We demonstrate numerically the diffraction-free propagation of sub-wavelength sized optical beams through simple elements built of metal-dielectric multilayers. The proposed metamaterial consists of silver and a high refractive index dielectric, and is designed using the effective medium theory as strongly anisotropic and impedance matched to air. Further it is characterised with the transfer matrix method, and investigated with FDTD. The diffraction-free behaviour is verified by the analysis of FWHM of PSF in the function of the number of periods. Small reflections, small attenuation, and reduced Fabry Perot resonances make it a flexible diffraction-free material for arbitrarily shaped optical planar elements with sizes of the order of one wavelength.
The optical nonlocality in symmetric metal-dielectric multilayer metamaterials is theoretically and experimentally investigated with respect to transverse-magnetic-polarized incident light. A nonlocal effective medium theory is derived from the transfer-matrix method to determine the nonlocal effective permittivity depending on both the frequency and wave vector in a symmetric metal-dielectric multilayer stack. In contrast to the local effective medium theory, our proposed nonlocal effective medium theory can accurately predict measured incident angle-dependent reflection spectra from a fabric…
A metallo-dielectric multilayer structure is proposed as a novel approach to the analysis of lactose malabsorption. When lactose intolerance occurs, the bacterial overgrowth in the intestine causes an increased spontaneous emission of H2 in the human breath. By monitoring the changes in the optical properties of a multilayer palladium-polymeric structure, one is able to detect the patient's disease and the level of lactose malabsorption with high sensitivity and rapid response.
We present the design, fabrication and performance test of a quasi three-dimensional carpet cloak made of normal dielectric in the microwave regime. Taking advantage of a simple linear coordinate transformation we design a carpet cloak with homogeneous anisotropic medium and then practically realize the device with multilayer of alternating normal dielectric slabs based on the effective medium theory. As a proof-of-concept example, we fabricate the carpet cloak with multilayer of FR4 dielectric slabs with air spacing. The performance of the fabricated design is verified through full-wave numer…
Epsilon-near-zero (ENZ) metamaterial slabs at visible frequencies based on metal-dielectric multilayers are experimentally realized. Transmission, reflection and absorption spectra are measured and used to determine the complex refractive indices and the effective permittivities of the ENZ slabs, which agree with the results obtained from both the numerical simulations and the optical nonlocalities analysis. Furthermore, light propagation in ENZ slabs and directional emission from ENZ prisms are also analyzed. The accurate determination of the ENZ wavelength for metal-dielectric multilayer met…
We show that a stack of metal-dielectric nanolayers, in addition to the long- and short-range plasmons, guides also an entire family of modes strongly confined within the multilayer - the bulk plasmon modes. We propose the classification scheme that reflects specific properties of these modes. We report experimental verification of the bulk plasmon modes by measuring modal indices in a structure made of three pairs of silica(29nm)/gold(25nm) layers.
Very weak left/right asymmetry in reflection and transmission is offered by a layer of a topological insulator on top of a layer of an anisotropic dielectric material, but it can be enhanced very significantly by using a periodic multilayer of both types of materials. This is an attractive prospect for realizing one-way terahertz devices, because both types of materials can be grown using standard physical-vapor-deposition techniques.
The concept of the broadband epsilon-near-zero meta-atom consisting of layered stacks with specified metallic filling ratio and thickness is proposed based on the Bergman spectral representation of the effective permittivity. The step-like metal-dielectric multilayer structures are designed to achieve realistic broadband epsilon-near-zero meta-atoms in optical frequency range. These meta-atoms can be integrated as building blocks for unconventional optical components with exotic electromagnetic properties over a wide frequency range, such as the demonstrated broadband directional emission and …
We analyse theoretically for the first time to our knowledge the perfect phase matching of guided TE and TM modes with a multilayer waveguide composed of linear isotropic dielectric materials. Alongside strict investigation into dispersion relations for multilayer systems, we give an explicit qualitative explanation for the phenomenon of mode matching on the basis of the standard one-dimensional homogenization technique, and discuss the minimum number of layers and the refractive index profile for the proposed device scheme. Direct applications of the scheme include polarization-insensitive, i…
The giant optical nonlocality near the Dirac point in lossless metal-dielectric multilayer metamaterials is revealed and investigated through the analysis of the band structure of the multilayer stack in the three-dimensional omega-k space, according to the transfer-matrix method with the optical nonlocal effect. The position of the Dirac point is analytically located in the omega-k space. It is revealed that the emergence of the Dirac point is due to the degeneracy of the symmetric and the asymmetric eigenmodes of the coupled surface plasmon polaritons. The optical nonlocality induced epsilon…
We investigate the resolution and absorption losses of a Ag/GaP multilayer superlens. For a fixed source to image distance the resolution is independent of the position of the lens but the losses depend strongly on the lens placement. The absorption losses associated with the evanescent waves can be significantly larger than losses associated with the propagating waves especially when the superlens is close to the source. The interpretation of transmittance values greater than unity for evanescent waves is clarified with respect to the associated absorption losses.
