it is necessary to match the normal component of the electric field ExM of the SPP vortex on the metal surface and the longitudinal mode component Ezw on the nanowire surface.
Consider the process of formation of surface plasmon-polariton modes upon excitation of a nonmagnetic metal nanowire with a circular radius a of cross section by monochromatic electromagnetic radiation with frequency ω . Suppose the boundary of the nanowire is corrugated in the form of a spiral with the deep d and step Λ along the axis z of the nanowire, then the radius of spiral is a-d=const . Corrugation of the nanowire boundary leads to disturbance of its permittivity εd=ε+Δεz , where ε=const . The perturbation of the permittivity of the nanowire is represented in the cylindrical coordinate system as Δε=-ε daexpis2πΛ z , then εd=ε1-d¯expisKz , where d¯=d/a , K=2π/Λ , and s=±1,±2,... is the index which characterizes the direction of rotation (helicity) and the number of corrugation branches.
(8) where ζs=s2k0d¯εKexpisKz2 , Ks=sd¯K2 , ν=2β/K . The longitudinal components of the electric field (7) and (8) attenuate when the SPP modes propagate along the nanowire.
The phase of the longitudinal component of the ℓ th mode of the SPPs on the nanowire surface r=a has the form ϕz0=arctanImfℓs/Refℓs , where fℓs=exp iℓφ+KszJνζs . From the expression for the phase of the longitudinal component, we can determine the “helicity” of the nanowire SPP mode σz=12π∮dr∇ϕz0 , that is σz=12π∫02πdφ∂ϕz0∂φ+∫0Λdz∂ϕz0∂z . Suppose the perturbation of the permittivity of nanowire is small d¯<<1 , and the propagation mode constant of the SPP mode for the nanowire is β=K , i.e. ε=εM' . Then taking into account only the first term of the series for the Bessel function J2ζ≈k02d¯ε2K2expisKz , we obtain the expression fℓs≈k02d¯ε2K2exp iℓφ+s1+d¯2Kz . The phase of the ℓ th SPP-mode of the nanowire has the form ϕz0=ℓφ+s1+d¯/2Kz in this approximation. The helicity of the longitudinal component of the ℓ th SPP-mode of the nanowire in this case is equal to σz=ℓ+s1+d¯/2 . Thus, we get the maximum or minimum signal in the readout probe depending on the helicity of the nanowire and the topological charge of the vortex under the readout probe.
The SPPs generated at the boundary of the homogeneous dielectric medium and the metal layer form the TM-mode propagating along the surface of the metal and having the plane wavefront. The inhomogeneities of the metal layer permittivity cause the reflection of the SPPs, while the modal composition of the surface waves changes. There is the interference of the TM-modes when the SPPs are reflected from the straight line boundary, but the SPP vortices do not arise. If the boundary of the inhomogeneity area is curvilinear, then the vortex lattice arises as a result of interference of the SPP-modes.
The distribution of the singular points at the minima of the SPP interference field, in which vortices are formed on the metal surface, depends on the curvature of the boundary of the inhomogeneity area. The curvature of the boundary of the inhomogeneity area in the metal layer can be changed by means of the external electrostatic field of negative charges at the control probes. It is possible to control the distribution of the minima of the SPP interference field by changing the voltage on the control probes located above the metal surface, i.e. varying the value of the negative charges of the control probes, we can change the configuration of the vortex grating. In the readout probes, which are nanowires with the spiral thread, the signals are effectively excited when the helicity of the thread coincides with the topological charge of the plasmon-polariton vortex.
W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature, vol. 424, pp. 824-830, 2003.
A. V. Zayats, and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” Journal of Optics A: Pure and Applied Optics, vol. 5, pp. S16-S50, 2003.
S. A. Maier, Plasmonics: Fundamental and Applications, New York: Springer Science+Bussines Media, 2007.
P. Berini, “Long-range surface plasmon polaritons,” Advances in Optics and Photonics, vol. 1, 484-588, 2009.
D. K. Gramotnev, and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit”, Nature Photonics, vol. 4, pp. 83-91, 2010.
M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Optics Express, vol. 19, pp. 22029-22106, 2011.
