Far-field position-tunable trapping of dielectric particles using a graphene-based plasmonic lens

Saeed Hemayat; Sara Darbari

doi:10.1364/OE.451740

1. Introduction

The magnificent work of Arthur Ashkin on optical trapping [1] has led to an ever-expanding interest in this field of research, since 1970 [2,3]. Although conventional optical trapping techniques were successful in trapping micrometer-sized objects, these techniques strongly suffer from the diffraction limit, preventing the optical tweezers from trapping of nano-scale objects. One of the significant breakthroughs in addressing this challenge was the advent of plasmonic tweezers [4–18]. In this type of tweezers, either the surface plasmon polaritons (SPPs) or the localized surface plasmons (LSPs) are used to trap subwavelength nano-scale objects. However, due to the evanescent nature of surface plasmons, and their inherent high-gradient field intensity, generally the plasmonic trapping occurs in the near field region, leading to a short-range trapping behavior for plasmonic tweezers. This inherent short-range trapping behavior limits the spatial affecting board of the tweezers to a small portion of the depth of general microfluidic channels. Hence, to benefit from plasmonic tweezing for manipulating particles in microfluidic channels, researchers have proposed some additional auxiliary manipulation techniques in the designed microfluidic configurations, such as electrostatic attraction of target particles towards the plasmonic structures [19], adding a top fluidic inlet to focus the z-position of the input particles near to the plasmonic structures [20], or utilizing thermophoretic forces to enhance approaching the particles to the plasmonic structures [21–23]. In this regard, we have recently proposed and designed a dual-mode plasmonic tweezers based on a closed-packed gold disc array, in which we benefit from the leaky surface plasmons on the discs to increase the affecting spatial board of tweezing [24].

The other general concerning issue in plasmonic tweezers is the proximity of the trapped particles with the plasmonic structure, which may lead to some thermal-induced stability issues for the trapped target particles [4,11,17,25]. In other words, formation of convection currents in the fluid may lead to escaping of the target particle from the plasmonic trap [26,27], or plasmonic heating can increase the Brownian motion of target particles, allowing them to escape from the plasmonic trap [28]. To surpass the aforementioned drawbacks and limitations, it is required to separate the plasmonic trapping site from the hot spots of the plasmonic structures, or design a plasmonic configuration so that the relating trapping site is located far away from the plasmonic structure. It is well known that the focal spot in plasmonic lenses is formed in the far-field, so that one can expect trapping of the target particle at the focal point, far away from the metallic plasmonic structure, owing to the gradient forces. Two of the best structures that can be used as candidates for the plasmonic lens are proposed in [29,30], where SPPs are excited at the input side of each of the slits and propagate between the two neighboring metallic walls through coupling of SPPs, forming an annular metal-insulator-metal (MIM) waveguide. Using the above-mentioned plasmonic lens structures leads to formation of a far field subwavelength focal point, which allows trapping nanoparticles far away from the metallic structures [31]. Another far-field trapping structure using the plasmonic lens concept is also proposed in [32], where forces on the order of nano-Newtons are achieved using a laser power of 0.5 W. It should be noted that none of the previously reported plasmonic lenses allow tunable focusing functionality, and different lens designs are needed to achieve different focal point positions.

Since the advent of epsilon-near-zero (ENZ) materials, they have found many applications in nanophotonics, and much research has been dedicated to this promising field [33–40]. It is well established that the mode attenuation increases drastically in graphene, once the magnitude of permittivity reaches values near zero, and as a result, graphene can be used as an efficient plasmonic switch [41–45]. Similar scheme can also be applied in transparent conductive oxides (TCOs), where a thin layer of ENZ material is sandwiched in a waveguide and exhibits a tunable absorption behavior between the low and high absorption levels [46].

In this paper, a graphene-based plasmonic lens with tunable far-field focusing behavior is proposed as a controllable plasmonic tweezers with long range operation, in which target particles can be trapped as floating particles at a considerable distance from the metallic lens structure. Moreover, the distance between the particle and the plasmonic lens can also be varied by changing the gate voltages applied to the graphene ribbons. Hence, we can prevent the proximity of the trapped particle and the plasmonic structure, and overcome the aforementioned thermal-induced issues, which are popular in conventional plasmonic tweezers. It should be considered that temperature distribution around trapping site depends on both on the heat generated from light dissipation in the gold structure, and thermal conductivity of the environment medium. Considering the lossy behavior of metals, and highly evanescent behavior of the plasmonic fields at the metal/dielectric interface, thermal hot spots occur generally in the vicinity of metallic structures. On the other hand, due to the high thermal conductivity of metallic structures, they serve as thermal heat sinks that can transfer the generated heat and reduce the peak temperature. Emergence of the electromagnetic hot spot at a distance far from the metallic structures in our design reduces the thermal conductivity from the trapping site and takes the trapping site away from the heat generation source in the metallic structure. For dielectric target particles, similar to this work, thermal generation at the trapping site is not significant, and forcing the trapping site away from the metallic lens structure can lower the peak temperature at around the target particles, and can relax conventional thermal issues in plasmonic tweezers. Furthermore, we can realize higher degrees of freedom in studying different characteristics of the floating trapped target particles, by the presented position-tunable 3D plasmonic tweezing. The achieved advantages were impossible in conventional plasmonic tweezers, because the highly evanescent nature of the plasmonic field at the metal interface leads to a strong vertical plasmonic gradient force towards the metal, and imposes a fixed trapping position in the vicinity of the plasmonic structures.

