1. Introduction

Mechanical action of light is among the most intriguing forms of light-matter interactions. In 1903, Nichols and Hull successfully conducted the first experiment to measure the pressure that light can apply on an object [1]. While these forces are typically insignificant on macroscopic objects, at the microscale they can be strong enough to accelerate and manipulate particles. This phenomenon forms the foundation of optical tweezers - a technology that uses focused laser beams to trap microscale beads and localize them in space [2,3].

Optical tweezers have been instrumental in advancing our understanding of fundamental biological and physical systems [4–10]. For example, optical tweezers revealed the hand-over-hand nature of the discrete 8 nm steps of kinesin motors as they walk along microtubules [11]. More recently, they have been used to study RNA transcription in real time [12]. To discern molecular motion, these studies relied on tethering micron-scale beads to the specimen of interest. This is because conventional optical tweezers are currently incapable of directly trapping nanoscale objects. Notably, to achieve trapping at the nanoscale, conventional optical tweezers would require impractically high optical powers leading to sample destruction. Further, the size of the optical trap created by conventional tweezers can not be smaller than the diffraction limit. Hence, even if localized, nanoscale particles would be confined within a trap hundreds of times their size.

To overcome the power and the trap-size limitations of conventional optical tweezers, plasmonic traps have been introduced as a next generation trapping technology [13]. Beside their ability of confining light below the diffraction limit, plasmonic traps achieve very strong field gradients that can localize nanoscale particles using reasonable optical power levels. The promise of plasmonic trapping was indeed proven through a wide range of plasmonic geometries that are capable of trapping dielectric particles smaller than 100 nm [14,15] - including 12 nm polymeric beads [16] and proteins with a 3.4 nm hydrodynamic radius [17].

To date, most plasmonic traps utilize planar geometries where the plasmonic trap is fabricated on a 2D substrate [14–16, 18–24]. This planar architecture allows for the integration of large arrays of traps that can be simultaneously illuminated achieving multi-particle trapping. More recently, optical fiber plasmonic traps have enabled 3D manipulation of the trapped particles. For example, in the pioneering work of Berthelot et. al. [25], a bowtie plasmonic aperture was patterned on a gold-coated tapered fiber tip and used as a plasmonic tweezer. The probe was able to trap and manipulate 50 nm dielectric particles in all directions. In a more recent work, Gelfand et. al. [26] introduced another optical fiber tweezers that utilized double nanohole apertures patterned on a cleaved non-tapered fiber tip to trap 40 nm dielectric particles. This non-tapered design exhibits increased mechanical sturdiness compared with tapered designs, but suffers from limited optical power coupling through the subwavelength aperture. Increasing mode overlap between the fiber and the plasmonic aperture remains an outstanding challenge and a critical consideration for future 3D plasmonic optical tweezers.

In this work, we design an optical fiber based plasmonic tweezer that utilizes a subwavelength coaxial geometry as the trapping aperture. Strong optical coupling through the coax is achieved without compromising the sturdiness of the fiber probe by using a grating to enable improved mode matching. As optical traps, coaxial apertures can trap dielectric particles as small as 2 nm [27]. Moreover, when illuminated with circularly polarized light, these apertures can, in theory, trap chiral nanoparticles based on their handedness [28]. Thus, integrating these coaxial tweezers on a fiber tip complements their unique trapping capabilities with three-dimensional manipulation.

The optical trap we investigate is illustrated in Fig. 1. The design uses a bull’s eye grating geometry to focus the incident optical power on the coaxial aperture. As the figure depicts, the grating has four key design parameters: the period (P), the distance from the center to the first groove (a), the duty cycle (w) and the grooves’ depth (s). Using full-field finite-difference time-domain simulations [29](FDTD Solutions, Lumerical Inc., Canada), we optimize the grating’s design parameters to maximize the optical power transmission through a given coaxial aperture. Then, we analyze the performance of the aperture as an optical trap by calculating the optical forces and the trapping potential on a dielectric particle interacting with the aperture. Our analysis shows that the optical power required to achieve stable optical trapping is reduce by up to eleven fold with the optimized grating.

 

Fig. 1 Cross-sectional schematic of the fiber-based optical tweezer, consisting of a nanoscale coaxial aperture on a metal-coated fiber tip. A grating is integrated at the fiber/metal interface around the coaxial aperture to focus the optical energy on the coaxial aperture. The grating period P, groove width w, groove height s, and offset from the coaxial aperture, a are optimized to maximally increase the transmission efficiency of the aperture and hence decrease the optical power required to trap a dielectric nanoparticle of diameter d_p.

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In this study, we consider two different coaxial designs, both integrated at the tip of a silica optical fiber. The first consists of an air-filled coaxial channel with inner and outer radii of 150 nm and 200 nm respectively in a 200 nm gold film. The second has a silica channel with inner and outer radii of 60 nm and 85 nm respectively in a 140 nm silver film. We chose these dimensions as they can be readily achieved with existing fabrication techniques [30–32].