L. Novotny, and B. Hecht, Principles of Nano-Optics, Cambridge: Cambridge University Press, 2012.
O. V. Shulika, and I. A. Sukhoivanov, Contemporary optoelectronics: Materials, metamaterials and device applications, Dordrecht: Springer Science+Bussines Media, 2016.
I. V. Dzedolik, Solitons and nonlinear waves of phonon-polaritons and plasmon-polaritons, New York: Nova Science Publishers, 2016.
A. B. Shesterikov, M. Yu. Gubin, M. G. Gladush, and A. V. Prokhorov, “Formation of plasmon pulses in the cooperative decay of excitons of quantum dots near a metal surface,” Journal of Experimental and Theoretical Physics, vol. 124, no. 1, pp. 18-31, 2017.
F. Pincemin, A. A. Maradudin, A. D. Boardman, and J.-J. Greffet, “Scattering of a surface plasmon polariton by a surface defect,” Physical Review B, vol. 50, pp. 15261-15275, 1994.
B. Hecht, H. Bielefeld, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Physical Review Letters, vol. 77, pp. 1889-1892, 1996.
P. Cheyssac, V. A. Sterligov, S. I. Lysenko, and R. Kofman, “Surface plasmon-polaritons 1. Interaction with 1D objects,” Physical Status Solidi (a), vol. 175, pp. 253-258, 1999.
H. Ditlbacher, J. R. Krenn, G. Schider, A. Leitner, and F. R. Aussenegg, “Two-dimensional optics with surface plasmon polaritons,” Applied Physics Letters, vol. 81, no. 10, pp. 1762-1764, 2002.
A. V. Krasavin, A. V. Zayats, and N. I. Zheludev, “Active control of surface plasmon–polariton waves,” Journal of Optics A: Pure and Applied Optics, vol. 7, pp. S85-S89, 2005.
V. N. Konopsky, and E. V. Alieva, “Long-range propagation of plasmon polaritons in a thin metal film on a one-dimensional photonic crystal surface,” Physical Review Letters, vol. 97, 253904, 2006.
P. A. Huidobro, M. L. Nesterov, L. Martin-Moreno, and F. J. Garcia-Vidal, “Transformation optics for plasmonics,” Nano Letters, vol. 10, pp. 1985–1990, 2010.
C. Zhao, J. Zhang, and Y. Liu, “Light manipulation with encoded plasmonic nanostructures,” European Physical Journal of Applied Metamaterials, vol. 1, 6, 2014.
Y.-G. Chen, Y.-H. Chen, and Z.-Y. Li, “Direct method to control surface plasmon polaritons on metal surfaces,” Optics Letters, vol. 39, pp. 339-342, 2014.
S.-Y. Lee, K. Kim, S.-J. Kim, H. Park, K.-Y. Kim, and B. Lee, “Plasmonic meta-slit: shaping and controlling near-field focus,” Optica, vol. 2, no. 1, pp. 6-13, 2015.
V. Coello, C. E. Garcia-Ortiz, and M. Garcia-Mendez, “Classic plasmonics: wave propagation control at subwavelength scale,” NANO, vol. 10, 1530005, 2015.
Q. Guo, C. Zhang, and X. Hu, “A spiral plasmonic lens with directional excitation of surface plasmons,” Scientific Reports, vol. 6, 32345, 2016.
H. Li, Y. Qu, H. Ullah, B. Zhang, and Z. Zhang, “Controllable multiple plasmonic bending beams via polarization of incident waves,” Optics Express, vol. 25, no. 24, pp. 29659-29666, 2017.
W.-B. Shi, T.-Y. Chen, H. Jing, R.-W. Peng, and M. Wang, “Dielectric lens guides in-plane propagation of surface plasmon polaritons,” Optics Express, vol. 25, no. 5, pp. 5772-5780, 2017.