2. Structure and operation principle of the proposed Au/SiO₂/graphene plasmonic lens

The proposed structure is composed of annular slits formed in an Au layer on top of a SiO₂ substrate, as shown in Fig. 1(a). The proposed structure is illuminated by an x-polarized normal incident plane wave with wavelength of 1550 nm, as shown in Fig. 1(b). The circular slits behave as the main components of the plasmonic lens, the ITO/SiO₂/graphene-ribbons stack serves as the electrostatic gate of graphene ribbons in order to switch ON/OFF the slits’ radiations, and the underlying SiO₂ layer below graphene, isolates the graphene ribbons from each other. It is notable that we have assumed graphene mono-layers in our design. Figure 1(b) clarifies the layers separately at the cross section of the plasmonic lens structure. Figure 1(c) and 1(d) represent the top views of the proposed two different circular slits as the radiating elements for the plasmonic lens structure, each composed of two sets of circular slits (S₁ and S₂) with different radii (b and a). Figure 1(c) shows the first lens design, in which slit-sets of S₁ and S₂ are composed of two concentric circular slits, while Fig. 1(d) displays the second lens design with each slit-set composed of four concentric circular slits. The inset in Fig. 1(d) illustrates the magnified scheme of S₂, wherein W_d1, W_d2, W_d3, and W_d4 correspond to the slits’ widths (shown in red), with equal inter-slit-spacing or inter-element distances of d. It is clear that slit dimensions and inter-slit-spacing are the same for the slit-set of S₁ in Fig. 1(c), but slit width values are limited to W_d1 and W_d2. To avoid confusion, in Fig. 1 we have shown only two of the possible slit-sets with slit numbers of N = 2 and N = 4. However, we have investigated the radiation behavior of the plasmonic lenses with slit numbers from N = 2 to 7, the results of which are presented in section 4. It is also notable that the distance between the slits in each set (d) are assumed to be larger than half of the effective wavelength (λ_eff) of the propagating plasmonic mode in the relating MIM waveguides in order to prevent the coupling between neighboring waveguides [47]. In Table 1 we have defined the values of geometrical parameters in the proposed plasmonic lens structures to switch the z-position of their central focal spot from z = 5000 nm (S₁:ON, S₂:OFF) to z = 9800 nm (S₁:OFF, S₂:ON). It should be noted that the fabrication method for realizing high aspect ratio trenches, similar to the proposed structure, is generally focused ion beam milling (FIB) as it supports etching procedures as small as 10 nm [48–50]. However, there are recent reports that propose powerful alternative techniques for ultra-high aspect ratio high-resolution nanofabrication [51–53]. W_Au1, W_Au2, W_Au3, and W_Au4 in Fig. 1(b) correspond to the widths of Au contacts, labeled as Au₁, Au₂, Au₃, Au₄ on individual graphene ribbons of g₁, g_2, g_3, g₄ in Fig. 1(a). Figure 1(e) show the cross section view of the proposed structure along the first graphene ribbon (yz-plane in Fig. 1(b)), in which electric biasing of graphene is clarified by applying the gate voltage of V_g4 between the top ITO layer and the gold contact on the right end of graphene ribbon (Au₄). Because the buried gold layer of the lens structure is patterned to isolated circular slits, applying a uniform electrostatic field to the graphene layer is not possible by biasing the underlying gold layer. Hence, we have used a top uniform ITO layer, as a transparent conducting layer, in an ITO/SiO₂/graphene stack for electrostatic gating of graphene ribbons, and control the Fermi level of each graphene ribbon individually, while keeping a high light transmission for trapping as well. Thus, a gate voltage (say V_g4) is applied to the end contact of each graphene ribbon (say Au₄) with respect to the top common ITO contact, so that the optical absorption of each graphene ribbon can be tuned by an individual uniform electrostatic field. The thickness of the Au layer (L_Au) is assumed to be 800 nm to ensure a phase shift of π/6 between the slits, as will be discussed in section 4.1. L_Au must be increased for N values higher than 7, to maintain the equal amount of phase difference between the slits’ radiations. Thicknesses of the top ITO layer, and the SiO₂ layers that have sandwiched the graphene ribbons, are shown in the magnified cross section scheme of the stack in Fig. 1(e). The mono-layer graphene ribbon is shown by a dashed red line in the cross section views of part (e). It is also notable that we have designed rectangular graphene ribbons on the circular slits structure, in order to achieve a radiation switching behavior with a simple and feasible biasing configuration for graphene, in which simple external accessibility to each ribbon is allowed individually, as shown in Fig. 1(e). It is also notable that refractive index (n) of SiO₂, polystyrene particles, and the water environment are assumed to be 1.55, 1.57, and 1.33, respectively. Moreover, the refractive index and permittivity of indium tin oxide (ITO) and Au are extracted from Refs. [54] and [55], respectively.

Fig. 1. Schematic structure of the proposed position-tunable focal plasmonic lens. (a) The overall view of the complete structure; (b) Layer by layer presentation of the cross section structure; (c), (d) Top views of the metallic lens designs with N = 2 and N = 4. The magnified scheme shows the widths of the concentric circular slits (W_d1, W_d2, W_d3, and W_d4) in part (d). (e) The side view of the proposed structure along graphene ribbon g4, showing the applied electrostatic biasing configuration (V_g4).

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Table 1. Geometrical parameters of the proposed structure

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To achieve a qualitative insight to the radiation behavior of the proposed circular slit-sets, we have approximated each radiating circular slit by two small sub-wavelength radiating rectangular apertures at the rightmost and the leftmost sides of the ring (as shown in Fig. 2(a)), owing to the incident x-polarized illumination and the resulting plasmonic mode shape, and considering the slit width (W_d) much smaller than the circular slit radii. This approximation is inspired by an extensive research [56], in which a clear connection between the dispersion behavior of a circular coaxial MIM waveguide and a simple MIM waveguide has been presented. Considering the x-direction of the electric field in the plane of apertures, one can approximate each ring as two magnetic currents, directed toward y direction (M = -2n^×E_x [57]). Here, M is the magnetic current, E_x is the x component of the electric field vector, and n^ is the unitary normal vector facing outward from the aperture plane.

Fig. 2. The illustrative schematic figures, showing the tunable radiation patterns of the simplest proposed lens design with N = 2. (a) Top view scheme of the metallic lens structure, with the magnified views of the left and right side effective radiating zones of a single circular slit, when illuminated by a x-polarized backside source. The effective radiating zones are approximated by rectangular left-side and right-side apertures, in order to achieve a qualitative insight to the radiation pattern of the whole structure. The qualitative radiation patterns of the tunable lens structure, when (b) S₁:ON, S₂:OFF, and (c) S₁:OFF, S₂:ON. Switching ON/OFF the radiation states of the slit-sets (S₁ and S₂) is done by electrostatic gating of the corresponding graphene ribbons ((g2, g3) and (g1, g4)).