2. Coaxial transmission optimization

We begin our study by evaluating the performance of the coaxial aperture when directly integrated on a fiber tip. To do so, we use the absolute transmission efficiency as a figure of merit. The absolute transmission efficiency is defined as the ratio between the total power transmitted through the aperture to the total input power delivered to the fiber tip. The inset of Fig. 2 shows the schematic of our FDTD simulation model. It consists of an optical fiber tip with a thin metal film (silver or gold) at the tip-end-facet. At the center of the tip, we introduce a coaxial aperture. The optical constants of silver and gold are adopted from Johnson and Christy [33].

 

Fig. 2 The transmission spectra of two coaxial apertures integrated onto fiber tips, including an Ag/SiO₂ coaxial aperture with a 120 nm core and 25 nm thick channel (blue) and an Au/air coaxial aperture with a 300 nm core and 50 nm thick channel (red).

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Figure 2 shows the transmission spectra of the two apertures under consideration. As seen, the resonance wavelength of the silver tip occurs at 695 nm while the gold tip has its resonance at 770 nm. These resonances are the Fabry-Perot resonances of the coaxial apertures. Note that the fiber-core diameters of the two tips are 4 μm for the silver tip and 5 μm for the gold tip. These are the typical core diameters of commercially available single-mode fibers operating at visible and near-infrared wavelengths. Beside the main resonance peak, additional peaks arise from the interference of the surface plasmons that propagate at the metal/fiber interface. These surface waves are excited at the coaxial aperture and propagate along the interface to the fiber edge where they are reflected back. The interference between the incident and reflected surface plasmons modulates the transmission spectra of the two apertures and give rise to minor resonances.

As Fig. 2 shows, both probes exhibit relatively low transmission efficiencies. For the silver tip the maximum absolute transmission is 1.48%, while for the gold tip it is 2.45%. Such low transmission efficiency leads to inefficient optical fiber based tweezers where large input powers are required to achieve stable trapping of a given particle. The main reason for such low transmission is the weak coupling between the incident light through the few micron diameter optical fiber core and the subwavelength coaxial aperture at the center of the fiber. Instead of being coupled through the aperture, most of the optical power is reflected back from the metal surface and lost.

In order to increase the optical coupling efficiency through the subwavelength coaxial aperture we consider a circular grating positioned at the metal/fiber interface, as shown in Fig. 1. The grating acts as an antenna that couples the incident optical power into surface plasmons and focuses them onto the coaxial aperture. In studies of subwavelength holes, it has been shown that optical transmission increases significantly when the resonance wavelengths of the grating and the aperture match each other. [34–38]

The optimization of the coupling grating involves four design parameters as explained in Fig. 1. For a given coaxial aperture we can tune these four parameters to maximize the optical transmission through the aperture. In order to find the optimal grating design, we perform an extensive search over this four dimensional parameter space using full FDTD analysis. It is worth noting that the period of an optimal grating can be estimated for a given coaxial aperture. In general, the grating period determines the wavelength at which the maximum coupling of the incident light into surface plasmons takes place. Hence, in order to maximize the transmission efficiency through a coaxial aperture with resonance wavelength λ_res, we need to tailor the grating period P such that it satisfies the following relation $P={\lambda }_{\mathit{res}}\sqrt{\left({∊}_m+{∊}_d\right)/{∊}_m{∊}_d}$ [39] where ∊_m and ∊_d are the relative permittivities of the metal (silver or gold) and the glass respectively. Consequently, we predict that the optimal grating periods for our two coaxial apertures should be 452 nm for the silver aperture and 502 nm for the gold one. However, the actual optimal period is expected to be blue shifted from this value due to Fano-type resonances [39].

Figures 3(a)–3(b) show the maximum transmission of the two tips under consideration as a function of the period (P) and the distance (a). The maps highlight the dramatic effect of both P and a on the light transmission through our coaxial apertures. These two maps are calculated at the optimal values of the duty cycle (w) and the grating depth (s). The optimal duty cycle for both tips is found to be w = P/2. For the grating depth (s), we constrain our optimization such that s is kept less than 100 nm. We find that the maximum transmission enhancement of the gold tip is achieved when s = 60 nm, while the optimal value of s for the silver tip is found to be s = 80 nm. The maps indicate that the optimal value of the the grating period for the gold and the silver tips are 430 nm and 460 nm respectively. As expected, they are both smaller than the values based purely on the coaxial resonance. In addition to the period P, the distance (a) is crucial to improve the transmission efficiency. We find from the transmission maps that the optimal values for the distance a are 320 nm and 270 nm for the gold and silver tips respectively. Note that for these values of P and a, we fit only four grating grooves at the metal/fiber interface of the silver tip; because of the larger diameter of the gold tip, we fit five grating grooves. It is worth noting that the effect of grating depth on the transmission efficiency is less significant once the period and the distance to the first groove are optimized. For example, detuning the depth of the optimized silver tip grating from 80 nm to 100 nm or 60 nm causes the maximum transmission to drop by small amount from 16.7% to 15.2% and 15% respectivley.

