Tuning of nanofocused vector vortex beam of metallic granary-shaped nanotip with spin-dependent dielectric helical cone

Fang Cheng; Deng-Feng Kuang; Li-Qun Dong; Yan-Yan Cao; Pan-Chun Gu

doi:10.1364/OE.25.017393

1. Introduction

The coupling of electromagnetic waves and electrons at metallic surfaces produce surface plasmon polaritons (SPPs). Efficient excitation and manipulation of SPPs generates nano-localized optical fields which have various prospective applications, such as optical nano-sensors [1–3], nano-manipulation [4, 5], and Tip Enhanced Raman Spectroscopy (TERS) [6–8]. Typical structures to generate plasmonic nanofocusing are nano metallic tips that have been theoretically investigated and experimentally demonstrated, such as pyramids [10, 11], campanile [12], cone [3, 13–16], conical nano-antenna [17] and V-shaped metal groove structure [18]. Furthermore, researchers have put forward structures with nano-gratings milled onto the metal tip [14, 19, 20] and a photonic-crystal cavity at the base of a tapered nano-wire [21] to enhance the intensity of SPPs. These geometries just can produce electric field components in radial direction, pointing perpendicular to the metal surface [22].

At the same time, planar spiral plasmonic lens, such as concentric circular grooves [23], subwavelength apertures [24, 25] and Archimedes’s spiral-shaped grooves or slits milled in a noble metal film [26–30] have been presented for coupling incident circularly polarized light to plasmonic vortices at the scale of several micrometers in the far field. Recently, a single-layer plasmonic vortex lens structure with a smoothed-cone tip at its center is demonstrated to couple a circularly polarized light to a plasmonic vortex and to efficiently transmit it to the far-field [31, 32]. On the other hand, a probe that integrates a planar spiral plasmonic lens and a sharp cone has also been presented to achieve electric field enhancement factor of 366 at 633 nm [33]. However, most manipulations of plasmonic vortex are now developed mainly in two dimensions, so there is much room for new developments by moving it into three dimensions [34].

In our previous study, we presented the metallic helical nano-cone [35] illuminated by radially polarized light to produce hybridly polarized strong localized electromagnetic field near the apex. We also presented non-linear metallic helical nanocone to control the intensity and polarization of plasmonic nanofocusing [36]. But these methods did not provide nano-focused plasmonic vortex. Here, we propose the combined configuration of dielectric helical cone (DHC) and metallic granary-shaped nano-tip (MGSN) innovatively to achieve and manipulate three-dimensional vector vortex optical field at the scale of several tens nanometers near the apex just with linearly polarized light vertical illumination. We also consider the tuning effect of the intensity, phase, and Poynting vector of three-dimensional vector vortex of MGSN with spin-dependent DHC.

2. Combined configuration and simulation method

The combined structure of dielectric helical cone and metallic granary-shaped nanotip investigated here is shown in Fig. 1, where, in the cylindrical coordinates, the function of dielectric helical cone can be expressed as:

h_{1} (ρ, θ) = h_{0} \cdot [(1 - \frac{ρ_{1}}{R}) + \frac{θ}{2 π}]

and the function of metallic granary-shaped nanotip is:

Fig. 1 The schematic diagram of dielectric helical cone and metallic granary-shaped nanotip in the Cartesian coordinate system. The blue structure is dielectric helical cone (DHC), the gray one is the symmetric metallic granary-shaped nanotip (MGSN).

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h_{2} (ρ, θ) = h_{m} \cdot r \cdot (1 - \frac{ρ_{2}}{r}) + (h {}_{n}- h_{m})

For numerical simulations, the cylindrical coordinate is changed to Cartesian coordinate by using the transformations: x = ρ·cosθ and y = ρ·sinθ, and h₀ is the initialized height of DHC, $0 \leq ρ_{1} \leq R$ , $0 \leq θ \leq 2 π$ , and $0 \leq ρ_{2} \leq r$ . h_n is the height of MGSN, (h_n- h_m) is the height of cylinder in MGSN, R and r are the total radius of the bottom of DHC and MGSN, respectively.

The electromagnetic (EM) field distributions of the beam generated by the combined configuration are calculated with three-dimensional finite-difference time domain (FDTD) method [37] when the DHC is illuminated by linearly polarized light. The mesh size is 20 nm and perfectly matched layers are used as simulation boundaries.