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In order to achieve a far-field focal spot with the assumed radiating rectangular apertures, the direction of the leftmost radiation beam should be directed toward the northwest (assuming that north is located along the + z axis), while the rightmost radiated beam should be directed toward the northeast, as shown in Fig. 2(b, c) schematically. Figure 2(a) manifests top view scheme of the simplest proposed plasmonic lens with N = 2, corresponding to Fig. 1(c), while Fig. 2(b) and 2(c) display the relating radiation behavior. Thus, we can replace each ring in the slit sets, S₁ or S₂, with two radiating rectangular apertures of widths W_d and effective length of L_d, as shown in the magnified view of Fig. 2(a). Figure 2(b) and (c) indicate that by switching ON/OFF the radiation state of S₁ and S₂, we can achieve different positions for the resulting far-field focal spot. This radiation state switching of slit-sets S₁ and S₂ can be obtained by electrostatically gating of the graphene ribbons g2 and g3, or g1 and g4, respectively. As can be seen in Fig. 2(b), when S₁ is ON and S₂ is OFF, graphene ribbons g1, g4 are biased in the ENZ sate, and graphene ribbons g2, g3 are biased in the non-ENZ state. However, Fig. 2(c) indicates the case of which S₁ is OFF and S₂ is ON, by biasing graphene ribbons g1, g4 in the non-ENZ sate, and biasing graphene ribbons g2, g3 in the ENZ state. It is obviously observed that the radiation patterns of S₁ and S₂ are similar, but the distance between the radiating slit-sets along the x-axis changes in Fig. 2(b) and 2(c). As a result, changing the ON state from S₁ to S₂, the distance between the radiating patterns and the resulting far-field interference changes, which in turn alter the z-position of the focal point. In other words, switching the bias voltage applied to the graphene ribbons between proper values, enables us to change the absorption behavior of each graphene ribbon between two different states: (i) as a nearly transparent medium that allows transmitting the radiation of the slits; (ii) act as an extremely lossy medium that selectively blocks the slit radiation, when operating in the ENZ condition for a certain wavelength. As a consequence of switching the absorption behavior of graphene ribbons in the proposed structure, vertical position of the focal point, and the corresponding trapping site position can be tuned.

3. Simulation method

3.1. Modeling graphene layers

Here, we have modeled the graphene mono-layer by inserting surface conductivity from Kubo’s formula, consisting of interband and intraband conductivity terms [58]:

(1)$${\sigma _s} = {\sigma _{\textrm{interband}}} + {\sigma _{\textrm{intraband}}} = {\sigma _{s\_real}} + i{\sigma _{s\_imag}},$$

wherein σ_{s_real} and σ_{s_imag} are the real and imaginary parts of the surface conductivity, and σ_interband and σ_intraband are defined as [58]:

(2)$${\sigma _{\textrm{interband}}} = \frac{{i{e^2}}}{{4\pi \hbar }}\ln \left[ {\frac{{2{E_F} - \hbar ({\omega + i{\tau^{ - 1}}} )}}{{2{E_F} + \hbar ({\omega + i{\tau^{ - 1}}} )}}} \right],$$

(3)$${\sigma _{\textrm{intraband}}} = \frac{{i{e^2}{k_\textrm{B}}T}}{{\pi {\hbar ^2}({\omega + i{\tau^{ - 1}}} )}}\left[ {\frac{{{E_F}}}{{{k_\textrm{B}}T}} + 2\ln \left( {{e^{ - \frac{{{E_F}}}{{{k_\textrm{B}}T}}}} + 1} \right)} \right],$$

where e is the electron charge, ℏ is the reduced Planck constant, E_F is the Fermi energy of graphene, ω is the angular frequency, τ is the carrier relaxation time, k_B is the Boltzmann constant, and T represents the temperature. It is evident from Eq. (2) and Eq. (3) that graphene conductivity is a function of the Fermi energy, and can be controlled by changing E_F. Moreover, the permittivity of graphene can be defined as [58]:

(4)$${\varepsilon _g} = {\varepsilon _b} + i\frac{{{\sigma _s}}}{{\omega {\varepsilon _0}\varDelta }},$$

wherein ɛ_b is the effective permittivity of the background medium, Δ is the thickness of the graphene sheet, which is mostly considered to be 0.34 or 0.7 nm for single and double layers of graphene. Then, replacing σ_s from Eq. (1) in Eq. (4) leads to the final description for the graphene permittivity as:

(5)$${\varepsilon _g} = {\varepsilon _b} - \frac{{{\sigma _{s\_imag}}}}{{\omega {\varepsilon _0}\varDelta }} + i\frac{{{\sigma _{s\_real}}}}{{\omega {\varepsilon _0}\varDelta }}.$$

3.2. Calculating plasmonic forces using Maxwell’s stress tensor (MST)

It is well established that the average force exerted by an electromagnetic field on an object enclosed in volume V, can be achieved by [59]:

(6)$$\left\langle \mathbf{F} \right\rangle = \oint\limits_V {\mathbf{T}(\mathbf{r},t).\hat{\mathbf{n}}ds} ,$$

where, F is the exerted force, S is the surface enclosing volume V, n is the normal vector of surface S, and T(r, t) is the Maxwell’s stress tensor, which can be written as [59]:

(7)$$\mathbf{T}(\mathbf{r},t) = \varepsilon \mathbf{E}\textrm{(}\mathbf{r}\textrm{)} \otimes {\mathbf{E}^\ast }\textrm{(}\mathbf{r}\textrm{)} + \mu \mathbf{H}\textrm{(}\mathbf{r}\textrm{)} \otimes {\mathbf{H}^\ast }\textrm{(}\mathbf{r}\textrm{)} - \frac{1}{2}({\varepsilon |{\mathbf{E}{{\textrm{(}\mathbf{r}\textrm{)}}^\textrm{2}}} |+ \mu |{\mathbf{H}{{\textrm{(}\mathbf{r}\textrm{)}}^\textrm{2}}} |} ),$$

where, ɛ and µ are the permittivity and permeability of the environmental medium, E is the electric field, H is the magnetic field, r is the position, and t is the time. Moreover, the potential energy (U) experienced by the target object in x direction, can be written as [10]:

(8)$$U(x )={-} \int_\infty ^x {{F_x}(x^{\prime})dx^{\prime}} .$$

It is well established that a plasmonic trap can be considered as a stable trap, wherein the potential energy depth exceeds 10k_BT, as the threshold margin for stable trapping [60].