 

Fig. 3 (a,b) The maximum transmission efficiency of the two coaxial apertures as a function of the grating period, P, and the distance to the first groove, a. (c) The normalized electric field amplitude 5 nm away from the input side of the aperture for three different grating dimensions of the silver tip: (i) a = 270 nm, P = 430 nm (Optimal), (ii) a = 230 nm, P = 430 nm and (iii) a = 270 nm, P = 480 nm. All fields are normalized to the maximum field amplitude of (iii).

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To further explain the role of the grating coupler, we plot the distribution of the electric field amplitude at the input side of the aperture as shown in Fig. 3(c). The figure illustrates the distribution of the normalized field amplitude 5 nm away from the input side of the silver aperture. At the optimal dimensions of the grating (i), the field is strongly localized at the center of the fiber on top of the aperture and quickly decays in the radial direction. Such distribution indicates that the incident optical power is efficiently coupled into surface plasmons which are then strongly focused on the aperture, leading to significantly higher transmission. When either the distance to the first groove (ii) or period of the grating (iii) are detuned from their optimal values, the field is no longer strongly concentrated in the center. Instead, the field disperses to the neighboring grooves and even scatters outside the fiber tip itself.

Figures 4(a)–4(b) illustrate the full transmission spectra of the two tips before and after introducing the optimized grating. The figure highlights the significant boost in the transmission efficiency after integrating the grating. For the gold tip, the maximum transmission is increased by eleven fold from 2.5% to 19.6%. Similarly, the maximum transmission of the silver tip increases eight fold from 1.5% to 16.8%. Such transmission values are maintained even when a particle is introduced in the near-field of the aperture. For example, peak transmission remains within 0.5 absolute percent of these values for a 30nm diameter particle with refractive index of 2, regardless of particle position.

 

Fig. 4 (a,b) The transmission spectra of the two fiber tips under consideration before (red curve) and after (blue curve) introducing the optimized coupling grating.(c,d) The electric field intensity enhancement 20 nm away from the two coaxial apertures before and after introducing the optimized grating.

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Figures 4(c)–4(d) depicts the electric field intensity enhancement 20 nm above the coaxial aperture before and after introducing the optimized grating at their respective resonances. As seen in the figure, the field intensity profile has two hot spots, a typical nearfield intensity profile for a plasmonic coaxial aperture illuminated with a linearly polarized light [27]. The figure also demonstrates the significant increase in the the field intensity enhancement achieved with the coupling grating. For the gold aperture, the intensity enhancement increases from 38.5 to 254. Similarly, the electric field intensity enhancement of the silver aperture increases from 27.5 to 385. This is an important result for optical trapping applications, given the direct dependence of the optical forces on the electric field intensity gradient, as will be explained in the following section.

3. Grating-enhanced coaxial optical forces

In principle, stable optical trapping is achieved when the depth of the optical trapping potential is larger than 1kT, where k is the Boltzmann constant and T is the temperature. In Ashkin’s seminal optical trapping paper, it was suggested that this threshold should be set at 10kT to overcome any excess energy the trapped particle might gain upon stochastic interaction with energetic molecules in the surrounding medium [3].

To calculate the optical forces acting on a particle interacting with the nearfield scattered from our plasmonic coaxial tweezers we use Maxwell’s stress tensor formalism [40, 41]. In this formalism, the total optical forces acting on a an object is determined by calculating the time-averaged Maxwell’s stress tensor 〈T〉 given by

To benchmark our plasmonic tweezers, we consider dielectric particles with a refractive index of 2 - an intermediate value between the typical refractive index of biological species such as proteins (n = 1.6) [42] and II–VI quantum dots such as CdSe quantum-dots (n = 2.4) [43]. The refractive index of the background medium is taken to be 1.

Figure 5(a) shows the transverse optical force F_x on a 30 nm particle located 25 nm away from the gold tip. The particle is swept from the center of the coaxial aperture along the x-axis where the force is calculated at each indicated point. The figure depicts the force exerted on the particle before (red curve) and after (blue curve) introducing the optimized grating normalized to 100 mW of input power. As expected, the enhancement in the transmission efficiency is translated into significant increase in the optical force for the same input power, with peak forces increasing by a factor of eight. As can be seen in the figure, this transverse force localizes the targeted particle on top of the coaxial dielectric channel (between x = 150 nm and x = 200 nm where the origin (x = 0) is at the center of the aperture).

