1. Introduction

When free space light is incident on the sub-wavelength scale metal structures or penetrates through metal holes, it will produce some interesting phenomena in the metal/dielectric interface, such as the excitation of SPPs. The resonances produced by the evanescent waves and electrons generated at the metal/dielectric interface result in SPPs, which propagate along the metal/dielectric interface [1]. In recent years, many SPP-related studies have been developed, such as: plasmonic dichroic splitter [2], unidirectional launching [3,4], tunable directional excitation [5–7], plasmonic structured illumination microscopy [8] and plasmonic splitter [9].

Using light to capture or manipulate particles has been a subject of much study [10,11], and has lately been applied to microfluidic particle sorting systems [12–15]. Scientists have recently used SPPs to capture particles, such as single-particle capture [16] or multi-particle capture using a one-dimensional array of light fields [17], or using one-dimensional array light fields for particle manipulation and separation [18,19]. Many studies also use the design of special micro/nano structures for single-particle [20,21] or two-dimensional multi-particle capture [22–24]. One of the advantages of plasma optical tweezers compared to conventional optical tweezers is it provides higher spatial precision.

Currently, the use of SPPs for two-dimensional multi-particle capture mainly depends on the design of the micro/nano structure to decide the state of particle trapping. Some studies designed micro/nano structures for generating two-dimensional SPP array interference patterns [25]. However, in order to achieve the dynamic manipulation of two-dimensional multi-particles through the SPP field, a dynamically manipulated two-dimensional SPP structured light field is necessary. This paper proposes a method to generate a dynamically manipulated two-dimensional SPP structure light field. This study generates a 2D SPP interference field within a four-slit structure, and achieves control of the transverse location of the entire two-dimensional SPP field by adjusting the phase difference between the four incident fundamental Gaussian beams (i.e. TEM_0,0 laser beam). The method proposed in this study can also be used to dynamically manipulate one-dimensional SPP interference fields. For the one-dimensional SPP interference field manipulation, this paper also provides relevant theoretical description and numerical simulation.

This article contains three main parts. First, the proposed design for the SPP interference field generation is explained according to the theory. Then, the one-dimensional and two-dimensional SPP interference patterns this study proposes are demonstrated and verified by using the FDTD simulation method. Finally, the manipulation of single-particle and multi-particles using two-dimensional SPP interference field are demonstrate by simulations.

2. Design and theory

This study uses a four-slit metal structure arranged in a rectangular shape and four fundamental Gaussian beams (i.e., TEM_0,0 laser beam,) with specific phase difference to each other as the incident lights to generate the interference pattern of specific SPPs. The SPP interference field is laterally moved by adjusting the phase difference between the four beams. Figure 1 shows the schematic diagram of the four-slit metal structure and the direction of the incident beams. Four metal slits are arranged in a rectangular shape on a transparent substrate. The thickness of the metal layer should be sufficient to prevent the penetration of incident light to avoid affecting the observation of the resulting SPPs. The beam is incident perpendicularly from beneath the substrate, creating SPPs at the metal/air interface for the SPP interference patterns between the four slits.

 

Fig. 1 Four-slit structure and axis diagrams: (A) Top view of the four-slit structure and definition of the coordinate axes, (B) Side view of the four-slit structure and the relationship between the incident light and structure.

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The generating SPP interference field of this study and the theory for its dynamic manipulation is described below. The SPPs excited at the metal/air interface are dominated by the component of the electric field vertical to the surface (here, the z-direction). The z-direction component of an SPP point source propagating along a metal/air interface can be expressed as [26–29]:

The light source used in this study is four coherent fundamental Gaussian beams with fixed and tunable phase relationships. Due to the spatial locality of the Gaussian beam, when the four beams enter the four slits respectively, each of the four beams does not affect the other three slits. When the fundamental Gaussian beam is denoted as $E(x,y,z;t)$, the mathematical form of the incident beam used in this study can be expressed as:

The phase constant ${{\displaystyle \varphi }}_i$ of the four SPP line sources excited by the incident light beams at the four slits is correlated to the phase term ${{\displaystyle \varphi }}_i\text{'}$ of the incident fundamental Gaussian beam [5,28], respectively. From Eq. (1), it can be seen that the amplitude variation of the four SPP line sources within the four metal slit ranges can be ignored when $r<<{{\displaystyle L}}_{spp}$ is established. In this situation, the SPP interference field in the middle region of the four slits can be expressed as:

Referring to Fig. 1(A), the symbols d₁ and d₂ in Eq. (4) represent the widths of the two-direction metal slits, respectively. From Eq. (4) it can be seen that the influence on the SPP interference field of the d₁ and d₂ values is the same as the influence of the phase constant ${{\displaystyle \varphi }}_i$ of the four SPP line sources, i.e. both of them have a linear relationship with the phase of an SPP line source. It can also be seen here that, regardless of the value of d selected, the phase value of the SPP line sources at a specific point can be adjusted to the desired value using the adjustment phase constant ${{\displaystyle \varphi }}_i$ of the SPP line sources. In this study, ${{\displaystyle d}}_1={{\displaystyle d}}_2={{\displaystyle d}}_{}$ is selected; after substituting it into Eq. (4), we obtained:

Figure 2 shows the result of plotting two-dimensional interference patterns generated between four slits using an analytical solution Eq. (5) when the four SPP line sources are in different phase relationships, that is, have different $\left({{\displaystyle \varphi }}_1+{{\displaystyle \varphi }}_3\right)-\left({{\displaystyle \varphi }}_2+{{\displaystyle \varphi }}_4\right)$ values. Figure 2(A) shows the case where the SPP interference field is the bright spot array when $\left({{\displaystyle \varphi }}_1+{{\displaystyle \varphi }}_3\right)-\left({{\displaystyle \varphi }}_2+{{\displaystyle \varphi }}_4\right)$ = 0, and the SPP interference field in Fig. 2(E) shows as the dark spot array when$\left({{\displaystyle \varphi }}_1+{{\displaystyle \varphi }}_3\right)-\left({{\displaystyle \varphi }}_2+{{\displaystyle \varphi }}_4\right)=\pi$is satisfied. The interference field shown in Figs. 2(B)-2(D) are transition SPP interference fields from the bright spot array to the dark spot array with the phase constant values $\left({{\displaystyle \varphi }}_1,{{\displaystyle \varphi }}_2,{{\displaystyle \varphi }}_3,{{\displaystyle \varphi }}_4\right)$of the four SPP line sources are (0, 0, 0, 0), (π/8, 0, π/8, 0), (2π/8, 0, 2π/8, 0), (3π/8, 0, 3π/8, 0), and (4π/8, 0, 4π/8, 0), respectively.

 

Fig. 2 The SPP interference field between the four metal slits when four ideal SPP line sources are in different phase relationships: (A) SPP interference bright spot array; (B) - (D) several transitional SPP interference fields between the SPP bright spot array and the SPP dark spot array; (E) SPP interference dark spot array. The (ϕ₁, ϕ₂, ϕ₃, ϕ₄) values of Figs. (A)-(E) (0, 0, 0, 0), (π/8, 0, π/8, 0), (2π/8, 0, 2π/8, 0), (3π/8, 0, 3π/8, 0) and (4π/8, 0, 4π/8, 0), respectively.

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By synthesizing the above two points, it can be seen that while maintaining $\left({{\displaystyle \varphi }}_1+{{\displaystyle \varphi }}_3\right)-\left({{\displaystyle \varphi }}_2+{{\displaystyle \varphi }}_4\right)$ as a specific constant, the lateral shift of a fixed SPP interference pattern between four slits can be achieved by changing the phase constant value ${{\displaystyle \varphi }}_i$($i=1~4$) of the four SPP line sources from the four slits. It is known that the phase constant value of the four SPP line sources can be changed by adjusting the phase of the four fundamental Gaussian beams incident on four slits. That is, by adjusting the phase values of the four incident fundamental Gaussian beams, the SPP interference pattern between the four slits can be laterally moved arbitrarily.