All numerical simulations are carried out using the finite-difference time-domain (FDTD) method.

4. Results and discussions

4.1. Designing the plasmonic lens structure

First, we design the graphene-less plasmonic Au/SiO₂ lens structure to achieve the desired radiation pattern. As discussed in section 2, the proposed plasmonic lens is illuminated by an x-polarized beam with wavelength of λ₀= 1550 nm (Fig. 1(b)). Surface plasmon polaritons are generated at the entrance of the slits, coupled to the MIM waveguide, propagate and radiate from the other side of the slits [30]. The lens design is optimized for formation of a single focal spot either at around z≈5 µm or z≈10 µm. If we assume the desired x-position of 5 µm for the center of the effective radiating zones of each slit sets (center of the small, dotted rectangle in Fig. 1(d)), and z-position of 5 µm for the desired focal spot, the main radiation lobes of the slits should be directed toward θ=45° in φ=0° plane of spherical coordinates. For this purpose, if we assume an inter-slit-spacing of d = 460 nm for N concentric circular slits, the total normalized electric field in the φ=0° plane should have maximum at θ=45°. The normalized electric field component can be obtained by [57]:

(9)$${E_{\varphi n}} = \left[ {\cos (\theta )\cos (\varphi )\frac{{\sin X}}{X}\frac{{\sin Y}}{Y}} \right].\left[ {\frac{{\sin \left( {\frac{N}{2}\psi } \right)}}{{\sin \left( {\frac{\psi }{2}} \right)}}} \right],$$

(10)$$X = \frac{{\pi {L_d}}}{{{\lambda _{eff}}}}\sin \theta \cos \varphi ,$$

(11)$$Y = \frac{{\pi {W_d}}}{{{\lambda _{eff}}}}\sin \theta \sin \varphi ,$$

(12)$$\psi = \frac{{2\pi d}}{{{\lambda _{eff}}}}\cos \theta + \varDelta \phi ,$$

wherein E_φn is the φ-component of the total normalized electric field in the far-field, L_d is the effective length of the aperture, W_d is the width of the equivalent rectangular apertures (as shown in Fig. 2(a)), and Δϕ is the phase difference between the slits in each slit-set. It has been proved in the literature that for the far field approximation E_θn= 0, in plane of φ=0°, because E_θn $\propto$sinφ [57]. Considering Eq. (10), since |sinX|≤1, φ=0° (cosφ=1), and 0≤θ≤π (in spherical coordinates), and taking into account that we assumed L_d≤λ_eff in our design, thus X = (πL_d/λ_eff)sinθcosφ≤π/2. On the other hand, the phase difference between the slits in the proposed design is controlled through changing the slit widths (W_di). For the design purpose, first we can estimate the effective index (n_eff) of the MIM waveguides using [61]:

(13)$${n_{eff}} = \sqrt {{\varepsilon _d}} \left( {\sqrt {1 + \frac{{{\lambda_0}}}{{\pi {W_d}\sqrt { - {\varepsilon_m}} }}\sqrt {1 + \frac{{{\varepsilon_d}}}{{ - {\varepsilon_m}}}} } } \right),$$

wherein W_d is the width of the MIM plasmonic waveguide, and SiO₂ is assumed as the dielectric layer. The achieved effective index of the MIM waveguide is plotted in Fig. 3(a) versus varying W_d, revealing that n_eff approximately changes from 2.6 to 1.6, while W_d is increased from 10 nm to 100 nm. The inset in Fig. 3(a) shows the top view of a single circular slit, schematically. It should be noted that the phase of the plasmonic modes on the surface of the gold slit lens structure can be obtained using (2π.n_eff.L_Au)/(λ₀). Thus, it is obvious that we can introduce the desired phase shifts between the slits’ radiations by adjusting n_eff through changing W_d. To introduce a phase shift of π/6 between the slits, the difference between the effective indices of two adjacent slits (Δn_eff= n_eff1-n_eff2= (π/6×λ₀)/(2π×L_Au)) should equal to 0.16, which can lead to selection of effective indices equal to 2.6, 2.44, 2.28, and 2.12 to support the desired phase shift in the plasmonic lens designs with N = 4, corresponding to W_di values of 10 nm, 15 nm, 23 nm, 35 nm, as shown in Table 1. It should be noted that the slit widths in the proposed design are not limited to the values in Table 1. We can choose larger width values (for the lens with four slits) to create a focal spot at the same location, as long as Δn_eff= 0.16 is satisfied, according to Fig. 3(a). Complete design parameters of different lens structures with N = 2 to 7, are presented in Table 1 in detail. We will show that increasing the number of radiating slits (N) in each set leads to higher radiation directivity, and narrower beam, as shown in Fig. 4. However, increasing N further after 7 in the proposed design, requires increasing the length of the MIM plasmonic waveguide (L_Au) to maintain the desired phase shift between the slits. For the present design, we assumed L_Au= 800 nm, so that it supports the π/6 phase difference up to seven slits (N = 7). It is notable that if we limited N to 4, then L_Au= 400 nm would be also sufficient to support the phase difference of π/6 between the neighboring slits, considering Δn_eff= (π/6×λ₀)/(2π×400) = 0.32 and the corresponding refractive indices of 2.6, 2.28, 1.96, 1.64 for the slits. Assuming a circular single MIM waveguide with W_d= 100 nm, Fig. 3(a) shows that n_eff= 1.6. Thus, we can plug in the value of λ_eff=λ₀/n_eff in the aforementioned criteria of (πL_d/λ_eff)sinθcosφ≤π/2, to achieve L_d≥484.37 nm. Similarly, using Fig. 3(a) for our proposed slit-set with N = 4 and the slit width values (W_di) of those presented in Table 1, the corresponding effective length values (L_di) are obtained as 298 nm, 317 nm, 339 nm, and 365 nm, respectively. Since the differences between the obtained L_d values are indistinguishable, especially when we are observing the far-field radiation behavior of the circular slits, we approximate the calculated L_di values by their average value, which is L_d= 330 nm. As discussed previously, it is desired that the maximum normalized electric field occur at θ=45° in φ=0° plane, to achieve a z-position of the focal point at about 5000 nm. Thus, by substituting the achieved X, Y, Ψ values from Eqs. (10)–(12) into the derivative of Eq. (9), and equating it to zero, we force the far-field electric field to maximize at θ=45° in φ=0°, and find that the required phase difference between the slits, which is achieved Δϕ=π/6. On the other hand, it will be shown in section 4.3 that N = 4 is the minimum slit number for the proposed lens structure that leads to acceptable radiation directivities, resulting in a highly gradient field at the focal spot, allowing a stable localized plasmonic trapping.

