In order to investigate the dependence of optical force on LSPR mode in the gap, we vary the structural parameters and compare forces acting on it. The PS particle in these simulations is located at coordinate of (0, 0, 45 nm) (Fig. 2-4 (a)). For lower z positions, the surface will touch the bowtie causing an impractical situation with overlap of particle and the bowtie. So the nearest position we select is at z = 45 nm. For comparison we also simulate the reference case with one triangle of the bowtie eliminated. Results corresponding to this reference case are noted by “single”. Red curve in Fig. 2-4 (b) shows the spectral response of vertical optical force Fz exerted upon the PS as a function of wavelength when it is trapped by LSPR mode of the bowtie structure. In the figure, negative sign of Fz is consistent with the trapping nature along -z direction. One can see that Fz becomes stronger significantly when wavelength gets closer to resonance near 1.55 μm. The PS particle will experience maximum vertical force as strong as 362 pN under 1 W excitation. Comparing to the reference case as shown by black curve in Fig.
2-4 (b), that force is over thirty times larger and is also much stronger than that achieved in reference [28].
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Fig. 2-4 (a) Illustration of particle position in cross-section view and top view. (b) Fz acting on the PS particle as a function of incident wavelength for single triangle and bowtie.
Fig. 2-5 (a) shows the field enhancement distributes from z = 30 nm to 60 nm (x = y = 0) when excitation wavelength is fixed at 1.55 μm. The insets show the cross-section view of the electric field intensity distribution. Fig. 2-5 (b) shows field intensity distribution from y = -40 nm to 40 nm when keeping z = 30 nm and the excitation wavelength is also fixed at 1.55 μm.
The insets show the top view of electric field intensity distribution. Comparing to single triangle, bowtie structure has stronger field confinement and enhancement provided by gap effect which causes a very concentrated LSPR mode within the gap. Therefore it is right to use gold bowtie as a structure for serving particle trapping.
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Fig. 2-5 (a) The field enhancement from (0, 0, 30 nm) to (0, 0, 60 nm) for single triangle and bowtie. Insets show cross-section view of the electric field intensity (b) The field enhancement from y = -40 to y = 40nm (x = 0, z = 30 nm) for single triangle and bowtie. Insets show top view of the electric field intensity. The incident wavelength is 1.55 μm.
To examine the spatial distribution of the trapping force, we fix the wavelength at 1.55 μm and move the PS particle along z axis (x = y = 0) as shown in Fig. 2-6 (a). Obviously, the force will become stronger when the PS particle gets closer to the gold bowtie. And the force will die out fast when it moves few tens of nanometers away. This is the evanescent characteristic of near-field optical trapping. Therefore only neighboring particles will be trapped. This characteristic will provide selectivity for trapping in microfluidic systems.
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Fig. 2-6 (a) Fz acting on the PS particle as a function of position along the z axis (x = y = 0). (b) Fx as a function of location along the axis of (x, 0, 45 nm) (indicated by red arrows in top-view schematic). (c) Fy as a function of position along the axis of (0, y, 45 nm) (indicated by green arrows in top-view schematic).
The other two paths are along x and y directions when z = 45 nm as indicated by red and green arrows in the insets of Fig. 2-6 (b) and (c), respectively. The evolution of Fx and Fy as functions of location along x and y directions are shown in Fig. 2-6 (b) and (c), respectively.
We see that Fx (Fy) is positive when the particle is at x (y) < 0 and is negative when it is at x (y) > 0. This is the trapping nature. Finally the particle will be trapped at x (y) = 0 nm (equilibrium position), where it will experience no optical force in x (y) directions. In other words, Fx (Fy) changes sign when the PS particle moves across center of the bowtie. This means that naturally the PS particle will be driven toward the center as it falls into the trapping zone. Notably, Fx has maximum force 46 pN/W at x = 11 nm and the maximun for Fy is 50 pN/W at y = 11 nm. And we obtained a theoretical stiffness of 4.35 pN/ (nm•W)
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along x direction and 4.96 pN/(nm•W) along y direction. This is because the field is more concentrated along y direction than x direction. Finally, the resultant effect of Fx, Fy, and Fz will produce a stable trapping site near the gap.
Actually LSPR mode of the bowtie not only can trap particles on top of the gap but also can trap particle from either side of the gap. Fig. 2-7 (a) illustrates the possible trapping position when the particle is nearest to side of the gap. We show the horizontal force acting on particle along x direction when z = 15 nm as shown in Fig. 2-7 (b). The dash line represents the region in which the particle cannot approach because of size limitation. We see that Fx is positive when the particle is at x < 0 and is negative when it is at x > 0. The Fx will attract the particle getting closer to the gap and finally the particle will be stuck at either side of the gap.
Maximum horizontal force the particle will experience is 1389 pN/W. This force is three times higher than that when the particle is trapped on the top of gap. Therefore the trapping is not limited from the top.
Fig. 2-7 (a) Illustration of particle position in cross-section view and top view.(b) Fx acting on the PS particle as a function of location along the axis of (x, 0, 15 nm). The position in which particle can’t enter is fitting by dash line.
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Fig. 2-8 plots the potential distributions in the three directions corresponding to Fig. 2-7.
The potential is calculated by integrating parallel component of the trapping force along the path in each direction. Here the zero potential point is set at position 2 μm far from the bowtie gap center because field distribution decays exponentially to negligible there. The highly concentrated field distribution and resonant enhancement leads to a depth of the trapping potential much larger 10 kBT even for particles with sizes down to 20 nm. Since trapping force on the particle at sides of the gap is much stronger than that at top center of the gap, it can be predicted that the potential depth at sides of the gap must also be larger than 10 kBT.
That means the particle coming from any direction can be trapped stably around the gap without being disturbed by Brownian motion.
Fig.2-8 The potential along three axis by integrated the trapping force Fz, Fx and Fy along the z, x, and y axis.
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