The method of moving the two-dimensional SPP interference pattern by adjusting the phases of the incident light beams is also suitable for the movement of one-dimensional SPP interference fringes. When only two incident fundamental Gaussian beams are used to excite the SPPs in two lateral (e.g. x-direction) slits, only two SPP line sources propagating along the y-direction exist. Equation (3) reduces to:

The SPP interference field between metal slits is obtained by simplifying Eq. (5):

3. Simulation results and discussion

This study employs simulations to verify the proposed method for the SPP interference pattern shifting and to demonstrate the particle manipulating using the control of the SPP interference pattern. This section contains two parts. The first part demonstrates the method of shifting SPP interference patterns by changing the phase of each incident fundamental Gaussian beam. The second part demonstrates manipulating particles by laterally shifting the SPP interference array patterns that resulted from the first part.

In the first part of the simulation calculation, the finite-difference time-domain (FDTD) method [34] was used to simulate the SPP excitation of the incident light on the four-slit metal structure and the propagation and interference of the SPPs between four slits. The FDTD method obtains the spatial distribution of the electromagnetic fields at a point in specific time by discretizing the Maxwell’s time-domain equations in space and time domains, and has been widely used in simulations of various types of electromagnetic-related academic research and engineering applications. The FDTD simulation program in this paper is a self-written Matlab program. After verifying the correctness of the calculation program, it was used to simulate various surface plasma distributions in this study. The parameters used in the calculation are as follows: the incident light wavelength is 1064nm, the incident Gaussian beam waist width w₀ is 3μm, and the beam spacing along two axes, D, is 6μm. The material of the transparent substrate is SiO₂. The material of the metal is Ag, where the dielectric coefficient of Ag used in this study is −58.488 + 1.172i [35]. The calculated wavelength of the SPPs generated at the Ag/air interface is ${\lambda }_{spp}\approx 1.055\mu m$. The SPP propagation length $L_{spp}\approx 482\mu m$. The length and the width of the slits are 6 μm and 200 nm, respectively. The thickness of the Ag layer is set to 200 nm, which is sufficient to prevent the incident light from transmitting through the metal and interfering with the SPPs.

First, the SPP interference field generated by only two incident fundamental Gaussian beams at two lateral metal slits is discussed. That is, the light field of Eq. (6) is used to verify the correspondingly generated one-dimensional SPP interference fringes described by Eq. (7). In the simulation calculation, each of the two incident Gaussian beams is perpendicularly incident on the center of a metal slit, and the spatial relationship between the incident beam and the metal slit structure is shown in Fig. 3(A). Figures 3(B)-3(F) plot the area within the green frame in Fig. 3(A), which is the resulting SPP interference field when adjusting the phase term ${{\displaystyle \varphi }}_i\text{'}$ of two incident Gaussian beams with the values listed in Table 1. The phase values of the two light sources listed in Table 1 correspond to the case where the phase constant difference between the two SPP line sources induced, $\Delta {{\displaystyle \varphi }}_y={{\displaystyle \varphi }}_2-{{\displaystyle \varphi }}_4$, is 1.0π, 0.75π, 0.5π, 0.25π, and 0.0π, respectively. The phase constant differences of two SPP line sources $\Delta {{\displaystyle \varphi }}_y$ are marked on the upper right of Figs. 3(B)-3(F).

 

Fig. 3 (A) the schematic diagram of the relationship between the two incident Gaussian beams; (B)-(F) SPP interference fields between the slits when the phases of two incident Gaussian beams are as the values listed in the columns (B)-(F) of Table 1, respectively. The translucent white box in Fig. (A) shows the four metal slits, and the white double arrows indicate the polarization direction of the incident light. The range of the displayed SPP field in Figs. (B)-(F) is the area within the green box in Fig. (A). The Δϕ_y values in the upper right corner of the five pictures shows the difference of the phase constant of the two SPP line sources, ϕ₂ - ϕ₄, in Eq. (7). The white dashed line shows the line y = 0 in the defined coordinate of Fig. 1(A), which is plotted for convenient observing of the movement of the SPP interference field.

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Table 1. Phase values of two incident Gaussian beams used in a simulation calculation when two Gaussian beams are used to generate SPP interference fringes between slits. The values listed in columns (B)-(F) of the table correspond to the phase values of two Gaussian beams used in the simulation results in Figs. 3(B)-3(F), respectively.