Fig. 3. (a) Effective index of the propagating plasmonic mode versus the slit width for a MIM waveguide at λ₀= 1550 nm. Simulation results of the electric field profile on the Au layer surface at λ₀= 1550 nm, for a single circular slit with the slit width of W_d= 100 nm, and gold thickness of L_Au= 800 nm: (b) in xy plane, and (c) along x-direction at y = 0 nm. (d) The simulated reflection coefficient of the investigated single circular slit, versus different wavelengths, confirming a reflection dip at λ₀= 1550 nm.

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Fig. 4. Radiation patterns corresponding to the half structure of slit-sets with (a) N = 2, (b) N = 3, (c) N = 4, (d) N = 5, (e) N = 6, (f) N = 7 slits, all designed with the same inter-slit phase shift of π/6, and inter-slit spacing of 460 nm, leading to similar main lobe direction and similar z-position of the focal point, consequently. The color bars represent the linear-scale directivities. (g) Superimposed linear-scale Directivity patterns of different lens structures in φ=0° plane (in spherical coordinates), allowing a precise comparative insight. Numbers on gray-color circular contours represent the corresponding Directivity values.

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As stated before, we designed the proposed position-tunable plasmonic lens in a way that the focal point can be tuned from around z≈5 µm to z≈10 µm, by changing the x-position of the radiating slit-sets, as described in Fig. 2 schematically. This vertical movement of the focal point is allowed, because the radiation pattern and orientation of the slit-sets do not change by changing the x-position of the radiating slit, which is realized through switching ON/OFF the absorbing state of the up graphene ribbons.

After describing an initial designing principle for the proposed lens structure, first we present our numerical simulation results for a typical circular slit with W_d= 100 nm, at λ₀= 1550 nm. Figure 3(b) shows the transverse component of the electric field (E_x) for this single circular slit in the xy-plane, at z = 0 nm on top of the Au layer. Although the propagating plasmonic mode in an MIM waveguide is TM in nature, it can be approximated as a TEM mode under certain conditions in the near-infrared (NIR) regime. This fact is shown in Fig. 3(c) by plotting the calculated field components for a single circular slit, where the transverse component of the electric field is approximately one order of magnitude larger than its longitudinal counterpart (E_z), and the mode can be approximated as TEM in the left-side and the right-side of the circular slit. This approximation has been discussed in [47], where the transverse-to-longitudinal ratio of the electric field can be achieved by:

(14)$$\frac{{|{{E_t}} |}}{{|{{E_l}} |}} = \frac{{|{{E_x}} |}}{{|{{E_z}} |}} \approx \frac{{|{{K_z}} |}}{{|{{K_x}} |}} = \frac{{\frac{\omega }{c}\sqrt {\frac{{{\varepsilon _m}{\varepsilon _d}}}{{{\varepsilon _m} + {\varepsilon _d}}}} }}{{\frac{\omega }{c}\sqrt {\frac{{{\varepsilon _d}^2}}{{{\varepsilon _m} + {\varepsilon _d}}}} }} = \left|{\sqrt {\frac{{{\varepsilon_m}}}{{{\varepsilon_d}}}} } \right|,$$

wherein E_t and E_l are the transverse and longitudinal components of the electric field, K_x and K_z are the transverse and longitudinal components of the wave vector, ω is the angular frequency, c is the free space velocity of light, ɛ_m and ɛ_d are permittivities of metal and dielectric, respectively. This TEM behavior is emerged due to the fact that the magnitude of the metal permittivity is significantly larger than the permittivity of dielectrics like SiO₂ in NIR wavelengths (ɛ_Au=-115.13 + i11.259 at λ₀= 1550 nm [55], and ɛ_d= 2.33). Plugging in the permittivity values for Au and SiO₂ in Eq. (14) leads to |(ɛ_m/ ɛ_d)^0.5|=10.7554, showing that the transverse component of the electric field is approximately one order of magnitude larger than the longitudinal component, and the mode can be approximated as TEM on the aperture plane. Moreover, Fig. 3(d) displays the magnitude of the reflection coefficient (Γ) of the investigated single circular slit versus wavelength, which shows a significant drop at λ₀= 1550 nm, approving successful excitation of SPPs in the slit.

