# Nanoscale Optical Trapping by Means of Dielectric Bowtie

^{*}

## Abstract

**:**

^{2}. The trapping of a virus with a diameter of 100 nm is demonstrated with numerical simulations, calculating a stability S = 1, and a stiffness k = 0.33 fN/nm, within a footprint of 0.96 µm

^{2}, preserving the temperature of the sample (temperature variation of 0.3 K).

## 1. Introduction

## 2. Dielectric Bowtie

_{2}) layer. The TTSWs show a wedge shape, with a height h and a wedge tip angle α.

_{2}O), used to transport biological species. Although small wavelengths are preferred in order to enhance the trapping efficiency by reducing the scattering force [33], this benefit is mitigated by a gradient force proportional to the operating wavelength and by a strong field confinement within the trapping region, in nanoscale, in a large spectrum region. Moreover, the low cost and easy availability of the light sources at telecommunication wavelengths paves the way to their use in several applications, including optical trapping [34]. Therefore, an operating wavelength λ

_{0}equal to 1550 nm is assumed, as in [34]. The relative refractive indexes of Si, SiO

_{2}, and water, at 1550 nm, are n

_{Si}= 3.48, n

_{SiO2}= 1.444, and n

_{H2O}= 1.318 + 9.86 × 10

^{−5}i, respectively [35,36,37].

## 3. Design to Enhance the Energy within the Trapping Site

_{norm}is shown in Figure 2a, with an impinging optical power P

_{norm}= 1 mW/µm

^{2}polarized along the y-direction. The inconstant behavior of the electric energy confinement in the trapping site is due to the spreading of the electromagnetic energy in the in-plane cladding layers, for configurations with h < 260 nm and α < 110°, or in the silicon wedges, for configurations with h > 260 nm and α > 110°, due to the additional high-permittivity antislot effect. In the remaining regions, the electromagnetic field moves from high to low, or vice versa, in the permittivity region.

_{norm}, (white empty circle in Figure 2a), as shown by Figure 2b, with a resulting total footprint A of about 0.96 µm

^{2}. The major component of the electric field is along the y-direction, representative of the quasi-TE polarized mode in the nanocavity. The electromagnetic energy density is mainly confined within the gap region, without spread in the cladding or dielectric material, typical of the slot effect, ensuring a stable trapping of the target particle.

^{2}. The temperature rises in correspondence with the trapping site, where the electromagnetic density is highly confined, as expected, with a maximum temperature increase ΔT of 0.07 K. Moreover, the ΔT change created by varying the input optical power P

_{norm}, and then the total optical power P

_{tot}(=P

_{norm}× A) has been calculated (see Figure 3b), measuring a slope of 7.51 × 10

^{−2}K/(mW/µm

^{2}). Similar values confirm the suitability of the dielectric nanoantennas for biological species trapping, because of the negligible temperature increase in contrast with the plasmonic nanoantennas where an increase in temperature of about 100 K has been obtained in [29], caused by the thermophoresis effect. The small temperature growth also preserves the vitality of the trapped biological species.

## 4. Numerical Results on the Optical Trapping and Discussions

^{®}has been used for e.m. and thermal simulations by combining the heat transfer and optical physics. According to the method described in [45], we have evaluated the optical force as the surface integral, on a surface with a radius few nm larger than the particle radius, of the vectorial product of the Maxwell stress tensor and the outgoing vector normal to the surface.

_{2}interface (see Figure 4). In order to estimate the trapping efficiency of the designed structures, the optical forces F

_{i}(i = x, y, z) on the target bead have been calculated by varying the position of the bead, as shown in Figure 5.

_{z}was calculated, observing a linear increase in the optical forces by varying the input power with a peak at z = z

_{eq}= 213.5 nm, close to the top surface of the bowtie. z

_{eq}is the z-coordinate of the equilibrium point, where the field gradients are the largest ones. Optical forces F

_{x}and F

_{y}were calculated by varying the bead position in both axes at z = z

_{eq}. In particular, both forces show symmetry with respect to the position x = x

_{eq}= 0 and y = y

_{eq}= 0, that represent the x- and y-coordinate of the equilibrium point. Along the y-direction, the movement of the bead is limited by the gap g, while along the x-direction, the bead undergoes the optical forces from x = −58 nm to x = +58 nm. For outer values, the bead can be considered free to move. An efficient and stable trapping requires the optical forces to be larger than the repulsive thermal force that could push the particle away from the equilibrium point. The stability of the nanocavity has been estimated as S = U/(k

_{B}·T

_{c}

_{)}, where U is the potential energy that corresponds to the work required to bring the nanoparticle from a free position to the trapping site [J], k

_{B}is the Boltzmann constant [J/K], and T

_{c}is the temperature in Kelvin [K], also taking into account the temperature rise due to the trapping effect [16]. The stability is usually referred to as the potential depth [46]. An increase in the stability corresponds to a longer trapping time, according to the Kramers theory [47,48]. In particular, the trapping time t

_{trap}is expressed as ${\mathit{t}}_{\mathit{trap}}\propto {\mathit{exp}(U/k}_{B}T)$. Therefore, the potential energy should be greater than or equal to k

_{B}T. Stability values larger than 1 are required to ensure the dominance of the optical forces with respect to the thermal one, in order to increase the trapping time. A trapping time of several seconds has been observed with other dielectric nanotweezers of S~1 [29]. Larger S values lead to a longer trapping time.