We experimentally demonstrate the effect of anomalous breakdown of the effective medium approximation in all-dielectric deeply subwavelength thickness ($d \simλ/160-λ/30$) multilayers, as recently predicted theoretically [H.H. Sheinfux et al., Phys. Rev. Lett. 113, 243901 (2014)]. Multilayer stacks are composed of alternating alumina and titania layers fabricated using atomic layer deposition. For light incident on such multilayers at angles near the total internal reflection we observe pronounced differences in the reflectance spectra for structures with 10-nm versus 20-nm thick layers, as we…
Chemical synthesized silver nanowires have been proved to be the efficient architecture for Plasmonic waveguides, but the high propagation loss prevents their widely applications. Here, we demonstrate that the propagation distance of the plasmons along the Ag NW can be extended if the Ag NW was placed on a dielectric multilayer substrate containing a photonic band gap, but not placed on a commonly used glass substrate. The propagation distance at 630 nm wavelength can reach 16 um even that the Ag NW is as thin as 90 nm in diameter. Experimental and simulation results further show that the pola…
Extremely anisotropic metal-dielectric multilayer metamaterials are designed to have the effective permittivity tensor of a transverse component (parallel to the interfaces of the multilayer) with zero real part and a longitudinal component (normal to the interfaces of the multilayer) with ultra-large imaginary part at the same wavelength, including the optical nonlocality analysis based on the transfer-matrix method. The diffraction-free deep-subwavelength optical beam propagation with near-zero phase variation in the designed multilayer stack due to the near-flat iso-frequency contour is dem…
Multilayered structures consisting of alternating negative-permittivity and dielectric layers are explored to obtain high-resolution imaging of subwavelength objects. The peaks with the smallest |ky| (ky is the transverse wave vector) on the transmission curves, which come from the guided modes of the multilayered structures, can not be completely damped by material loss. This makes the amplitudes of the evanescent waves around these peaks inappropriate after transmitted through the imaging structures, and the imaging quality is not good. To solve such a problem, the permittivity of the dielec…
Metal-dielectric layered stacks for imaging with sub-wavelength resolution are regarded as linear isoplanatic systems - a concept popular in Fourier Optics and in scalar diffraction theory. In this context, a layered flat lens is a one-dimensional spatial filter characterised by the point spread function. However, depending on the model of the source, the definition of the point spread function for multilayers with sub-wavelength resolution may be formulated in several ways. Here, a distinction is made between a soft source and hard electric or magnetic sources. Each of these definitions leads…
We provide theoretical and numerical tools to quantitatively study the impact of nonlocality arising from free electrons in metals on the optical properties of metallo-dielectric multilayers. Though effects due to nonlocality are in general quite small, they nevertheless can be important for very thin (typically below 10 nm) metallic layers - as are used in structures characterized by relatively flat dispersion curves. Such structures include those with negative refractive index; hyperbolic metamaterials; and materials with index near zero. We find in all cases that the inclusion of nonlocal e…
Multiple p- and s-polarized compound surface plasmon-polariton (SPP) waves at a fixed frequency can be guided by a structure consisting of a metal layer sandwiched between a homogeneous isotropic dielectric (HID) material and a periodic multilayered isotropic dielectric (PMLID) material. For any thickness of the metal layer, at least one compound SPP wave must exist. It possesses the p-polarization state, is strongly bound to the metal/HID interface when the metal thickness is large but to both metal/dielectric interfaces when the metal thickness is small. When the metal layer vanishes, this c…
In this article, it has been theoretically shown that broad angle negative refraction is possible with asymmetric anisotropic metamaterials constructed by only dielectrics or loss less semiconductors at the telecommunication and relative wavelength range. Though natural uniaxial materials can exhibit negative refraction, the maximum angle of negative refraction and critical incident angle lie in a very narrow range. This notable problem can be overcome by our proposed structure. In our structures, negative refraction originates from the highly asymmetric elliptical iso-frequency.This is artifi…
Imaging with a layered superlens is a spatial filtering operation characterized by the point spread function (PSF). We show that in the same optical system the image of a narrow sub-wavelength Gaussian incident field may be surprisingly dissimilar to the PSF, and the width of PSF is not a straightforward measure of resolution. FWHM or std. dev. of PSF give ambiguous information about the actual resolution, and imaging of objects smaller than the FWHM of PSF is possible. A multiscale analysis of imaging gives good insight into the peculiar scale-dependent properties of sub-wavelength imaging.