Here, Directivity is used as a measure of how directive a radiation beam is, which is defined as the ratio of the maximum radiation intensity per steradian to the radiation intensity of an isotropic radiator. For a slit-set consisting of N slits, Directivity is [57]:

(15)$$Directivity = \frac{1}{{\frac{1}{2}{{\int_0^\pi {\left[ {\frac{{\sin \left( {\frac{N}{{{\lambda_{eff}}}}\pi d\cos \theta } \right)}}{{\frac{N}{{{\lambda_{eff}}}}\pi d\cos \theta }}} \right]} }^2}\sin \theta d\theta }}.$$

Radiation patterns of slit-sets with N = 2 to 7 with phase shift of Δϕ=π/6, are shown in Fig. 4. Considering the mirror symmetry between the radiation patterns of halves of each circular slit-set, for instance N = 4 (Fig. 4(c)), the left-side and right-side counter propagating waves interfere in the far-field, forming a focal spot. It can be observed that the Directivity of the slit-sets increases from 6.78 for N = 2, to 13.3 for N = 7. It is obvious that increasing number of slits in each slit-set can lead to better field confinement, higher gradient field and stronger localized trapping behavior at the resulting focal point, at the expense of more complex lens structure. It should be noted that, as previously mentioned in section 2, the proposed structure supports the π/6 phase difference up to seven slits, and further increasing the number of slits requires larger L_Au values, in order to maintain the same amount of phase difference between the slits. Figure 4(g) illustrates a better comparative demonstration of the radiation patterns for different lens structures with N = 2 to 7, in φ=0 plane, confirming higher values of directivity for N = 7, in the expense of higher complexity of the structure.

4.2. Tuning the focal spot location using graphene ribbons

As mentioned before, graphene ribbons are used here to tune the z-position of the focal spot, and the resulting trapping site. For this purpose, first we investigate the real part, imaginary part, and the magnitude of the graphene’s permittivity using Kubo’s formula, as plotted in Fig. 5(a-c), respectively. It can be seen in Fig. 5(c) that for a fixed wavelength, the magnitude of the graphene’s permittivity becomes nearly zero at a specific E_F value. For instance, for λ₀= 1550 nm, ENZ occurs at E_F ≈0.535 eV, while for λ₀= 1064 nm, ENZ occurs at E_F≈0.715 eV. The field switching behavior of the graphene at ENZ condition is the direct consequence of the continuity of normal electric flux density at the boundaries [46], which can be written as:

(16)$${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\mathbf n}} }.({{{\mathbf D}_{\mathbf 1}} - {{\mathbf D}_{\mathbf 2}}} )= \rho ,$$

(17)$${ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\mathbf n}} }.({{\varepsilon_1}{{\mathbf E}_{\mathbf 1}} - {\varepsilon_2}{{\mathbf E}_{\mathbf 2}}} )= \rho ,$$

(18)$${\varepsilon _2}{E_{2n}} = {\varepsilon _1}{E_{1n}} - \rho ,$$

(19)$${E_{2n}} = \frac{{{\varepsilon _1}{E_{1n}} - \rho }}{{{\varepsilon _2}}},$$

wherein D₁ and D₂ are the electric flux densities in the dielectric medium and graphene, n^ is the normal unitary vector at the graphene/dielectric interface, E₁ and E₂ are the electric fields, E_1n and E_2n are the normal components of the electric field, and ɛ₁, ɛ₂ represent the permittivities in the dielectric medium and graphene. It is obvious in Eq. (19) that as soon as ɛ₂ approaches values near zero (the narrow dark blue band in Fig. 5(c)), E_2n (the electric field in graphene) rises significantly, and the switching behavior occurs. It is notable that here we have neglected the phonon interactions for the graphene plasmon.

Fig. 5. (a) Real part, (b) imaginary part, (c) and magnitude of the graphene permittivity, as a function of wavelength and Fermi energy, according to the Kubo’s formula.

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The ENZ behavior of graphene is further clarified in Fig. 6(a), where the simulated mode intensity in the graphene increases significantly in a single-aperture/graphene/SiO₂ structure, as E_F approaches 0.535 eV, meeting the ENZ condition for graphene at λ₀= 1550 nm. Moreover, Fig. 6(b) shows that Directivity of a single-aperture/graphene/SiO₂ structure at θ=0° and in φ=0° plane is suddenly decreased once the Fermi energy approaches 0.535 eV, which is consistent with the previously shown ENZ condition for a graphene layer at λ₀= 1550 nm (Fig. 5(c)). It is also notable that changing E_F of graphene ribbons for about 80 meV, from 0.535 eV to 0.615 eV, allows a radiation switching functionality for graphene, with ON/OFF radiation ratio of about 49 as shown in Fig. 6(b). This ENZ aspect of graphene has been well established in designing electro-absorption modulators [41–43], optical phased array nano-antennas [44], and our previously published plasmonic force switch reports [45]. Here, graphene ribbons are used as independent electrostatic switches to turn ON/OFF the radiation of slit-sets of the lens, and change the z-position of the resulting focal point, and the vertical distance of the floated 3D trapping site, consequently. The proposed tunable z-position of the floated trapping site can be a promising degree of freedom to investigate the trapping stiffness, and mechanical properties of the target particles, which has not been allowed in conventional plasmonic tweezers.

Fig. 6. (a) The simulated normalized mode intensity on the graphene layer, and (b) the simulated Directivity of a graphene/SiO₂/single-aperture structure for λ₀= 1550 nm, confirming a sharp dip at E_F≈ 0.535 eV.

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As discussed in the previous section, using the switchable behavior of the graphene ribbons, here we propose a position-tunable graphene-based lens structure (as shown in Fig. 1(a)), to achieve a 3D plasmonic trapping, for the first time. At this stage, two independent radiating slit-sets of S₁ and S₂ with N = 4 are proposed (as shown in Fig. 1(d)), which can be switched ON/OFF individually by electrical gating of the corresponding graphene ribbons. Figure 7(a) and 7(b) show the electric field intensities of our lens design in the xz-plane and along the x-direction at the z-position of the relating focal point, when S₁ is switched ON and S₂ is switched OFF for the incident wavelength of λ₀= 1550 nm. Similarly, Fig. 7(c) and 7(d) show the planar (in xz-plane) and linear distribution (along the x-direction) of the electric field intensity around the second focal point, when S₁ is switched OFF and S₂ is switched ON. Comparing Fig. 7(a) with Fig. 7(c), it is obvious that the z-position of the far-field focal point center has moved from z≈5000 nm to z≈9800 nm by inverting the radiation state of S₁ and S₂.

Fig. 7. Normalized electric field intensity in xz-plane and along x-direction at z-position of the relating focal points, resulting from radiation of slit-sets: (a, b) S₁, (c, d) S₂, with N = 4. Comparing parts (a) and (c) confirms the tunable z-position of the focal hot spot.