_{B}T) has been calculated with an input optical power P

_{norm}of 6 mW/µm

^{2}(P

_{tot}= 5.77 mW), where F

_{z}shows a peak of 27.45 fN. This performance demonstrates the high stability of a nanoparticle in the trapping site, which corresponds to a long trapping time (tens of seconds), with low input power. However, the experimental demonstration of optical trapping could introduce several physical factors that affect the optical trapping time. In particular, the pointing stability and the power fluctuations of the laser, the resolution of the motion stages, and the environment, in terms of temperature changes, acoustic noise, and mechanical vibrations, are crucial factors for the success of experimental trapping [9].

_{i}, expressed as ${k}_{i}=\partial {\mathrm{F}}_{\mathrm{i}}/\partial \mathrm{i}$ (i = x, y, z), has been evaluated (Figure 6b) at the equilibrium point. A high stiffness value ensures a stable position of the particle under test at the equilibrium point. For P

_{norm}= 6 mW/ µm

^{2}, k

_{x}= 0.27 fN/nm, k

_{y}= 0.93 fN/nm, and k

_{z}= 0.33 fN/nm have been calculated. The strong electric field confinement along the y-axis leads to the higher value of stiffness.

_{x}and F

_{y}with respect to x = 0 and y = 0, respectively, allows the particle to place itself in the trapping site. Although a direct comparison between the different trapping techniques is difficult due to the dependence of the performance on the particle size, the proposed device shows a stiffness (0.0572 pN/nm/W) significantly higher than other dielectric cavities for trapping (e.g., Z. Xu et al. report 0.0027 pN/nm/W for the trapping of 20 nm bead [29]), with a significant saving in terms of power (e.g., H. J. Yang et al. report a power of 250 mW for a stable trapping (S = 10) of 75 nm bead [49]) and footprint, with respect to the resonant cavities [11], also preserving the sample temperature, in contrast with the plasmonic nanocavities [50].

_{z}at the equilibrium point by varying the polarization direction of the input beam, represented as the angle γ between the y-axis and the beam. As expected, the optical forces became negligible for γ = 45° and the bead is repelled from the trapping site for γ > 45°. The second-order shape of the F

_{z}vs. γ trend makes the proposed device very robust with respect to the polarization direction: a change in the optical force of less than 10% has been calculated for γ, ranging from 0° to 15°.

_{z}vs. operating wavelength λ

_{0}is shown in Figure 8b. For λ

_{0}< 1300 nm, the antislot effect arises. As expected, according to the Rayleigh theory [33], the optical forces increase as the operating wavelength decreases due to the mitigation of the scattering effect. However, although a change in the scattering force of more than 80% is expected within the range 1300–1700 nm, since the scattering force is directly proportional to (1/λ

_{0})

^{4}, the force gradient attenuates the wavelength dependence.

_{z}change of ±7% over a range of 400 nm, with respect to λ

_{0}= 1550 nm, has been obtained, with a resulting negligible stability change of about ±0.23. At the same time, this result demonstrates an attenuation of the force dependence by the wavelength and a very high robustness of the device to λ

_{0.}

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic of the investigated nanobowtie dimer configuration in SOI platform (t: silicon thickness; g: width of the trapping site, sketched with a dotted cylinder; α: wedge angle; h: wedge height).

**Figure 2.**(

**a**) 3D representation of EC

_{norm}, by considering P

_{norm}= 1 mW/µm

^{2}; (

**b**) electric field distribution for the proposed dielectric bowtie with t = 220 nm, α = 110°, h = 260 nm, and g = 120 nm.

**Figure 3.**(

**a**) Top view of the temperature distribution ΔT (K) (ΔT = T − T

_{0}, T

_{0}= 293.15 K) for the designed configuration at z = 110 nm, with P

_{norm}= 1 mW/µm

^{2}; (

**b**) temperature increase ΔT in the trapping region as a function of the normalized input optical power P

_{norm}and then P

_{tot}(A = 0.96 µm

^{2}).

**Figure 5.**Optical forces F

_{x}, F

_{y}, and F

_{z}, varying the input power P

_{norm}and considering a particle with d = 100 nm.

**Figure 6.**Stability (

**a**) and stiffness k (

**b**) of the trapping event by varying the input optical power P

_{norm}.

**Figure 7.**F

_{x}(

**a**), F

_{y}(

**b**), F

_{z}(

**c**), and |F| (

**d**) distribution within x-y plane with P

_{norm}= 6 mW/µm

^{2}and z = 275 nm.

**Figure 8.**Optical force F

_{z}behavior at the equilibrium point, varying the polarization direction γ (

**a**), operating wavelength λ

_{0}(

**b**), and the radius R of the wedge fillet shape (

**c**) with P

_{norm}= 6 mW/µm

^{2}.

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**MDPI and ACS Style**

Brunetti, G.; Sasanelli, N.; Armenise, M.N.; Ciminelli, C.
Nanoscale Optical Trapping by Means of Dielectric Bowtie. *Photonics* **2022**, *9*, 425.
https://doi.org/10.3390/photonics9060425

**AMA Style**

Brunetti G, Sasanelli N, Armenise MN, Ciminelli C.
Nanoscale Optical Trapping by Means of Dielectric Bowtie. *Photonics*. 2022; 9(6):425.
https://doi.org/10.3390/photonics9060425

**Chicago/Turabian Style**

Brunetti, Giuseppe, Nicola Sasanelli, Mario Nicola Armenise, and Caterina Ciminelli.
2022. "Nanoscale Optical Trapping by Means of Dielectric Bowtie" *Photonics* 9, no. 6: 425.
https://doi.org/10.3390/photonics9060425