 

Fig. 5 (a) The transverse confining force (F_x) exerted on a 30 nm dielectric particle as the particle moves across the gold aperture along the x-axis normalized to 100 mW of input power. The red curve shows the force in the absence of the coupling grating while the blue curve shows the force after introducing the grating. (c) The resulting transverse confining potential along the x-axis due to the confining force (F_x). (b,d) The transverse confining force (F_x) exerted on a 10 nm dielectric particle as the particle moves across the silver aperture along the x-axis normalized to 100 mW of input power and the corresponding trapping potential. (e,f) The evolution of the pulling force (F_z) as the particle moves away from the aperture in the z-direction.

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Similarly for the silver tip, we calculate the transverse force on a 10 nm particle located 7 nm away from the tip as shown in Fig. 5(b). As seen, for the same amount of input power the optical force increases by a factor of eleven. It is worth noting that the force directly over the dielectric channel (between x = 60 nm and x = 85 nm) exhibits significant spatial variation. This is because the gradient of the electric field intensity is maximized at the two interfaces of the dielectric channel compared to that in the middle of the channel. Because of the small size of the particle (here, 10nm diameter), it can accurately sample the change in the field gradient, and consequently the corresponding optical forces, as it is swept over the channel.

To confirm that these optical forces are strong enough to stably trap the particle, we calculate the optical trapping potential. In general the trapping potential is obtained from

The trapping potential along the x-axis for the two tips is depicted in Figs. 5(c)–5(d). For the gold tip with the 30 nm particle, we see that the trapping potential increases from less than 2.8kT to over 20kT after including the optimized grating. In a similar fashion, the trapping potential on the 10 nm particle interacting with the silver tip increases from 1kT to 11kT. Hence, the condition for stable optical trapping is achieved with only 100 mW of input power after introducing the coupling grating, compared to over 1W without.

While the transverse force F_x confines the particles to the coaxial channel, there is also a pulling force F_z that drags the particle closer to the aperture [27]. In Figs. 5(e)–5(f) we study the evolution of the optical forces as the targeted particles approach the aperture in the vertical direction along the z-axis. These forces are calculated along the z-axis at x = 155 nm for the gold tip and x = 67 nm for the silver tip, i.e., the two positions along the x-axis where the pulling force F_z is maximum. The figure illustrates the exponential decay of the optical forces away from the aperture which is a characteristic of nearfield optical tweezers. Moreover, it shows that introducing the grating extended the range of pulling force (F_z) in addition to providing a stronger grip on the targeted particle. In particular, F_z for 100mW of input power remains greater than 1 pN for distances up to 50nm (25nm) for the gold (silver) apertures with the grating. Without the grating, the force F_z is less than 1 pN even when the particle is right on top of the aperture.

4. Outlook and conclusion

In this paper we introduced an alternate design for a fiber-based plasmonic tweezer. The design combines a plasmonic coaxial aperture with a bull’s eye grating to boost the optical transmission, near-field intensities, and optical trapping forces. With an optimized coupling grating, transmission increases by a factor of eight and eleven respectively for the gold and silver structures. Accordingly, both the near-field intensities and the optical trapping potential depths increase by the same factor. Such grating-flanked apertures can thus reduce the total input power required for stable nanospecimen trapping by an order of magnitude. Importantly, the investigated optical tweezers can be readily fabricated with existing techniques. For example, it has been shown that direct focused ion beam milling with either Ga⁺ or He⁺ ions can be used to fabricate air-filled coaxial apertures [30]. Alternatively, e-beam lithography can, with high fidelity, be used to make apertures with thin silica-filled channels and even smaller coaxial diameter [31]. More recently, fabrication processes based on atomic layer deposition have been used to fabricate large scale arrays of coaxial apertures with dielectric channels as thin as 2 nm [32].

Our coaxial optical tweezer facilitates 3D trapping and manipulation of nanoscale dielectric particles, typically inaccessible with conventional optical tweezers, while maintaining trapping powers below 100 mW. This is achieved while avoiding the use of fragile tapered fiber tips, though further power reduction could be envisioned when used in combination with tapering. Besides reducing the total optical power required for trapping, our design can be combined with various thermal coatings to further reduce the effects of heating associated with plasmonic tweezers. For example, indium tin oxide has a thermal conductivity that is ten times higher than that of silica [44] and could be used to improve heat dissipation at the gold/glass interface, similar to the copper heat sinks used in prior plasmonic tweezers [20]. Looking forward, such grating-flanked apertures could not only be used for 3D near-field tweezing, but also for precise far-field beam-shaping [45] and far-field optical trapping [46]. Such probes have the potential to enable a wide range of applications including 3D nanoparticle assembly and 3D nanoprinting in addition to direct trapping and 3D manipulation of fundamental biological specimens.