Letter pubs.acs.org/NanoLett
Spinning Light on the Nanoscale Jingbo Sun, Xi Wang, Tianboyu Xu, Zhaxylyk A. Kudyshev, Alexander N. Cartwright, and Natalia M. Litchinitser* Electrical Engineering Department, University at Buffalo, The State University of New York, Buffalo, New York 14260, United States ABSTRACT: Light beams with orbital angular momentum have significant potential to transform many areas of modern photonics from imaging to classical and quantum communication systems. We design and experimentally demonstrate an ultracompact array of nanowaveguides with a circular graded distribution of channel diameters that coverts a conventional laser beam into a vortex with an orbital angular momentum. The proposed nanoscale beam converter is likely to enable a new generation of on-chip or all-fiber structured light applications. KEYWORDS: Optical vortex, cylindrical nanowaveguide, phase control, orbital angular momentum
T
he synergy between structured light and nanostructured materials opens entirely new opportunities in fundamental and applied science. Potential applications include imaging, increasing capacity of communication systems through mode division multiplexing, micromanipulation, probing of atomic forbidden states and building higher dimensional quantum encryption systems.1 Many of these new applications call for ultracompact sources of structured light, or beams carrying orbital angular momentum (OAM) that can be integrated on a chip or directly on an optical fiber. However, until now a majority of approaches to generating OAM beams have been based on macroscopic bulky optical components such as spiral phase plates, cylindrical lens converters, q-plates, spatial light modulators (SLM), or specialty fibers.2−5 Recent progress in nanostructured optical materials has enabled new ways of manipulating intensity, polarization and phase distribution of light beams.6−26 Nanostructures and metamaterials facilitate a new class of planar optical elements where their optical properties originate from spatial distribution of their refractive index rather than their shape.7−26 Moreover, arrays of nanoslits or nanoholes milled in a metallic film have been demonstrated to produce converging or diverging lenses by realizing different propagation constants in the slits or holes of different sizes.27−36 These structures open new paths toward the realization of ultracompact optical components for beam manipulation on the nanoscale. In this work, we design and fabricate an array of nanowaveguides with a circular graded size distribution, which changes the propagation constant in a prescribed way as schematically shown in Figure 1. Each nanowaveguide introduces a specific phase change determined by its radius, and therefore, by carefully choosing the spatial distribution of the nanowaveguide radii, a total phase change of 2π can be imposed on the wavefront of the beam upon its propagation through © 2014 American Chemical Society
Figure 1. Schematic of a nanowaveguide array that induces wavefront shaping.
such an array. As a result, a conventional (Gaussian) laser beam transmitted through such a structure acquires an OAM and is transformed into a vortex beam. Such structures are compact, versatile and can be readily integrated with optical fibers or on a chip. First, in order to design the proposed structure we discuss the properties of a single nanowaveguide composed of a cylindrical hole milled in a thin silver film and filled with dielectric material. Such a nanowaveguide can be considered as a cylindrical waveguide with a dielectric core and metal cladding whose relative dielectric permittivity can be described by the Drude model εAg = ε∞ − ω2p/[ω(ω + iγ)], where ε∞ is the relative dielectric permittivity at an infinite frequency, ωp is the plasma frequency, and γ is the collision frequency. The theory of wave propagation in cylindrical waveguides is well understood,37 so here we only summarize the main points. The waveguide modes in cylindrical coordinates for electric and magnet ic fields can be written as ψ(r,ϕ,z,t) = ψn(r)exp(imϕ)exp[i(βz − ωt)], where ω and β are the angular Received: February 19, 2014 Revised: March 27, 2014 Published: April 3, 2014 2726
