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Radial Polarization of Periodically Focused Modes in Chains of Dielectric Spheres Arash Darafsheh, Vasily N. Astratov* Department of Physics and Optical Science, Center for Optoelectronics and Optical Communications, University of North Carolina at Charlotte, 9201 University City Blvd., Charlotte, NC 28223-0001, USA * Tel: 1 (704) 687 8131, Fax: 1 (704) 687 8197, e-mail:
[email protected], http://maxwell.uncc.edu/astratov/astratov.htm ABSTRACT Recently we showed that light focusing and transport properties of chains of dielectric spheres with D>>10λ, where D is the diameter of the sphere and λ is the wavelength, are dominated by periodically focused modes (PFMs) which have extremely small propagation losses [12]. In this work we show that along with a special case of PFMs in chains of spheres with index n = 3 which propagates in such structures without losses, similar periodic modes exist in a broad range of indices from 1.4 to 2.0. For each n such generalized PFMs have various radial extents in the regions between the neighbouring focused beams. We show that for 10-sphere long chains with 1.68 < n < 1.80 such modes have total propagation losses smaller than 1 dB. Using numerical ray tracing, we demonstrate that such chains filter radially polarized beams. Using collimated incident beams, a polarization degree in excess of 0.9 is demonstrated for 10-sphere long chains for 1.68 < n < 1.80 range. These properties make chains of microspheres instrumental for developing novel focusing and polarization components. Keywords: microsphere, focusing, Laguerre-Gaussian beams, radial polarization, degree of polarization, geometrical optics design. 1. INTRODUCTION Chains of microspheres emerged as a paradigm in microphotonics due to unusual combination of periodic light focusing with progressively smaller beam sizes and small optical propagation losses [1-12]. These properties have been observed in mesoscale (wavelength-scale) spheres [5-9] and in sufficiently large spheres with D>>10λ, where D is the diameter of the sphere and λ is the wavelength, where the approximations of geometrical optics can be applied [11,12]. Recently we showed that in the limit of geometrical optics these properties are explained by filtering periodically focused modes (PFMs) which have a period equal to the size of two spheres, as schematically illustrated in Fig. 1. These modes have a radial polarization and propagate through the chain without attenuation under Brewster angle conditions [12] if the following criteria are met: n = 3 , θi = π/3, and r/D = 3 /4, where n is the refractive index of the sphere, θi is the angle of incidence, D is the diameter of the sphere, and r is the axial offset of a given ray.
nb=1 n
θi r
θr
D
0
Figure 1. Ray tracing in touching spheres under conditions of periodic focusing [12]. In the limit of physical optics, the cylindrical symmetry of these modes suggests that they belong to Laguerre-Gassian beams [13,14]. However, unusual and interesting PFM properties are connected with their sharp focusing with 2D period and with their radial state of polarization (SOP). It is well known that, generally, the radially polarized beams can be focused to smaller dimensions compared to diffraction limited spot sizes [14-16]. This means that the radial SOP of PFMs should favor the sharper focusing of light in such structures. It should be noted, however, that the functioning of these structures as polarization filters, to the best of our knowledge, has not been studied previously. This work is aimed at understanding the SOP evolution as the beam propagates through the chain within the framework of geometrical optics. First, we show that along with a special case of lossless PFMs at n = 3 introduced in our previous work [12] similar 2D periodic modes exist in a broad range of indices from 1.4 to 2.0. For each n such generalized PFMs have various radial extents (r) in the regions between the neighboring focused beams in order to provide the 2D periodicity. Using numerical ray tracing, we show that for 10-sphere long
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chains with 1.68 < n < 1.80 such modes have propagation losses smaller than 1 dB. Second, we study SOP of the output beams for collimated incident beams. We show that due to coupling to low loss PFMs such structures can be used as filters of radial polarization. High polarization degrees are demonstrated using 10-sphere chains with 1.71 < n < 1.78. 