Black phosphorus (BP) as a promising two-dimensional material with extraordinary optical properties constitutes an excellent building block in multilayer hyperbolic metamaterials. In this work, we design a multilayer structure composed of BP/dielectric layer stacking unit cells patterned on a gold mirror and theoretically demonstrate the tunable anisotropic absorption in the infrared regime. The electric dipole resonance between the adjacent unit cells drives the structure in the critical coupling state, and impedance of the structure matches to that of free space, showing the perfect absorpti…
We experimentally and theoretically investigate the second order nonlinear optical response of metallo-dielectric multilayer structures composed of Ag and Ta2O5 layers, deposited by magnetron sputtering. Second harmonic generation measurements were performed in reflection mode as a function of incidence angle, using femtosecond pulses originating from a Ti:Sapphire laser system tuned at 800 nm. The dependence of the generated signal was investigated as a function of pump intensity and polarization state. Our experimental results show that the conversion efficiency from a periodic metallo-diele…
In this Letter, we investigate the group delay of optical pulses through multilayer dielectric mirrors (MDM) combined with gravitational wave (GW). We find that the delay increases linearly with MDM length for the transmitted wave packet, whereas the Hartman effect disappears. Our study provides insight into the nature of both quantum tunnelling and GW.
The optical characterization of dielectric periodic line-gratings for which the period of the grating is significantly larger than the probing wavelength is investigated experimentally, numerically, and analytically. The experimental diffraction efficiencies are analyzed in transmission with the help of a closed-form approximate solution within a single scattering picture. In particular, the angular width of significant scattering and the haze are analyzed for samples of increasing grating profile amplitude.
A single-order transmission diffraction grating based on dispersion engineered all-dielectric metasurfaces is proposed and its wavelength discriminating properties have been theoretically described and confirmed using numerical simulations. The metasurface is designed using a 2D array of all-dielectric resonators, which emulates a Huygens source configuration to achieve a perfect match to free-space in broad bandwidth. Using a holey dielectric nanodisk structure as the unit cell, the resonant wavelength is tapered across the metasurface to engineer the wavelength dependent spatial phase gradie…
We analyze the impact of nonlocality on the waveguide modes of metallo-dielectric multilayers and optical patch antennas, the latter formed from metal strips closely spaced above a metallic plane. We model both the nonlocal effects associated with the conduction electrons of the metal, as well as the previously overlooked response of bound electrons. We show that the fundamental mode of a metal-dielectric-metal waveguide, sometimes called the gap-plasmon, is very sensitive to nonlocality when the insulating, dielectric layers are thinner than 5 nm. We suggest that optical patch antennas, which…
Small form-factor, narrowband, and highly directive antennas are of critical importance in a variety of applications spanning wireless communications, remote sensing, Raman spectroscopy, and single photon emission enhancement. Surprisingly, we show that the classical directivity limit can be appreciably surpassed for electrically small multilayer spherical antennas excited by a point electric dipole even if limiting ourselves to purely dielectric materials. Experimentally feasible designs of superdirective antennas are established by using a stochastic optimization algorithm combined with a ri…
We theoretically investigate general existence conditions for broadband bulk large-wavevector (high-k) propagating waves (such as volume plasmon polaritons in hyperbolic metamaterials) in subwavelength periodic multilayer structures. Describing the elementary excitation in the unit cell of the structure by a generalized resonance pole of a reflection coefficient, and using Bloch's theorem, we derive analytical expressions for the band of large-wavevector propagating solutions. We apply our formalism to determine the high-k band existence in two important cases: the well-known metal-dielectric,…
The general features of the light scattering resulting in the so-called resonant Wood's anomalies in the reflection and transmission spectra are described using the effective parameters of a quasi-guided mode. The expression determining the spectral angular dependence of Wood's anomaly in the case of plasmonic grating structures is given and compared to the analogous expression for dielectric grating structures. Comparison of the resonant Wood's anomalies (RWA) with the surface plasmon-polariton resonances (SPPRs) is discussed as well.