J. Wang, C. Chen, and Z. Sun, “Creation of multiple on-axis foci and ultra-long focal depth for SPPs,” Optics Express, vol. 25, no. 2, pp. 1555-1563, 2017.
Z. Wang, G. Ren, Y. Gao, B. Zhu, and S. Jian, “Plasmonic in-plane total internal reflection: azimuthal polarized beam focusing and application,” Optics Express, vol. 25, no. 20, pp. 23989-23999, 2017.
J. F. Nye, and M. V. Berry. “Dislocations in wave trains,” Proceedings of the Royal Society of London A, vol. 336, pp. 165-190, 1974.
M. R. Dennis, K. O'Holleran, and M. J. Padgett, “Singular optics: Optical vortices and polarization singularities,” Progress in Optics, vol. 53, pp. 293-363, 2009.
M. Soskin, S. V. Boriskina, Y. Chong, M. R. Dennis, and A. Desyatnikov, “Singular optics and topological photonics,” Journal of Optics, vol. 19, no. 1, 010401, 2017.
H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Letters, vol. 10, pp. 529-536, 2010.
S. V. Boriskina and B. M. Reinhard, “Adaptive on-chip control of nano-optical fields with optoplasmonic vortex nanogates,” Optics Express, vol. 19, no. 22, pp. 22305-22315, 2011.
H. Zhou, J. Dong, Y. Zhou, J. Zhang, M. Liu, and X. Zhang, “Designing appointed and multiple focuses with plasmonic vortex lenses,” IEEE Photonics Journal, vol. 7, 4801007, 2015.
A. M. Kamchatnov, and N. Pavloff, “Interference effects in the two-dimensional scattering of microcavity polaritons by an obstacle: phase dislocations and resonances,” European Physical Journal D, vol. 69: 32, 2015.
G. Yuan, Q. Wang, and X. Yuan, “Dynamic generation of plasmonic Moiré fringes using phase-engineered optical vortex beam,” Optics Letters, vol. 37, no. 13, pp. 2715-2717, 2012.
I. V. Dzedolik, and V. Pereskokov, “Formation of vortices by interference of surface plasmon polaritons,” Journal of the Optical Society of America A, vol. 33, no. 5, pp. 1004-1009, 2016.
I. V. Dzedolik, S. Lapayeva, and V. Pereskokov, “Vortex lattice of surface plasmon polaritons,” Journal of Optics, vol. 18, no. 7, 074007, 2016.
I. V. Dzedolik, and V. S. Pereskokov, “Topology of plasmon-polariton vortices on an adaptive mirror,” Atmospheric and Oceanic Optics, vol. 30, no. 2, pp. 203-208, 2017.
V. N. Konopsky, “Operation of scanning plasmon near-field microscope with gold and silver tips in tapping mode: demonstration of subtip resolution,” Optics Communications, vol. 185, pp. 83-93, 2000.
D. V. Kazantsev, and H. Ryssel, “Scanning head for the apertureless near field optical microscope,” Modern Instrumentation, vol. 2,pp. 33-40, 2013.
D. V. Kazantsev, E. V. Kuznetsov, S. V. Timofeev, A. V. Shelaev, and E. A. Kazantseva, “Apertureless near-field optical microscopy,” Uspekhi Fizicheskikh Nauk, vol. 187, no. 3, pp. 277-295, 2017.
D. I. Yakubovsky, A. V. Arsenin, Yu. V. Stebunov, D. Yu. Fedyanin, and V. S. Volkov, “Optical constants and structural properties of thin gold films,” Optics Express, vol. 25, no. 21, pp. 25574-25587, 2017.
F. Ruting, F. I. Fernandez-Dominguez, L. Martin-Moreno, and F. J. Garcia-Vidal, “Subwavelength chiral surface plasmons that carry tunable orbital angular momentum”, Physical Review B, vol. 86, 075437, 2012.
K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity to nanostructures,” Physical Review Letters, vol. 110, 143603, 2013.

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