1. Introduction

Vortices can be observed on microscopic as well as macroscopic scales. Vortex-like structures within Bose-Einstein condensates [1] and tornados are two common examples of vortex formation on micro- and macro-scales, respectively. Vortices could also be found in nano-optics, for example, in electromagnetically-excited metallic nanoparticles [2–4], within sub-wavelength slits [5], in nonlinear Kerr media [6], and in a super lens [7]. In the 1970s, Nye and Berry theoretically pointed out that optical vertices can be created within phase singularities; that is, points at which the Poynting vector vanishes, while there exists a circulating component of the Poynting vector in the surrounding region [8]. This fascinating phenomenon has attracted widespread attention [9–11], and has been developed into a new field of optics, called singular optics [12,13].

In this paper, we analyze the optical phenomena associated with sub-wavelength Taiji patterns. The Taiji pattern originated thousands of years ago in Eastern culture. It is a very old schematic representation of two opposing but complementary patterns in oriental civilization [14]. Figure 1(c) shows the diagram of a Taiji mark for the design of our experimental sample. To the best of our knowledge, the optical properties of this interesting pattern have never before been studied. At first, we investigate the transmittance spectra of the Taiji pattern, where we show that the simulated and experimentally observed spectra are in good agreement. We have found that, due to the special structure of the Taiji mark, the vortex-like pattern of the time-averaged Poynting vector exhibits one or more dips at plasmonic resonance(s). Furthermore, the plasmonic resonance could be adjusted by modifying the tail structure and the periodicity of the Taiji pattern.

 

Fig. 1 (a) Schematic diagram showing the process of e-beam lithography. Our samples were fabricated without any adhesion layer (such as Ti or Cr) between the gold film and the glass substrate. A 3 nm film of gold was sputter-deposited on the developed resist layer, followed by the thermal evaporation of another 27 nm of gold. (b) SEM image of a small region from the fabricated sample, showing diameter of tails. (c) The ideal Taiji mark designed for the studies reported in this paper. (d) Magnified view of a single cell in the SEM image of frame (b), showing the fitting curves of a fabricated Taiji mark (R ₁ = 275 nm, R ₂ = 250 nm, D ₁ = 100 nm, diameter of the tangent circle D ₂ = 60 nm).

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2. Fabrication

A 70 × 70 array of gold Taiji marks with 30 nm thickness covering a total area of around 50 × 50 µm² was fabricated on a fused silica substrate using electron beam lithography. The diameter of each Taiji mark is 500 nm, and the periodicity is 700 nm. Figure 1(a) shows the schematic diagram of the Espacer (Kokusai Eisei Co., Showa Denso Group, Japan)/poly methyl methacrylate (PMMA-495K) bi-layer resist fabrication process. Espacer is a material which can eliminate the static charge problem during e-beam exposure. A 200 nm-thick PMMA layer was spin-coated on the fused silica wafer, then backed on a hot plate for 3 minutes at 180°C. Espacer was spin-coated at 1500 rpm over the PMMA layer. The sample was written using an e-beam lithography system (Elionix ELS-7000) at the acceleration voltage of 100 kev with 30 pico-ampere of current. After exposure, the sample was rinsed with de-ionized water to remove Espacer, then developed in solution of methyl isobutyl ketone (MIBK) and isopropyl alcohol (IPA) of MIBK:IPA = 1:3 for 60 seconds, rinsed again, this time with IPA, for 20 seconds, and blow-dried with nitrogen gas. Once the development of the resist was complete, a 3 nm-thick gold (Au) film was sputter-deposited on the patterned substrate. This was followed by thermal evaporation of another 27 nm of Au to bring the total thickness of the gold film to 30 nm. (It is worth mentioning here that we could make our samples without sputtering any adhesion layer, such as Ti or Cr, between the gold film and the glass substrate. This fabrication method is especially suited to our purposes, as the Ti or Cr layers usually change the resonance features of the Taiji mark, yet they are too thin to be accurately modeled with our simulation software.) The sample was eventually soaked in acetone for over 12 hours, then the un-patterned regions were lifted off in an ultrasonic cleaner.