2. THEORETICAL MODELING 2.1 Periodically focused modes for a broad range of indices We consider a collimated incident beam which can be obtained directly from a laser source or from single mode fibers. In geometrical optics such incident beams can be modelled as a collection of meridional rays parallel to the axis of the chain with a broad range of random axial offsets (r), as illustrated in Fig. 1. Modes with 2D periodicity tend to have smaller propagation losses in such structures due to the fact that their external (θi) and internal (θr) angles of incidence are periodically reproduced. So, if these angles are close to the Brewster angle the losses can be minimized for TM polarized incident rays. The exact conditions for Brewster angle incidence can be satisfied at n = 3 , θi = π/3, and r/D = 3 /4 [12]. It is interesting, however, to consider different spheres’ indices and to study if some periodic modes are possible under these conditions. The rays are periodically reproduced (with a period of 2D) when the refracted ray passes through the point where the spheres touch leading to the following equation:
θ i = 2 cos −1 ( n / 2nb ) ,
(1)
or, equivalently, for axial offset of the ray, r:
r / D = 0.5 sin(2 cos−1 ( n / 2nb )) ,
(2)
where nb is the refractive index of the background medium. We consider nb = 1 in this work. The results of calculations of the index dependences of the angle of incidence (θi) and of the axial offset (r/D) for the modes with 2D periodicity are represented in Figs. 2 (a) and 2 (b), respectively. It is seen that PFMs exist in a very broad range ( 2 < n > 10λ. Using a collection of collimated rays with random polarization as a model for the collimated incident beams, we demonstrate high radial polarization degrees (> 0.9) of the components formed by the 10-sphere long chains. These properties make chains of microspheres instrumental for developing novel focusing elements with the polarization filtering capability. 4. ACKNOWLEDGMENTS The authors gratefully acknowledge support for our work from the National Eye Institute under Award Number R41EY019598. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Eye Institute or the National Institutes of Health. This work was also supported by the U.S. Army Research Office (ARO) through Dr. J. T. Prater under Contract No. W911NF-09-1-0450 and from the National Science Foundation (NSF) under grant ECCS-0824067. REFERENCES [1] V.N. Astratov: Fundamentals and applications of microsphere resonator circuits, in Photonic Microresonator Research and Applications, I. Chremmos, O. Schwelb, and N. Uzunoglu, Eds., New York: Springer Series in Optical Sciences, vol. 156, 2010, pp. 423-457, ISBN: 978-1-4419-1743-0. [2] V.N. Astratov, J.P. Franchak, S.P. Ashili: Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder, Appl. Phys. Lett., vol. 85, pp. 5508-5510, Dec. 2004. [3] Z. Chen, A. Taflove, V. Backman: Highly efficient optical coupling and transport phenomena in chains of dielectric microspheres, Opt. Lett., vol. 31, pp. 389-391, Feb. 2006. [4] S.P. Ashili, V.N. Astratov: The effects of inter-cavity separation on optical coupling in dielectric biospheres, Opt. Express, vol. 14, pp. 9460-9466, Sept. 2006. [5] A.M. Kapitonov, V.N. Astratov: Observation of nanojet-induced modes with small propagation losses in chains of coupled spherical cavities, Opt. Lett. vol. 32, pp. 409-411, Feb. 2007. [6] S. Yang, V.N. Astratov: Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 dB per sphere, Appl. Phys. Lett., vol. 92, 261111, July 2008. [7] T. Mitsui et al.: Observation of light propagation across a 900 corner in chains of microspheres on a patterned substrate, Opt. Lett., vol. 33, pp. 1189-1191, June 2008. [8] T. Mitsui et al.: Micro-demultiplexer of coupled resonator optical waveguide fabricated by microspheres, Adv. Mater., vol. 22, pp. 3022-3026, July 2010. [9] O. Lecarme et al.: Colloidal optical waveguides with integrated local light sources built by capillary force assembly, J. Vac. Sci. Technol. B, vol. 28, pp. C6011-C6015, Nov. 2010. [10] V. N. Astratov: Focusing multimodal optical microprobe devices and methods, International patent publication No. WO/2011/005397 (priority date 17 June 2009). [11] A. Darafsheh et al.: Contact focusing multimodal microprobes for ultraprecise laser tissue surgery, Opt. Express, vol. 19, pp. 3440-3448, Feb. 2011. [12] A. Darafsheh, V.N. Astratov: Periodically focused modes in chains of dielectric spheres, Appl. Phys. Lett, vol. 100, 061123, Feb. 2012. [13] B.E.A. Saleh and M.C. Teich: Fundamentals of Photonics, New Jersey: John Wiley & Sons, 2007. [14] Q. Zhan: Cylindrical vector beams: from mathematical concepts to applications, Advances in Optics and Photonics, vol. 1, pp. 1-57, Jan. 2009. [15] S. Quabis et al.: Focusing light to a tighter spot, Opt. Commun., vol. 179, pp. 1-7, May 2000. [16] R. Dorn, S. Quabis, and G. Leuchs: Sharper focus for a radially polarized light beam, Phys. Rev. Lett., vol. 91, 233901, Dec. 2003. [17] E. Hecht: Optics, 4th Ed., Reading, MA: Addison-Wesley, 2002. [18] A. Al-Qasimi et al.: Definitions of the degree of polarization of a light beam, Opt. Lett., vol. 32, pp. 1015-1017, May 2007.
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