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eq 1 suggest that the working wavelength range for the structure described in this paper extends from 487 to 537 nm. Next, we performed numerical simulations of the nanowaveguide array with the assumptions that all of the channels were identical and separated by 550 nm, and that the nanowaveguide radius ranged from 76 to 132 nm. The inset in Figure 2a shows the unit cell used in the numerical simulations. Periodic boundary conditions were applied to the side walls of the unit cell. To understand the effect of the proposed structure on different polarization states of the incident light, both linearly and circularly polarized plane waves at 532 nm wavelength were considered as incident beams. The simulation results corresponding to the case of linearly polarized beam are also shown in Figure 2a by black soliddotted line and for the case of circularly polarized beam by red dot-dashed lines. Good agreement between numerical and analytical results with respect to the phase change as a function of nanowaveguide radius indicates that (i) the nanowaveguides in the array are weakly coupled and can be considered as independent waveguides; and (ii) the initial state of polarization does not affect the phase change imposed by propagation in an array of nanowaveguides. Finally, Figure 2b shows the electric field distributions inside the nanowaveguide for both linearly and circularly polarized incident beam cases, confirming that the performance of the array is independent of the incident polarization state. The theoretical results discussed above were used to design the structure shown in Figure 1. In particular, we conclude from Figure 2a that designing the structure such that the nanowaveguides radii change from 76 to 132 nm in a circular array, we expect that an incident beam passed through this structure will acquire an OAM (topological charge of 1) and will be transformed into an optical vortex. The experimental samples were prepared using the focused ion beam (FIB) method employing a Zeiss AURIGA CrossBeam Workstation (FIB-SEM). The nanowaveguides were milled in an 800 nm thick silver film deposited on a glass slide with radii determined from the interpolation between the simulation results and the theoretical calculation in Figure 2, which are summarized in Table 1. The final
frequency in free space and the propagation constant inside the waveguide that is determined from the boundary conditions, respectively. From the requirement of continuity of the tangential components of the electric field E and the magnetic field H at the cylindrical surface r = a, we can obtain the dispersion relation by solving the equation38,39 2 ⎡ ⎡ J ′ (au) kAg Hm(1)′(av) ⎤ H (1)′(av) ⎤⎢ kd2J ′m (au) ⎥ ⎢ m ⎥ − m(1) − (1) ⎢ uJ au ( ) ⎢⎣ uJm (au) ⎥ vHm (av) ⎦⎣ m vHm (av) ⎥⎦
=
2 m2β 2 ⎛ 1 1⎞ ⎜ ⎟ − a2 ⎝ v 2 u2 ⎠
(1)
where Jm(au) and Hm (av) are the mth-order first kind Bessel function and Hankel function, respectively, kd = (ω/c)(εd)1/2 is the wavenumber in the dielectric, and kAg = (ω/c)(εAg)1/2 is the wavenumber in the metal. The transverse propagation constants in the dielectric and in the metal are given by u = (k2d − β2)1/2, v = (k2Ag − β2)1/2, respectively. The transcendental eq 1 can be solved to find the phase change βl, where l is the waveguide length, as a function of the radius of the nanowaveguide a. In the experimental part of this study, we used a silver film with thickness l = 800 nm as the cladding and the fabricated nanowaveguides were filled with poly(methyl methacrylate) (PMMA) with relative dielectric permittivity εd = 2.07, which is used to increase the cutoff wavelength of the nanowaveguide. Using these parameters, we obtained the solution of eq 1 shown by the solid (blue) curve in Figure 2a. (1)
Figure 2. (a) Phase change as a function of the nanowaveguide radius. Solid blue line corresponds to theoretical predictions using eq 1; the dashed red line corresponds to results from numerical simulations for the case of a linearly polarized input beam; the black solid-dotted line corresponds to the case of circular polarization. (b) Vector plots for the electric fields inside the nanowaveguide for linearly and circularly polarized incident beams. The inset shows the unit cell used in numerical simulations.