3. Microfocused optical vortex of DHC

The structural parameters of DHC are initialized as follows: h₀ = λ/(n-1) = 1.1 μm, R = 2 μm, λ₀ = 550 nm, and the material is taken as glass with the refractive index n = 1.5. The unit of the intensity of the electric fields is taken as 1 a.u., which is normalized to the intensity of the incident electric field. We simulate the light field distributions of DHC in two conditions: right-hand DHC (RDHC) and left-hand DHC (LDHC), where, the observed direction is along –z axis. Figures 2(a)–2(c) and Figs. 2(f)–2(h) show the electric field intensity distributions near the focus of DHC in yz plane, xz plane and xy plane, respectively. Figures 2(d) and 2(i) show the simulation results of the Poynting vector near the focus of DHC in xy plane, respectively. Figures 2(e) and 2(j) show the simulation results of the E_x phase distributions near the focus of DHC in xy plane, respectively. The total intensity through RDHC is 11.88 a.u., and the position of the focus is z_f = 4.80 μm. The total intensity through LDHC is 11.79 a.u., and the position of the focus is z_f = 4.18 μm. It can be clearly seen from Figs. 2(a)–2(c) and Figs. 2(f)–2(h) that DHC has the microfocused effect and the focused spot is deviated from z axis, presenting asymmetrical distribution, because the cone in DHC makes the electric field focus and the helix in DHC makes the focus be off the center. Furthermore, the angular momentum density $\vec{M}$ is [38]:

M = ε_{0} {\vec{r}}_{0} \times (\vec{E} \times \vec{B})

where, ε_m is the permittivity of vacuum,

{\vec{r}}_{0}

is the direction of propagation of light,

\vec{E}

is electric field intensity, and

\vec{B}

is magnetic flux density. In an ideal condition,

\vec{E} \times \vec{B}

is along the axis of propagation, so we simulate the component of

\vec{E} \times \vec{B}

which is perpendicular to the direction of propagation of light

{\vec{r}}_{0}

to confirm DHC carries angular momentum. In view of the formula of Poynting vector

\vec{S}

, it is described as [38]:

\vec{S} = \frac{1}{μ_{0}} \cdot (\vec{E} \times \vec{B})

where, μ₀ is the permeability of vacuum. According to the Eq. (3) and Eq. (4), we can take Poynting vector in xy plane S_xy as the standard to investigate angular momentum. The Poynting vector S_xy distributions in xy plane at z = 4.80 μm of RDHC and z = 4.18 μm of LDHC are shown in Figs. 2(d) and 2(i), respectively. S_xy of DHC originates from the central point, showing the energy in the xy plane flows along the same direction as the chirality of DHC. The phase distributions in Figs. 2(e)–2(f) indicate the DHC carries angular momentum with topological charge l = ± 1, which both confirm the DHC generates focused vector vortex beam carrying angular momentum.

Fig. 2 Simulations of right-hand dielectric helical cone and left-hand dielectric helical cone (a, f) Electric field intensity distributions in yz plane when x = 0 of RDHC and LDHC, respectively. (b, g) Electric field intensity distributions in xz plane when y = 0 of RDHC and LDHC, respectively. (c, h) Electric field intensity distributions in xy plane when z = 4.8 μm of RDHC, z = 4.18 μm of LDHC. (d, i) The Poynting vector in xy plane S_xy when z = 4.8 μm of RDHC, and z = 4.18 μm of LDHC. The white arrows represent the direction of energy flow. (e, j) The E_x phase distributions in xy plane when z = 4.8 μm of RDHC, and z = 4.18 μm of LDHC.

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Then, we calculate the electric field intensity distributions at the microfocused spots to investigate their full widths at half maximum (FWHMs) of RDHC at z = 4.80 μm and LDHC at z = 4.18 μm, respectively. The electric field intensity distribution curve of RDHC along x axis when y = −0.38 μm is shown in Fig. 3(a) and the electric field intensity distribution curve of LDHC along x axis when y = 0.38 μm is shown in Fig. 3(c). Correspondingly, the distribution curve of RDHC along y axis when x = 0.14 μm is shown in Fig. 3(b) and the distribution curve of LDHC along y axis when x = 0.14 μm is shown in Fig. 3(d). The FWHMs of RDHC along x axis and y axis are about 0.90 μm and 0.53 μm, respectively. The FWHMs of LDHC along x axis and y axis are about 0.83 μm and 0.47 μm, respectively. Thus, we can put a metallic nano structure with appropriate size at the focuses of RDHC and LDHC to further discuss the effect of spin-dependent DHC on the metallic nano structure. Taking advantage of the characteristic that DHC can form microfocused vector vortex beam carrying angular momentum, we tune MGSN with the vector vortex beam generated by DHC to excite SPPs and obtain highly localized nano-focused vortex beam.