We design a plasmonic fiber waveguide (PFW) composed of coaxial cylindrical metal-dielectric multilayers in nanoscale, and constitute the corresponding dynamical equations describing the modes of propagation in the PFW with the Kerr nonlinearity in the dielectric layers. The physics is connected to the discrete matrix nonlinear Schrödinger equations, from which the highly confined ring-like solitons in scale of subwavelength are found both for the visible light and the near-infrared light in the self-defocusing condition. Moreover, the confinement could be further improved when increasing the …
Probing topological edge states in multilayer dielectric photonic nanostructures
Research School of Physics, Australian National University（澳大利亚国立大学物理研究学院）；Institute of Photonics and Quantum Sciences, Heriot-Watt University（赫瑞-瓦特大学光子与量子科学研究所）
We show that dielectric or even perfectly conducting cylinders can be cloaked by a uniform or a layered dielectric cladding, without the need of any exotic or magnetic material parameters. In particular, we start by presenting a simple analytical concept that can accurately describe the cloaking effect obtained with conical silver plates in the visible spectrum. The modeled structure has been originally presented in [S. A. Tretyakov, P. Alitalo, O. Luukkonen, C. R. Simovski, Phys. Rev. Lett., vol. 103, p. 103905, 2009], where its operation as a cloak in the optical frequencies was studied only…
Metasurfaces have attracted extensive interests due to their ability to locally manipulate optical parameters of light and easy integration to complex optical systems. Particularly, metasurfaces can provide a novel platform for splitting and diffracting light into several beams with desired profile, which is in contrast to traditional gratings. Here, we propose and experimentally demonstrate a novel method for generating independently selective diffraction orders. Our method is based on complex amplitude modulation with ultrathin dielectric metasurfaces. By tailoring the geometric parameters o…
Engineering plasmonic metamaterials with anisotropic optical dispersion enables us to tailor the properties of metamaterial-based waveguides. We investigate plasmonic waveguides with dielectric cores and multilayer metal-dielectric claddings with hyperbolic dispersion. Without using any homogenization, we calculate the resonant eigenmodes of the finite-width cladding layers, and find agreement with the resonant features in the dispersion of the cladded waveguides. We show that at the resonant widths, the propagating modes of the waveguides are coupled to the cladding eigenmodes and hence, are …
Existing methods for the localization of light at the nanoscale use either a structure with negative permittivity, by exploiting subwavelength plasmonic resonances, or a dielectric structure with a high refractive index, which reduces the wavelength. In this paper, we provide an alternative to these two methods based on a modification of the modes of dielectric resonators by means of an active medium. We show that an active medium can promote subwavelength light localization in the dielectric structure. We consider a dielectric layer of size substantially smaller than a half-wavelength of ligh…
Diffraction gratings with high upward diffraction efficiency and large effective length are required for chip-scale light detection and ranging. In this paper, we propose a diffraction grating based on a multilayer silicon nitride waveguide, which theoretically achieves an upward diffraction efficiency of 92$\%$, a near-field effective length of 376 $μm$ and a far-field divergence angle of 0.105$^{\circ}$ at a wavelength of 850 nm. The diffraction grating has a high tolerance to process variations based on Monte Carlo Analysis. When the conditions are $\pm$5$\%$ layer thickness variation, $\pm…
Generation of a photonic nanojet by a linearly polarized wave beam or a plane wave impinging a dielectric microcylinder implies partial conversion of the propagating waves into the evanescent ones. This conversion is manifested in nanojet waist of subwavelength effective width. However, this known near-field effect is relatively weak. We theoretically show that the incidence of a wave beam formed by two plane waves enables much stronger near-field effects: a deeply subwavelength focusing of the incident beam and a strong enhancement of the electric field on the whole cylinder surface and near …