Table 1. Design Parameters phase change hole radius (nm)
−160°
−105°
−50°
−10°
45°
90°
128°
165°
76
80
85
90
100
106
115
132
structure consisted of multiple circular arrays of nanowaveguides resulting in a total sample diameter of 32 μm as shown in Figure 3. Finally, the PMMA was coated on top of the sample which was then placed in vacuum so that it (PMMA) filled the nanowaveguides. The sample was studied using an optical interference system with a 532 nm laser diode, as shown in Figure 4a. Linearly polarized light from the laser was split into a reference beam and a probe beam using a prism. The probe beam was first passed through a spatial filter in order to improve its quality and then focused on the sample. Next, the probe beam from the sample was collimated by a ball lens and then recombined with the reference beam by using a thin-film beam splitter (FBS), forming an interference pattern that was recorded using a CCD camera. A quarter-wave plate was placed in front of the
The solution indicates that the phase change in the nanowaveguide increases along with its radius. For the purpose of this study, it is important to note that a nanowaveguide radius change of approximately 56 nm results in a 2π phase change. Therefore, arranging the nanowaveguides of varying radius in rings (as schematically shown in Figure 1) such that the radius difference between the smallest and the largest one is around 56 nm would result in twisting the phase of the beam and converting it into an OAM beam with topological charge 1. It is noteworthy that as long as the nanowaveguide array with a given hole size distribution imposes a 2π shift on a transmitted beam, the beam would acquire an OAM with charge 1 independent of its wavelength. Our theoretical studies based on 2727
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gradually changing nanowaveguide radii. Using the analytical solution for the metal-dielectric cylindrical waveguide, we showed that each nanowaveguide introduces a specific phase change determined by its radius. Therefore, we designed the spatial distribution of the nanowaveguide radii such that a total phase change of 2π was imposed on the wavefront of the beam upon its propagation through such an array. As a result, a laser beam passing through the designed array acquired an OAM and was transformed into a vortex beam with a topological charge of one, which was confirmed using optical interference experiments. Finally, we demonstrated that the proposed OAM mode converter is polarization independent in that both linearly and circularly polarized input beams could be converted into OAM beams. The proposed design is ultracompact and can be readily integrated on a chip or on a transverse cross-section of optical fiber.
Figure 3. SEM picture of the nanowaveguide array prepared by FIB.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: natashal@buffalo.edu. Author Contributions
J.S. and N.M.L. proposed the idea developed in this work. Z.K. and J.S. did the theoretical part of the work. J.S. and T.X. did the simulation. X.W. did the fabrication of the sample. J.S. did the optical characterization of the sample. A.N.C. evaluated the experimental data. N.M.L and J.S. wrote the paper. N.M.L. supervised this work.
Figure 4. (a) Experimental set up; (b−d) intensity, spiral, and fork interference patterns for the case of a linearly polarized input beam; (h−j) the same as (b−d) but for a circularly polarized beam; (e−g) and (k−m) show the numerical results for linear and circular polarization, respectively.
Funding
U.S. Army Research Office Award # W911NF-11-1-0333. Notes
The authors declare no competing financial interest.
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laser to repeat the experiment with a circular polarized incident beam. The experimental results for both linear and circular polarizations are shown in Figure 4. Figure 4b shows the output intensity for a linearly polarized output beam, which exhibits a doughnut shape with a phase singularity in the center. The nonuniform intensity distribution around the singularity results from the size differences between holes in the array, which does not affect the OAM loading. The spiral pattern in Figure 4c, resulting from the interference between the probe beam and the reference beam (coaxial Gaussian beam), reveals the presence of the singularity and the helical wavefront of the optical vortex with topological charge one. By rotating the thinfilm beam splitter to introduce a small angle between the probe beam and the reference Gaussian beam, a fork shaped interference pattern is observed, once again confirming that the output (probe) beam is an optical vortex with topological charge one. Figure 4e−g shows the results of numerical simulations for the array identical to that used in experiments. Similar results were obtained in experiments using circularly polarized beam, confirming that the proposed structure can work for any polarization, as shown in Figure 4h−j and corresponding numerical simulations shown in Figure 4k−m. Finally, as any metal-based optical device, the losses in the metal limit the overall transmittance. However, it should be mentioned that the transmittance of the nanowaveguide array can be greatly increased by properly designing the unit cell’s size or optimizing the spacing between the nanowaveguides.40 In summary, we proposed and experimentally demonstrated that a nanowaveguide array milled in a metal film can be used to control the wavefront of a light beam and that an optical vortex at 532 nm is produced by using such an array with
ACKNOWLEDGMENTS The authors would like to thank Scott Will for valuable suggestions on the manuscript. This research was supported by the U.S. Army Research Office Award # W911NF-11-1-0333.
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