Figure 1(b) shows an SEM micrograph of the fabricated pattern; a schematic drawing of the ideal Taiji pattern is shown in Fig. 1(c). Comparing Fig. 1(b) with Fig. 1(c), it is evident that the tail feature of the fabricated pattern deviates slightly from that of the ideal Taiji pattern. According to our calculations, the best fitting curve for the tail feature of the Taiji pattern is a tangent circle between an ellipse (long axis D ₁ = 275 nm, short axis D ₂ = 250 nm) and a circle (diameter = 250 nm). Figure 1(d) shows the fitting curve as well as the relevant dimensions for a single period of the Taiji pattern depicted in Fig. 1(b). We measured seventeen different Taiji patterns and determined the average radius of the tangent circle to be around 30 nm (standard deviation ~2 nm). The numerical simulation model was subsequently set up according to these SEM observations.

3. Measurement and simulation result

Figure 2 shows the experimental as well as simulated transmittance spectra of the sample illuminated at normal-incidence with an x–polarized light source (frame a) and a y-polarized light source (frame b). The transmission spectra from λ = 900 nm to λ = 1900 nm were measured using a Bruker VERTEX 70 Fourier-transform infrared spectrometer with a Bruker HYPERION 1000 infrared microscope (40 × Cassegrain objective, numerical aperture NA = 0.4, near-infrared polarizer, and an InGaAs detector). An iris was used to collect the incident light to a square area of about 50 × 50 µm². The transmission spectra are normalized by those of the fused silica wafer. The simulation spectra were obtained with the finite-element method (FEM) of a commercial finite-element solver COMSOL Multiphysics. The simulated structure had a 60 nm-diameter tangent circle at the tail of the Taiji pattern, in accordance with our SEM observations. The refractive index of the fused silica glass substrate is 1.4584. The permittivity of gold was described by the Drude-Lorentz model using a damping constant of 0.14 ev, and a plasma frequency of 8.997 ev [15,16]. A single Taiji pattern was simulated (with periodic boundary conditions) under normal-incidence illumination by linearly-polarized light in the wavelength range from 800 nm to 3000 nm.

 

Fig. 2 (Top row) Experimental and simulated transmittance spectra for (a) x-polarized illumination; (b) y-polarized illumination. The illumination wavelengths available in our experiments ranged from λ_min = 900 nm to λ_max = 1900 nm. (Bottom row) Evolution of the simulated transmittance spectra upon varying the diameter of the tangent circle: (c) x-polarized illumination; (d) y-polarized illumination.

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In Figs. 2(a) and 2(b), the intensity differences between experimental and simulated spectra are due, in part, to the presence of surface roughness in the actual sample (ignored in the simulations); the inaccuracies of the Drude-Lorentz model of the dielectric constant of Au may also be responsible for some of the observed differences. The experimentally-determined resonance wavelengths, however, are in good agreement with our simulations. The slight differences between experiment and simulation where these resonance wavelengths are concerned may be attributed to the tail structure of the Taiji pattern. Figures 2(c) and 2(d) show the evolution of the simulated spectra as the tail diameter of the Taiji pattern is changed; Clearly, the tail feature has a substantial influence over the resonant behavior of the structure. In our measurements of the transmission spectra, where the Taiji sample depicted in Fig. 1(b) was under investigation, the tail features of each Taiji pattern were slightly different from those of the other Taiji patterns in the sample. We believe these variations throughout the periodic array have been responsible for the observed deviations of the measured resonance wavelengths from the simulated results.

There are two dips and one peak in the experimental as well as simulated spectra of Fig. 2; these features of the transmission spectrum are shared between the cases of x-polarized illumination in Fig. 2(a) and y-polarized illumination in Fig. 2(b). Of particular interest in the case of the second dip from the left of transmission spectra, we found a vortex-like feature in the time-averaged Poynting vector profile within the near field region, i.e., 250 nm above and 150 nm below the surface of the Taiji pattern. The vortex is present in both cases of x- and y-polarization for the second dip of transmission spectra; see Fig. 3(a) for the case of y-polarized illumination at λ = 1615 nm, where the red arrows represent the computed Poynting vector. Furthermore, we have found that the resonance wavelength associated with this vortex-like Poynting vector profile can be controlled by adjusting the periodicity of the pattern as well as the tail structure of the individual Taiji marks. We can still observe energy vortex without the gold dot outside. However, the energy vortex disappeared if we remove the anti-dot (the hole inside the gold Taiji mark).