We demonstrate theoretically a strong local enhancement of the intensity of the in-plane microwave magnetic field in multilayered structures made from a magneto-insulating yttrium iron garnet (YIG) layer sandwiched between two non-magnetic layers with a high dielectric constant matching that of YIG. The enhancement is predicted for the excitation regime when the microwave magnetic field is induced inside the multilayer by the transducer of a stripline Broadband Ferromagnetic Resonance (BFMR) setup. By means of a rigorous numerical solution of the Landau-Lifshitz-Gilbert equation consistently w…
Normal-incidence transmission and dispersion properties of optical multilayers and one-dimensional stepwise potential barriers in the non-tunneling regime are analytically investigated. The optical paths of every constituent layer in a multilayer structure, as well as the parameters of every step of the stepwise potential barrier, are constrained by a generalized quarter-wave condition. No other restrictions on the structure geometry is imposed, i.e., the layers are arranged arbitrarily. We show that the density of states (DOS) spectra of the multilayer or barrier in question are subject to in…
We study dispersion properties of layered metal-dielectric media having different layers thicknesses ratios. Plotting dispersion diagrams and isofrequency contours, we find that strong nonlocality is inherent property of such periodic structures. We introduce different level of losses and analyze complex modes of the metamaterial demonstrating that it operates in a regime with infinite numbers of eigenmodes, with nonlocality slightly affected by losses.
Symmetry and reciprocity constraints on polarization state of the field diffracted by gratings of quasi-planar particles are considered. It is shown that the optical activity effects observed recently in arrays of quasi-planar plasmonic particles on a dielectric substrate are due to the reflection of the field at the air-dielectric slab interface and are proportional to this reflection coefficient.
The open nonlinear electrodynamic system - nonlinear transverse non-homogeneous dielectric layer, is an example of inorganic system having the properties of self-organization, peculiar to biological systems. The necessary precondition of effects of self-organization is the presence of a flow of energy acting in system from an external source, due to which the system gets ability to independent formation of structures. On an example of the transverse non-homogeneous, isotropic, nonmagnetic, linearly polarized, nonlinear (a Kerr-like dielectric nonlinearity) dielectric layer the constructive app…
Designing a low-loss high reflectivity mirror for gravitational waves detectors by combining a dielectric metasurface and a multilayer stack
Aix Marseille Univ, CNRS, Centrale Med, Institut Fresnel（艾克斯-马赛大学、法国国家科学研究中心、中央理工学院马赛分校、费斯尼尔研究所）；Laboratoire des Matériaux Avancés—IP2I, Université de Lyon, CNRS/IN2P3（先进材料实验室—IP2I、里昂大学、法国国家科学研究中心/法国国家核子和粒子物理研究所）
Optical forces acting on nano-sized particles are typically too small to be useful for particle manipulation. We theoretically and numerically demonstrate a mechanism that can significantly enhance the optical force acting on a small particle through a special type of resonant particle-substrate coupling. The resonance arises from the singular behavior of the particle's effective polarizablity in the presence of a metal-dielectric-metal multilayer substrate. We show that this phenomenon is closely related to the existence of a flat-band plasmon mode supported by the multilayer substrate.
We present a detailed study of light propagation in waveguides with anisotropic metamaterial cores. We demonstrate that in contrast to conventional optical fibers, our structures support free-space-like propagating modes even when the waveguide radius is much smaller than the wavelength. We develop analytical formalism to describe mode structure and propagation in strongly anisotropic systems and study the effects related to waveguide boundaries and material composition.
