Role of the normal polarization in the far-field subwavelength imaging by a dielectric microsphere or microcylinder R. Heydarian1, C. R. Simovski1;2 1 Department of Electronics and Nano-Engineering, Aalto University, P.O. Box 15500, FI-00076 Aalto, Finland 2 Faculty of Physics and Engineering, ITMO University, 199034, Birzhevaya line 16, Saint-Petersbug, Russia E-mail: [email protected] January 2020 Abstract. Role of the normal polarization in the far-field subwavelength imaging granted by a dielectric microsphere or microcylinder is discussed and the hypotheses explaining this experimental fact are suggested. One of these hypotheses is confirmed by exact numerical simulations. This mechanism of the magnifying superlens operation is based on the excitation of creeping waves at a curved dielectric interface by a normally polarized dipole. The set of creeping waves after their ejection from the surface creates an imaging beam which may mimic either a Bessel beam or a Mathieu beam depending on the microparticle radius. This mechanism corresponds to the asymmetric coherent illumination. arXiv:2001.08135v4 [physics.optics] 7 Apr 2020 2 1. Introduction several scienrific groups have explored this field since 2011 (see e.g. in [16, 15, 9, 18, 19]). In Nanoimaging of objects in real time { beyond [15] a direct lateral resolution on the level δ = scanning the substantial areas with strongly λ/6 was complemented by the interferometric submicron tips connected to cantilevers { is resolution δ = λ/10 in the normal direction. a very important branch of nanophotonics. Also, in this work it was shown via simulations A lot of top-level studies has been done in that two dipoles located at the surface of a this field recently and pioneering techniques glass microsphere of the dimensionless radius were developed, such as stimulated emission kR = 60 with the gap δ = λ/6 between them depletion [1], awarded by the Nobel prize can be resolved (in simulations) if and only if in chemistry (2014). However, in spite they are excited with nearly opposite phases. of advantages of this method, there are To have opposite phases is impossible for two applications, especially in the biomedicine, small closely located scatterers illuminated where fluorescent labels in the object area are by a plane wave. Moreover, the dielectric prohibited (see e.g. in [2]). Label-free optical scatterers resolved in [15] with the gap δ = nanoimaging still evokes a keen interest, and λ/6 were illuminated by an incoherent light. so-called superlenses (see e.g. in [4]) are still Further, the resolution δ = λ/15 was achieved a subject of an intensive research. In the using a glass MP for two plasmonic scatterers present work, we concentrate on a technique [16]. The theory of these papers could not which seems to be the most affordable and explain these experimental results. straightforward type of superlens - dielectric However, in order to properly exploit spherical or cylindrical microlens. a novel technique, one obviously needs to In work [3] it was experimentally revealed understand its physics. Initially, the authors that a simple glass microsphere operates as a of [3] assumed that this imaging is related far-field magnifying superlens { a device which with the phenomenon of so-called photonic creates a far-field magnified image of an object nanojet (PNJ) [5]. The PNJ maintains a with its subwavelength details detectable by slightly subwavelength ((0:3 − 0:5)λ) effective a conventional microscope. The imaged area width along a path that extends more than 2λ is rather small (several square microns) and behind the MP . In works [6, 7, 8] it was shown centered by the optical axis of the microscope that the PNJ is a non-resonant phenomenon passing through the microsphere center. Even and results from the constructive interference few square microns is an area much larger of cylindrical or spherical harmonics excited than the object field of a scanning near-field inside the MP by a plane wave. For the optical microscope (SNOM). Therefore, this (relative to the ambient) refractive index of technique promises a much faster imaging of the MP n = 1:4 − 2 a whatever MP radius the whole substrate than the use of SNOM. A from R = λ to R = 20λ corresponds to a direct analogue of the spherical or cylindrical sufficient amount of spatial harmonics which dielectric microparticle (MP) operating in the experience the constructive interference at the superlens regime is a metamaterial hyperlens rear extremity of the MP. The studies also [11, 12, 13, 14]. However, dielectric MPs are have shown that in the near vicinity of the available on the market and are incomparably rear edge point the package of evanescent cheaper than the hyperlenses. Therefore, waves is excited that grants to the waist 3 of the wave beam a high local intensity. PNJ model of a 