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If the SPP phase constant difference $\Delta {{\displaystyle \varphi }}_y$ is represented with the phases of the corresponding incident Gaussian beams, ${{\displaystyle \varphi }}_2\text{'}$ and ${{\displaystyle \varphi }}_4\text{'}$, it will be $\Delta {{\displaystyle \varphi }}_y=\pi +{{\displaystyle \varphi }}_2\text{'}-{{\displaystyle \varphi }}_4\text{'}$ [27]. The phases of the induced SPP line sources are correlated with the phases of the incident beams. However, due to the opposite sign of the charge distribution induced by the incident light on two opposite sides of a slit, an extra $\pi$ phase difference is introduced between the two SPP line sources that travel toward each other. The research results in [3] show an illustration of this phenomenon. When using a single linearly polarized plane wave incident on two slits (the entire light field is in phase), the SPP interference fringes generated between the two slits appear as interference patterns resulting from two SPP line sources with $\pi$ phase difference, rather than the situation resulting from two SPP line sources with the zero phase difference. The phase values in Table 1 reflect that when the phase of the Gaussian beam 2, ${{\displaystyle \varphi }}_2\text{'}$, is of a fixed value 0, gradually changing the phase value ${{\displaystyle \varphi }}_4\text{'}$ of the Gaussian beam 4 will cause a phase constant difference of two resulting SPP line sources: $\Delta {{\displaystyle \varphi }}_y=\pi +{{\displaystyle \varphi }}_2\text{'}-{{\displaystyle \varphi }}_4\text{'}$ decreases. It corresponds to the FDTD simulation results in Figs. 3(B)-3(F): the upward movement of the SPP interference fringes between two metal slits. That is, the fringes gradually shifted from the case where the center is dark lines to the case where the center is bright lines. In the simulations, no matter whether the metal structure adopted is four slits or two slits, the SPP interference results obtained will be the same because only the part of the incident light whose polarization direction is perpendicular to the slits can excite the SPPs [25,26]. Therefore, two incident Gaussian beams of y-direction polarization cannot excite SPPs in the other two longitudinal slits.

Next, the SPP interference field generated by the incident four fundamental Gaussian beams in four metal slits is discussed. That is, the light field of Eq. (3) is used to verify the correspondingly generated two-dimensional SPP interference array described by Eq. (5). Here we show the case of generating a bright spot interference array, so controlling the phase relationship between the four beams is fixed to satisfy $\left({{\displaystyle \varphi }}_1+{{\displaystyle \varphi }}_3\right)-\left({{\displaystyle \varphi }}_2+{{\displaystyle \varphi }}_4\right)=\text{0}$. Here, the relationship of the phase constant of four SPP line sources is set as $\left({{\displaystyle \varphi }}_1+{{\displaystyle \varphi }}_3\right)\text{=}\left({{\displaystyle \varphi }}_2+{{\displaystyle \varphi }}_4\right)=\text{2}\pi$. That is, the four incident Gaussian beams need to satisfy the phase relationship $\left({{\displaystyle \varphi }}_1\text{'}+{{\displaystyle \varphi }}_3\text{'}\right)\text{=}\left({{\displaystyle \varphi }}_2\text{'}+{{\displaystyle \varphi }}_4\text{'}\right)=\pi$. From Eq. (5), it can be seen that under this condition, $\left({{\displaystyle \varphi }}_1+{{\displaystyle \varphi }}_3\right)\text{=}\left({{\displaystyle \varphi }}_2+{{\displaystyle \varphi }}_4\right)=\text{2}\pi$, the exponential term of Eq. (5) will be 1, so the two sets of one-dimensional interference fringes produced by two of four metal slits are in-phase superimposed. In the simulation calculation, each of the four incident Gaussian beams is perpendicularly incident on the center of a metal slit, and the spatial relationship between the incident beam and the four-slit structure is shown in Fig. 4(A). Figures 4(B)-4(F) plot the area within the green frame in Fig. 4(A), which are the resulting SPP interference field when adjusting the phase term ${{\displaystyle \varphi }}_i\text{'}$ of the four incident Gaussian beams with the values listed in Table 2. The phase values of the four light sources listed in Table 2 correspond to the case where the phase constant difference of four SPP line sources induced ($\Delta {{\displaystyle \varphi }}_x={{\displaystyle \varphi }}_1-{{\displaystyle \varphi }}_3$, $\Delta {{\displaystyle \varphi }}_y={{\displaystyle \varphi }}_2-{{\displaystyle \varphi }}_4$), are (2.0π, 2.0π), (1.5π, 2.0π), (1.5π, 1.5π), (1.0π, 1.5π) and (1.0π, 1.0π), respectively. The phase constant differences of the four SPP line sources ($\Delta {{\displaystyle \varphi }}_x$, $\Delta {{\displaystyle \varphi }}_y$) are marked on the upper right of Figs. 4(B)-4(F).