 

Fig. 3 (a) Vortex-like profile of the time-averaged Poynting vector (red arrows) at different planes in the near-field region of the illuminated Taiji structure for the case of y-polarized illumination at λ = 1615 nm. (b) Schematic diagram of the physical model used to explain the origin of the vortex-like profile. The vortex in this diagram is produced by a horizontal electric dipole and a vertical surface current.

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4. Vortex-like Poynting vector profile

To understand the vortex-like Poynting vector profile, we analyze the simple physical model of an electric dipole perpendicular to the direction of the surface currents in the near field region of the Taiji marks. The near-field electromagnetic field distribution for the dipole and also for the surface currents could be seen in the electrostatics regime. In Fig. 3(b), we show the Poynting vector as solid red lines, the electric field (induced by the electric dipole) as dashed black lines, and the magnetic field (induced by the surface current) as blue symbols. It should be evident, in the cross-product of the electric and magnetic fields, that a vortex-like Poynting vector profile in the near field region could be achieved if an electric dipole perpendicular to the direction of the surface current were present. However, the oscillation direction of the electric dipole is typically the same as that of the current. That the Taiji pattern can form a vortex-like Poynting vector profile is rooted in its special structure.

As the simulation results of Fig. 4(a) , corresponding to y-polarized illumination at λ = 1615 nm, show the sharp tail of the Taiji pattern is associated with a high concentration of negative electric charge and, therefore, a lowered electric potential. It is in the wake of this lowered potential that an electric current flows to the tail. The dipole in Fig. 4(a) is seen to oscillate horizontally as a function of time, while the current flows vertically, in agreement with the proposed physical model of Fig. 3(b). We don’t see vortex-like energy profile at the first dip resonance for both x- and y- polarization. The resonant mode of charge isn’t dipole mode so that it cannot form vertex-like energy profile. We believe it is related to the Rayleigh anomalies of periodicity of Taiji marks. The mechanism of energy vortex for x-polarization is similar to the case of y-polarization; however, because of the resonant mode of charge, the direction of energy vortex for the case of x-polarization and y-polarization are not the same. The vortex is clockwise for horizontal (x-) polarization, and counterclockwise for perpendicular (y-) polarization.

 

Fig. 4 Simulated results corresponding to the interface region between air and gold (Taiji mark), obtained for the case of y-polarized illumination at λ = 1615 nm, i.e., second dip in the transmittance spectrum of Fig. 2(b). (a) Surface charge density in arbitrary units (color-coded) and surface current density (red arrows). (b) Vertical magnetic field in arbitrary units (color-coded) and horizontal electric field (red arrows). (c) Vertical electric field in arbitrary units (color-coded) and horizontal magnetic field (red arrows); shown on the right-hand-side in each case are the resulting time-averaged Poynting vector profiles.

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We found that the vortex could be separated into an inner part and an outer part, with the corresponding Poynting vectors expressed as E _|| × H _z and E _z × H _||, both fields being produced by surface charge and surface current; here E _|| = E_x x ^{^} + E_y y ^{^} and H _|| = H_x x ^{^} + H_y y ^{^}. In Fig. 4(b), the distribution of H_z is shown as a color bar and the direction of E _|| as red arrows; their cross-product gives rise to the inner vortex associated with the Taiji pattern. Similarly, E _z × H _|| produces the outer vortex.

5. Summary

The special geometrical features of the Taiji pattern, when illuminated under the conditions of plasmonic resonance, produce a vortex-like Poynting vector profile in the near-field region of the Taiji mark. We traced the origin of this vortex to an induced electric dipole on the surface of the Taiji mark that, in its orientation, is perpendicular to the induced surface current. The plasmonic resonance wavelength associated with the vortex can be fine-tuned by adjusting the tail structure of the Taiji mark.