2D MP superlens does not Authors of [3] assumed: since a plane wave work (it was clearly proved in [17]) why it will excites these evanescent waves in a MP, the work for the 3D superlens? evanescent waves excited by a closely located In works [18, 19] resonant mechanisms subwavelength scatterer should reciprocally of subwavelength imaging by a dielectric convert into propagating waves and form a MP were analyzed. One was related to PNJ. However, further studies (see e.g. in whispering gallery resonances, another { to [16, 15, 9, 10, 18, 19, 17] etc.) have shown that the Mie resonances. In both cases, the a scatterer located near a MP does not produce wave packages responsible for subwavelength a PNJ behind it. Moreover, in work [17] it was hot spots inside the particle experience the noticed that the explanation of the superlens leakage and partial (quite weak) conversion functionality of a dielectric MP via the into propagating waves. Two other resonant evanescent waves [9, 10] is disputable because mechanisms were recently reported in works in presence of evanescent waves the reciprocity [20] and [21]. However, all these resonant principle is not reducible to the inversion of the mechanisms do not explain why a dielectric wave propagation. Really, in both focusing and microsphere operates as a far-field superlens emitting schemes, the evanescent waves decay in a broad frequency range. In work [22] in the same directions { from the rear point a broadband subwavelength resolution in the of the sphere. Therefore, the evanescent waves incoherent light was theoretically obtained for responsible for the subwavelength width of the a glass MP. However it was as modest as PNJ waist in the focusing scheme cannot be δ = λ/4 and demanded the use of an exotic linked to those excited by the imaged object in microscope with a solid immersion lens as an the emitting scheme. To confirm this point in objective. This microscope has the f-number [17] the exact simulations of the point-spread smaller than unity. It obviously implies the function for the structures from [3, 15, 9, 10] reduction of the diffraction-limited image size were done and the subwavelength imaging of a point source compared to the finest size was absent, though the PNJ in the reciprocal granted by a usual microscope δ ≈ 0:5λ case manifested the subwavelength waist. The [26]. This size is the radius of the Airy only difference in these simulations from the circle in the image plane and it is equal to experiments and theoretical speculations in the finest possible resolution of a microscope the cited works was replacement of the 3D [26]. For microscopes with small f-numbers MP (sphere) by the 2D one (cylinder) that the Airy circle radius (and finest resolution) also implies the 2D dipole source (dipole line δ = λ/4 does not require additional imaging parallel to the cylinder axis). One may devices [27]. However, in all experiments believe that the 3D geometry grants specific with microspheres offering the subwavelength mechanisms of superlens operation compared resolution the standard microscopes were used. to the 2D one. However, this belief does not Moreover, the resolution was noticeably finer disable the argument against the explanation than δ = λ/4 predicted in [22] as a limit value. of the MP superlens operation involving the In the present paper, we suggest a PNJ and reciprocity. Moreover, the PNJ is hypothesis that the superlens operation of a formed by a dielectric MP in both 3D and 2D dielectric MP is related to its capacity to geometries with similar efficiency [8]. If the create a diffraction-free wave beam. This 4 property should be common for both 2D and dipoles transmits through the MP as it is 3D geometries. The imaging beam results described in work [17] and represents a weakly from the emission of a dipole source (a small directive wave beam experiencing the Abbe scatterer) which is polarized normally to the diffraction (the angular beam width grows surface of a MP. There are at least two with the distance). mechanisms which result in the formation of For a normally polarized dipole, the the diffraction-free beam by a dielectric MP. situation depicted in Fig. 1(b) is different. The first one is an incoherent mechanism The corresponding image dipole is in phase and corresponds to the creation of a radially with the real one and the two dipole sources polarized Gaussian beam. The second one { real and imaginary can be united into a demands the illumination by a laser light and dipole effectively located at the interface. In the nonzero phase shift between two dipoles this case the imaging beam turns out to be in order to resolve them. This mechanism of almost diffraction-free. Fig. 1(b) illustrates a subwavelength imaging results in either Bessel simplistic model of the imaging beam { that or Mathieu imaging beam.