 

Fig. 4 (A) The schematic diagram of the relationship between the four incident Gaussian beams; (B)-(F) SPP interference fields between the slits when the phases of four incident Gaussian beams are as the values listed in the columns (B)-(F) in Table 2, respectively. The translucent white box in Fig. (A) shows the four metal slits. The range of the displayed SPP field in Figs. (B)-(F) is the area within the green box in Fig. (A). The Δϕ_x and Δϕ_y values in the upper right corner of the five pictures show the difference of the phase constant between two of the four SPP line sources in Eq. (5), i.e., ϕ₁ - ϕ₃, and ϕ₂ - ϕ₄, respectively. The two white dashed lines show the lines y = 0 and x = 0 in the defined coordinate of Fig. 1(A), which are plotted for convenient observing of the movement of the SPP interference bright spot array.

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Table 2. Phase values of four incident Gaussian beams used in a simulation calculation when four Gaussian beams are used to generate an SPP interference bright spot array between slits. The values listed in columns (B)-(F) of the table correspond to the phase values of two Gaussian beams used in the simulation results in Figs. 4(B)-4(F), respectively.

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If the SPP phase constant differences ($\Delta {{\displaystyle \varphi }}_x$,$\Delta {{\displaystyle \varphi }}_y$) are represented with the phases of the corresponding incident Gaussian beams, ${{\displaystyle \varphi }}_{\text{1}}\text{'}$, ${{\displaystyle \varphi }}_{\text{2}}\text{'}$, ${{\displaystyle \varphi }}_{\text{3}}\text{'}$ and ${{\displaystyle \varphi }}_{\text{4}}\text{'}$, it will be$\Delta {{\displaystyle \varphi }}_x=\pi +{{\displaystyle \varphi }}_1\text{'}-{{\displaystyle \varphi }}_3\text{'}$ and $\Delta {{\displaystyle \varphi }}_y=\pi +{{\displaystyle \varphi }}_2\text{'}-{{\displaystyle \varphi }}_4\text{'}$ [27]. What facts that Figs. 4(B)-4(F) and the phase values listed in the Table 2 show are addressed as follows. When the phase value of the four incident Gaussian beams have the values of column (B) of Table 2, the phase constant difference of the four SPP line sources will be $\Delta {{\displaystyle \varphi }}_x={{\displaystyle \varphi }}_1-{{\displaystyle \varphi }}_3=\pi +{{\displaystyle \varphi }}_1\text{'}-{{\displaystyle \varphi }}_3\text{'}=2.0\pi$ and $\Delta {{\displaystyle \varphi }}_y={{\displaystyle \varphi }}_2-{{\displaystyle \varphi }}_4=\pi +{{\displaystyle \varphi }}_2\text{'}-{{\displaystyle \varphi }}_4\text{'}=2.0\pi$. From Eq. (5), it can be found that in this case, the electric field intensity at the center of the SPP interference field (x = 0, y = 0) is the maximum [see Fig. 4(B)]. That is, the center position of the SPP interference field at this time is a bright spot. In this bright spot array, the distance from the adjacent dark spots from a bright spot on the x-axis or y-axis is $\pm {\lambda }_{spp}/2$.