We have fabricated a layered nano-composite by alternating metal and gain medium layers, the gain dielectric consisting of a polymer incorporating optically pumped dye molecules. Exploiting an improved version of the effective medium theory, we have chosen the layers thicknesses for achieving a very small value of the real part of the permittivity epsilon_\| (parallel to the layers plane) at a prescribed visible wavelength. From standard reflection-transmission experiments on the optically pumped sample we show that, at a visible wavelength, both the real and the imaginary parts of the permitt…
Passive optical elements can play key roles in photonic applications such as plasmonic integrated circuits. Here we experimentally demonstrate passive gap-plasmon focusing and routing in two-dimensions. This is accomplished using a high numerical-aperture metal-dielectric-metal lens incorporated into a planar-waveguide device. Fabrication via metal sputtering, oxide deposition, electron- and focused-ion- beam lithography, and argon ion-milling is reported on in detail. Diffraction-limited focusing is optically characterized by sampling out-coupled light with a microscope. The measured focal di…
We present a type of grism, a series combination of transmission grating and prism, in which we reduce the number of diffraction orders and achieve a configuration with very high angular dispersion. The grism can be fabricated from a single dielectric material and requires no metallic or dielectric film layers for high transmission diffraction efficiency. One can reach 100% in the -1st transmission diffraction order and the equal damage threshold as the dielectric bulk material. We realized such an element in fused silica with an efficiency of more then 99%. The bevel backside reflection is re…
Multilayer hyperbolic metamaterials consisting of alternating metal and dielectric layers have important applications in spontaneous emission enhancement. In contrast to the conventional choice of at least dozens of layers in multilayer structures to achieve tunable Purcell effect on quantum emitters, our numerical calculations reveal that multilayers with fewer layers and thinner layers would outperform in Purcell effect. These discoveries are attributed to the negative contributions by an increasing layer number to the imaginary part of the reflection coefficient, and the stronger coupling b…
In this letter, we demonstrate a novel diffraction-free Bloch surface wave (DF-BSW) sustained on all-dielectric multilayers that does not diffract after being passed through three obstacles or across a single mode fiber. It can propagate in a straight line for distances longer than 110 μm at a wavelength of 633 nm and could be applied as an in-plane optical virtual probe, both in air and in an aqueous environment. The ability to be used in water, its long diffraction-free distance, and its tolerance to multiple obstacles make this DF-BSW ideal for certain applications in areas such as the biol…
We study light propagation in nanoscale periodic structures composed of dielectric and metal in the visible range. We demonstrate that diffraction can be tailored both in magnitude and in sign by varying the geometric features of the waveguides. Diffraction management on a subwavelength scale is demonstrated by numerical solution of Maxwell equations in frequency domain.
For dielectric multilayered metamaterials, the effective-parameter representation is known to be insensitive to geometrical features occurring at deeply subwavelength scales. However, recent studies on periodic and aperiodically ordered geometries have shown the existence of certain critical parameter regimes where this conventional wisdom is upended, as the optical response of finite-size samples may depart considerably from the predictions of standard effective-medium theory. In these regimes, characterized by a mixed evanescent/propagating light transport, different classes of spatial (dis)…
Using the transfer-matrix approach and solving time-domain differential equations, we analyze the loss compensation mechanism in multilayer systems composed of an absorbing transparent conductive oxide and dielectric doped with an active material. We reveal also another regime with the possibility of enhanced transmission with suppressed reflection originating from the resonant properties of the multilayers. For obliquely incident and evanescent waves, such enhanced transmission under suppressed reflection turns into the reflectionless regime, which is similar to that observed in the PT-symmet…
Conventional projection Talbot lithography usually employs opaque (amplitude) or transparent (phase) masks for creating a periodic array of Fresnel diffraction fringes in the photosensitive substrate. For particular mask design the longitudinal periodicity of Talbot carpet can be avoided producing quasi uniform striped pattern (Talbot stripes). We propose a novel hybrid amplitude-phase mask which is engineered for obtaining extremely smooth Talbot stripes and simultaneously high lateral optical contrast and extreme spatial resolution better than a third of laser wavelength. By means of the num…
We report a general description of quasi-phase-matched parametric process in nonlinear photonic crystals (NLPC) by extending the conventional X-ray diffraction theory in solids. Under the virtual wave approximation, phase-matching resonance is equivalent to the diffraction of the scattered virtual wave. Hence a modified NLPC Ewald construction can be built up, which illustrates the nature of the accident for the diffraction of the virtual wave in NLPC and further reveals the complete set of diffractions of the virtual wave for both of the air-dielectric and dielectric-dielectric contacts. We s…
We demonstrate bound states in the first TE and TM diffraction continua (BSC) in a linear periodic array of dielectric spheres in air above the light cone. We classify the BSCs according to the symmetry specified by the azimuthal number $m$, the Bloch wave vector $β$ directed along the array, and polarization. The most simple symmetry protected TE and TM polarized BSCs have $m=0$ and $β=0$ and occur in a wide range of the radius of the spheres and dielectric constant. More complicated BSCs with $m\neq 0$ and $β=0$ exist only for a selected radius of spheres at a fixed dielectric constant. We a…