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4.3. Plasmonic force and potential energies

In this section, we present the simulated plasmonic forces exerted on a spherical polystyrene (n = 1.57) particle with radius of 500 nm, and the relating potential energy. First, we investigate the radiation Directivity of each half slit-set, and the resulting potential energies around the focal point for different N values. It is observable in Fig. 8 that by increasing the number of radiating elements (slits) from N = 2 to 7, the linear-scale Directivity (blue curve) increases from 6.8 to 13.3. Higher Directivity and narrower radiation beam width leads to more confined focal point, higher gradient plasmonic field and deeper potential wells, consequently. It is observable in Fig. 8 that increasing N from 2 to 7 leads to deeper potential depths along all x, y, and z directions, for both slit-sets of S₁ (red curve) and S₂ (purple curve). Considering -10k_BT (dashed gray line) as the threshold value for the minimum potential energy for stable trapping of the target particles, it is observable that N = 4 is the required minimum slit number for stable trapping at both S₁:ON and S₂:ON states, for the investigated incident light intensity of I_inc= 96 µW/µm². On the other hand, the maximum number of slits is limited to N = 7, for L_Au= 800 nm in the present design, because increasing L_Au is inevitable in order to keep the desired phase shift of π/6 between the slits for higher N values, as discussed in section 4.1. Hence, we have chosen N = 4 for the designed tunable lens, which benefits from a simple and feasible structure, in addition to a stable 3D trapping behavior for the described polystyrene target particles.

Fig. 8. Variation of the calculated Directivity (blue curve), and the corresponding minimum potential energy resulted from S₁ (red curves) and S₂ (purple curve), versus increasing the number of slits from N = 2 to 7. The target particle has been assumed a polystyrene particle with radius of 500 nm, and incident intensity has been assumed I_inc= 96 µW/µm².

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The exerted forces and potential energies in x and y directions are plotted for the lens structure with N = 4, in Fig. 9(a) and 9(b) for S₁:ON, S₂:OFF, and in Fig. 9(c) and 9(d) for S₁:OFF and S₂:ON. It can be observed that once the polystyrene particle is located around the z-position of the focal spot (z≈5000 nm for S₁ and z≈9800 nm for S₂), at x≈-1000 nm, a gradient force drags the particle toward + x direction, attracting the particle toward the focal point at this state. When the particle passes the focal spot, the direction of the exerted gradient force reverses, so that the particle is attracted back to the focal spot at x = 0 nm. The described tunable trapping site is verified by the calculated potential depth, shown by the red dash-dotted curves in Fig. 9. Figure 9(a), 9(b), and (e) show a stable trapping at the 3D position of the focal spot (x = y = 0, z = 5000 nm), so that the potential depth exceeds 10k_BT, when S₁:ON, S₂:OFF. Similarly, Fig. 9(c), 9(d), and 9(f) reveal a stable 3D trapping at the focal point (x = y = 0, z = 9800 nm), when S₂:ON, and S₁:OFF. It is evident that the potential depth along y-direction is smaller than that along x-direction in our design with an x-polarized incident illumination. Hence, the incident illumination power should allow a potential depth along y-direction deeper than 10k_BT. Our simulations show that I_inc= 96 µW/µm² leads to a y-direction potential depth of 18.2k_BT for S₁:ON, and 12.5k_BT for S₂:ON in our designed structure. It can be seen in Fig. 9(e, f) that upon reaching the particle to z≈4000 nm (for S₁:ON) and z≈8000 nm (for S₂:OFF), the particle starts to feel positive values of force that drags it toward larger z-values. Upon passing the particle from the focal spots, z≈5000 nm and z≈9800 nm for S₁:ON and S₂:ON states, respectively, F_Z reverses and drags the particle back toward the focal spot. It can be observed that the depth of the potential well in z-direction reaches its minimum at the location of the focal spot, owing to the highly gradient field around the focal point. The observed shallower potential well along z-direction in Fig. 9(f) corresponds to the second weaker hot spot at z≈5.5 µm in Fig. 7(c), which has been emerged due to the constructive interference of radiating side lobes for N = 4. It should be mentioned that this weak potential well cannot lead to stable trapping and the affected particle is attracted to the main focal point trap. It is evident that side lobes can be diminished, using lens designs with larger N values and higher Directivity (Fig. 4), at the expense of more complicated structures. The other worthy point in the whole parts of Fig. 9 is that the minimum potential depth for stable trapping is emerged in part (f) for U_z of the state S₁:OFF, S₂:ON, wherein the potential depth slightly exceeds 10k_BT. This observation reveals that I_inc= 96 µW/µm² is the minimum source intensity required for stable and tunable 3D trapping of polystyrene particles with radius of 500 nm.

Fig. 9. The plasmonic force components (blue solid curve) and potential energies (red dash-dot curve) exerted on polystyrene particles in different directions, for N = 4 and I_inc= 96 µW/µm². Parts (a, b, e) correspond to the radiation state of S₁:ON, S₂:OFF, resulting in emergence of trapping site at z = 5000 nm. Parts (c, d, f) correspond to the radiation state of S₁:OFF, S₂:ON, resulting in emergence of trapping site at z = 9800 nm. For all the investigated states, the lateral position of the trapping site is fixed at x = y = 0, at the center of the concentric circular slit-sets due to the symmetry. Potential energy threshold for stable trapping (-10k_BT) is highlighted by dashed gray lines in potential energy plots.

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It is evident that we can also release the trapped particles by electrostatically switching OFF both S₁ and S₂ states, or turning OFF the incident laser beam.