Fig. 3 Electric field intensity distributions at the corresponding focused spots of RDHC and LDHC: (a, c) along x axis when y = −0.38 μm of RDHC and y = 0.38 μm of LDHC, respectively. (b, d) along y axis when x = 0.14 μm of RDHC and y = 0.14 μm of LDHC, respectively.

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4. Nano-focused vector vortex beam of the combined configuration

Silver is taken as the material of MGSN and the relative permittivity is considered as ε_m = −10.0452 + 0.8516i when the incident wavelength is 550 nm according to the Drude model [39]. Here, the structural parameters of MGSN are initialized as follows: h_m = 0.779 μm and h_n = 1.52 μm. The MGSN is located at the positions of the focuses of RDHC and LDHC, with z_Rf = 4.80 μm and z_Lf = 4.18 μm, respectively. The SPPs propagate along the surface of the MGSN with a wave vector k_spp, given by

k_{S P P} = \frac{w}{c} \cdot \sqrt{\frac{ε_{m}^{'} ε_{d}}{ε_{m}^{'} + ε_{d}}}

with the dielectric function of the metal and surrounding dielectric, ε_m and ε_d, respectively. From this

λ_{S P P} = λ_{0} \sqrt{\frac{ε_{m}^{'} ε_{d}}{ε_{m}^{'} + ε_{d}}}

the SPPs propagation length

δ_{s p p}

, the distance over which the power/intensity of the mode falls to 1/e of its initial value:

δ_{s p p} = λ_{0} \cdot \frac{{(ε_{m}^{'})}^{2}}{2 π ε_{m}^{''}}

where,

ε_{m}

is the real part and

ε_{m}^{'}

is the imaginary part of the relative permittivity, here,

ε_{m} = - 10.0452

and

ε_{m}^{'} = 0.8516

. So

λ_{S P P} = 0.521 μ m < λ_{0}

confirming the SPPs bound to the metal surface and

δ_{s p p} = 10.377 μ m > > λ_{s p p}

indicating we can use the bottom size of MGSN to further modulate and control SPPs.

Because the bottom center of MGSN is the original point of the xy plane, we calculate the distance between the center and the microfocused spot of DHC in xy plane in Fig. 2 and obtain the minimum diameter (i.e. internal diameter) of microfocused spot is r_min = 180 nm and the maximum diameter (i.e. external diameter) is r_max = 740 nm, respectively. Most importantly, the distance between the position of the maximum electric field intensity of the microfocused spot of DHC and the center of MGSN is r_best = 280 nm. Therefore, by taking the bottom radius r of MGSN as a variable, and r_min≤ r≤ r_max with the step being 10 nm, we calculate and discuss the influence of the microfocused spot of DHC on the maximum electric field intensity of the nanofocused vector vortex of MGSN with different bottom radius r. The variation of the maximum electric field intensity of the right-hand combined configuration and left-hand combined configuration with different r are shown in Fig. 4(a), respectively. We can see that the peak values are 13416 a.u. and 12149 a.u. respectively, both occurring at r = r_best = 280 nm, which indicate we can obtain the optimal nanofocusing of MGSN when the bottom edge of MGSN is just illuminated by the brightest location of the microfocused spot of DHC.

Fig. 4 (a)The variation of the maximum electric field intensity of the right-hand combined configuration and left-hand combined configuration with different r when the incident wavelength is 550 nm. Dashed line indicates the maximum intensity corresponding to the bottom radius. (b) The variation of the maximum electric field intensity of the right-hand combined configuration and left-hand combined configuration with different incident wavelength λ when the bottom radius of MGSN is 280 nm. Dashed line indicates the maximum intensity corresponding to the incident wavelength.

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When the bottom radius of MGSN is kept as 280 nm, we also calculate the variation of maximum electric field intensity of the right-hand combined configuration and left-hand combined configuration with different incident wavelength λ in steps of 10 nm, which are shown in Fig. 4(b), respectively. The peak values are 14728 a.u. and 13152 a.u. respectively, both occurring at λ = = 540 nm. From the curves, we can see that the combined configuration is wavelength sensitive when the materials and structural parameters are specified.

Taking right-hand and left-hand combined configurations with r = r_best = 280 nm to analyze, the positions of the maximum electric field intensity of plasmonic beams both occur at x = 2.24μm, y = 2.24μm, z = 6.88 μm. Figure 5 shows the electric field intensity distributions in xz plane, yz plane, and xy plane, the Poynting vector and phase distributions in xy plane near the apex. From Figs. 5(a)–5(c) and Figs. 5(f)–5(h), we can see that when the vortex beam generated by DHC illuminates MGSN, SPPs propagate along the metallic surface and achieves highly localized electric field enhancement near the apex of MGSN. Figures 5(d) and 5(i) show the direction of the Poynting vector is in accordance with the chirality of helix in DHC, and Figs. 5(e) and 5(j) further confirm the focused beam carries angular momentum, which is caused by the helical structure in DHC. The combined configuration indicates that the DHC can transfer the vortex characteristics to the MGSN structure and can make the electric field intensity increase up to four orders just under the condition of linearly polarized incidence.

Fig. 5 Simulations of right-hand combined configuration and left-hand combined configuration (a, f) Electric field intensity distributions in yz plane when x = 0 of right-hand combined configuration and left-hand combined configuration, respectively. (b, g) Electric field intensity distributions in xz plane when y = 0 of right-hand combined configuration and left-hand combined configuration, respectively. (c, h) Electric field intensity distributions in xy plane when z = 5.88 μm of right-hand combined configuration and left-hand combined configuration, respectively. (d, i) The Poynting vector in xy plane S_xy when z = 5.88 μm of right-hand combined configuration and left-hand combined configuration, respectively. The white arrows represent the direction of energy flow. (e, j) The phase distributions in xy plane when z = 5.88 μm of right-hand combined configuration and left-hand combined configuration, respectively.

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Similarly, we calculate the intensity distributions of the nanofocused spots near the apex of MGSN along x axis and y axis and are shown in Figs. 6(a)–6(d), respectively. The FWHMs of the nanofocused spots of right-hand combined configuration and left-hand combined configuration are both about 30 nm, which indicates the nanofocused vector vortex beam size is approximately 900 nm² in xy plane. At the same time, the longitudinal components of the maximum electric field of the nanofocused spots of right-hand combined configuration and left-hand combined configuration are 49.43% and 49.69%, respectively. These characteristics of the nanofocused vector vortex beam can be important for the applications of nano-sensing and nanofabrication.

Fig. 6 Electric field intensity distributions at the corresponding nanofocused spots of right-hand combined configuration and left-hand combined configuration: (a, c) along x axis when y = 0 of right-hand combined configuration and left-hand combined configuration, respectively. (b, d) along y axis when x = 0 of right-hand combined configuration and left-hand combined configuration, respectively.

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To further study the influence of DHC on the polarization variation of electric field near the apex of MGSN, we calculate the polarization angle of E_r along z axis from z = 6.0 to 6.2 μm at x = 20 nm, y = 0 of right-hand combined configuration and left-hand combined configuration, and show the results in Figs. 7(a) and 7(b), respectively. Accordingly, the direction of E_r rotates 49.1° and 67.0°, respectively. Therefore, the micro dielectric structure DHC plays an important role in rotation of E_r along z axis near the apex of the MGSN. This indicates the vector vortex beam generated by our structure can be applied in guiding, rotating, and sorting of microscopic particles at nano scale.

Fig. 7 The rotation of E_r along z axis near the apex of MGSN from z = 6.0 to 6.2 μm of (a) right-hand combined configuration and (b) left-hand combined configuration.

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5. Conclusions

In summary, we put forward the combined configuration of dielectric helical cone and metallic granary-shaped nanotip to produce three-dimensional vector vortex nanofocused optical field near the apex of the nanotip. The intensity and phase of the electric fields, and Povnting vector of the optical field generated by the combined configuration with linearly polarized illumination are studied by using 3D-FDTD method. We can obtain the optimal nanofocusing of MGSN when the bottom edge of MGSN is just illuminated by the brightest location of the microfocused spot of DHC. The DHC can also transfer the vortex characteristics to the MGSN. When the materials and structural parameters are specified, the combined configuration is wavelength sensitive. The localized vector electric field has the maximum intensity enhancement up to 10⁴ times and minimum size of about 900 nm², and the maximum radial electric field rotates 67.0° along z axis. The dielectric helical cone may be fabricated with grayscale direct laser writing and inductively coupled plasma etching [40], and the metallic granary-shaped nanotip may be fabricated with secondary electron lithography [31]. Therefore, the vector vortex beam generated by our combined configuration can be important for the applications of nanofabrication, nano-sensing and nano-manipulation.

Funding

National Natural Science Foundation of China (NSFC) (11274186)

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Tuning of nanofocused vector vortex beam of metallic granary-shaped nanotip with spin-dependent dielectric helical cone

Abstract

1. Introduction

2. Combined configuration and simulation method

3. Microfocused optical vortex of DHC

4. Nano-focused vector vortex beam of the combined configuration

5. Conclusions

Funding

References and links

Cited By

Figures (7)

Equations (7)

Optics Express