As the final step, we investigated the variation of plasmonic forces and the resulting potential energies along different directions, versus varying the Fermi energy of graphene and the particle’s position. As can be seen in Fig. 10, once E_F reaches the values near 0.535 eV (ENZ for λ₀= 1550 nm), all the force components and the corresponding potentials approach zero. The other worthy point here is that changing the Fermi energy of graphene ribbons in each state (S₁:ON, S₂:OFF or S₂:ON, S₁:OFF) does not change the trap location. It is clearly shown in Fig. 10 that changing the Fermi energy of graphene ribbons, only turns ON/OFF the applied force and the potential energy exerted on the particle. Figure 10(a-c) illustrate the amount of exerted force components on the target particle, versus changing both the particle's position and Fermi energy of the graphene ribbons g2 and g3 (corresponding to S₁:ON). As mentioned before, the focal spot, corresponding to S₁:ON is located around z≈5000 nm, so that the particle is dragged toward the focal spot through the gradient force, as shown in Fig. 10(c). As expected, it is observable in Fig. 10(a) and 10(b) that the particle is dragged toward the trapping location through the gradient force either in ± x or ± y directions. Moreover, this 3D and floated trapping behavior is confirmed in Fig. 10(d-f), wherein the trapping site margins are defined by the dashed curves, corresponding to the potential depth of 10k_BT. Moreover, Fig. 10(d-f) confirm the trapping/releasing behavior of the presented tweezers by switching E_F value of g2 and g3, for instance from 0.615 eV to 0.535 eV, wherein the U_z switches from -12k_BT to zero, for S₁:ON, S₂:OFF. It can be observed that at the focal spot (z≈5000 nm for S₁:ON), the potential energy depth exceeds 10k_BT for all E_F values, except for E_F≈0.535 eV. When E_F= 0.535 eV for all graphene ribbons, the slits radiations are turned off by graphene ribbons, and the focal spot and the trapping behavior disappear, and particle is released. Increasing E_F for a slight amount of about 80 meV (switching from 0.535 eV to 0.615 eV), the slits radiations turn ON, and the focal spot and trapping behavior emerge again. Similar discussions are applicable to the other state (S₁:OFF, S₂:ON), where the focal spot is emerged at z≈9800 nm (Fig. 10(g-l)). The worthy point here is that apart from the main potential well that is located at z≈9800 nm, a shallower potential well is also formed at z≈6000 nm, as shown in Fig. 10(l), due to the constructive interference between the radiations of side lobes of each slit-set. However, this shallow potential well cannot lead to a stable trapping because the achieved potential depth is about 6k_BT for the investigated incident intensity.

Fig. 10. (a-c) Variation of the plasmonic force components, (d-f) the resulting potential energies along different directions, versus varying the Fermi energy of graphene and the particle’s position for S₁:ON, S₂:OFF. (g-i) The plasmonic force components, (j-i) and the resulting potential energies, versus varying E_F and the particle’s position for S₁:OFF, S₂:ON. Here, the tunable trapping behavior is investigated for the lens structure with N = 4, for a target polystyrene particle with radius of 500 nm, at λ₀= 1550 nm and I_inc= 96 µW/µm². Dashed curves in (d-f) and (j-l) represent the -10k_BT criteria boundaries.

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5. Conclusion

In this report, we proposed and designed a position-tunable plasmonic tweezers to address the major drawbacks of conventional plasmonic tweezers, arising from their inherent near field trapping. Here, we designed a plasmonic lens with two slit-sets (S₁, S₂), each consist of four concentric circular slits, to achieve enough radiation directivity for creation of a subwavelength focal spot in the far-field region at incident wavelength of 1550 nm. Radiation of each slit-set leads to a constructive interference in the far field, emerging a confined focal point that allows 3D trapping due to the gradient force along x, y, and z directions. We showed that the total radiation pattern of each slit-set in our design allows surpassing the 10 k_BT potential depth criteria, required for stable trapping of target particles. Taking advantage of the strong electrostatically tunable optical property of graphene, we utilized graphene ribbons in our design to allow individual switching for each slit, with ON/OFF radiation switching ratio≈49 versus applying an electric pulse of 80 meV. Hence, we can change the Fermi energy of graphene ribbons to ENZ, increasing the field absorption on graphene and switching OFF the far field radiation of the corresponding slit. Hence, applying appropriate biasing we can switch the radiation state of the designed plasmonic lens from S₁:ON, S₂:OFF to S₁:OFF, S₂:ON, and increase the z-position of the 3D trapping site from z≈5 µm or z≈9.8 µm. The present configuration can be proposed as a new generation of electrostatically position-tunable 3D plasmonic tweezing, without the need for any external bulky optomechanical equipment, suitable for simple, compact, and efficient lab-on-a-chip applications.

Funding

Tarbiat Modares University (IG-39703); Iran National Science Foundation.

Disclosures

The authors declare no conflicts of interest.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Number of Slits	W_d1-d7 (nm)	L_Au (nm)	d (nm)	a (nm)	b (nm)	W_{Au1 & Au4} (nm)	W_{Au2 & Au3} (nm)
N = 2	10, 15	800	460	4520	4728	8040	3200
N = 3	10, 15, 23	800	460	4119	5217	8040	3200
N = 4	10, 15, 23, 35	800	460	3718	5707	8040	3200
N = 5	10, 15, 23, 35, 50	800	460	3317	6196	8040	3200
N = 6	10, 15, 23, 35, 50, 70	800	460	2916	6685	8040	3200
N = 7	10, 15, 23, 35, 50, 70, 98	800	460	2515	7174	8040	3200

Far-field position-tunable trapping of dielectric particles using a graphene-based plasmonic lens

Abstract

1. Introduction

2. Structure and operation principle of the proposed Au/SiO₂/graphene plasmonic lens

3. Simulation method

3.1. Modeling graphene layers

3.2. Calculating plasmonic forces using Maxwell’s stress tensor (MST)

4. Results and discussions

4.1. Designing the plasmonic lens structure

4.2. Tuning the focal spot location using graphene ribbons

4.3. Plasmonic force and potential energies

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Tables (1)

Equations (19)

Optics Express

Abstract

1. Introduction

2. Structure and operation principle of the proposed Au/SiO2/graphene plasmonic lens

3. Simulation method

3.1. Modeling graphene layers

3.2. Calculating plasmonic forces using Maxwell’s stress tensor (MST)

4. Results and discussions

4.1. Designing the plasmonic lens structure

4.2. Tuning the focal spot location using graphene ribbons

4.3. Plasmonic force and potential energies

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Tables (1)

Equations (19)

Optics Express

2. Structure and operation principle of the proposed Au/SiO₂/graphene plasmonic lens