To move this bright spot interference array, it is only necessary to adjust the phase of the four incident light beams, ${{\displaystyle \varphi }}_{\text{1}}\text{'}$, ${{\displaystyle \varphi }}_{\text{2}}\text{'}$, ${{\displaystyle \varphi }}_{\text{3}}\text{'}$ and ${{\displaystyle \varphi }}_{\text{4}}\text{'}$. When the phase of the incident light beam is modulated such that a decrease in ${{\displaystyle \varphi }}_{\text{2}}\text{'}$ is accompanied by an increase in ${{\displaystyle \varphi }}_{\text{4}}\text{'}$, the $\Delta {{\displaystyle \varphi }}_y$ can be reduced while maintaining the phase relationship $\left({{\displaystyle \varphi }}_2\text{'}+{{\displaystyle \varphi }}_4\text{'}\right)=\pi$. At this time, the center bright spot of the SPP interference array will move in the + y direction while maintaining the SPP interference pattern unchanged. Similarly, when the phase of the incident light beam is modulated such that a decrease in ${{\displaystyle \varphi }}_{\text{1}}\text{'}$ is accompanied by an increase in ${{\displaystyle \varphi }}_{\text{3}}\text{'}$, the $\Delta {{\displaystyle \varphi }}_x$ can be reduced while maintaining the phase relationship $\left({{\displaystyle \varphi }}_1\text{'}+{{\displaystyle \varphi }}_3\text{'}\right)=\pi$. At this time, the center bright spot of the SPP interference array will move in the + x direction while maintaining the SPP interference pattern unchanged. Instead, the SPP interference array can be moved in the opposite direction. In other words, the proposed method can use the adjustment of the phase difference between four incident Gaussian beams to move the entire SPP bright spot array under the condition of a fixed SPP interference pattern. It shows great potential that the method of generating the SPP bright spot array and movement of SPP interference pattern proposed in this study will be applicable to near-field multi-particle capture, manipulation, and microfluidic particle sorting.

In this study, the slit length was set to a small value of 6 μm due to the limited ability of the computer. In the experiment, the number of SPP bright/dark spots and the range of SPP interference field distribution can be increased by increasing the slit length. It is a good way to estimate the range limit of the SPP interference field generated by this method using the quantity of SPP propagation length $L_{spp}$. This is because the variation in the attenuation of the field intensity due to the propagation of a SPP source in the interference field is in the form of $\exp \left(-r_{\max }/L_{spp}\right)$, where r_max denotes the maximum distance that a SPP line source can propagate in the interference field. For an interference field with a square boundary, the value of r_max is $d$. It is found that the longer the slit length, the greater the interference field distribution will deviate from that described by the predicted SPP interference field, Eq. (5). However, we can see that even when the slit length is one-twentieth of $L_{spp}$ (i.e., $24.1\mu m$), the attenuation of the SPP field strength due to propagation throughout the interference field is still not high (i.e., less than 5%). The large range of SPP bright/dark spot interference fields generated by this study can enlarge the working range of SPP applications, such as the particle trapping [18], manipulation [19], microfluidic particle sorting [12,13,15], and etc.

Next, how to use dynamic particle manipulation by modulating the incident light phase to laterally shift the excited SPP interference bright spot array will be demonstrated by simulation calculations. In the simulation, the phase of the incident light is changed over time to control the bright spot position of the SPP interference bright spot array, thereby controlling the position of the particles. The situation discussed here is that the particles to be manipulated are in a static liquid with a perturbation of Browning motion, and the particle size is in the sub-micron scale. At this time, the viscous force affected by the particles is greater than that of the inertial force, so the light field can effectively control the particles. In this study, the recursive relationship between the position of a particle and the force it was subjected to was derived by the Langevin's equation [36–38]:

The calculation of the force for the particles situated in a fixed SPPs interference field (i.e., the phase difference between the four SPP line sources is fixed at this time) is as follows. Since the trapping particles are in the Rayleigh region and the SPPs interference field is a stable standing wave, the transverse force acting on the trapping particles to be considered is only the gradient force [40,41]. The long-time average gradient force of particles due to a fixed SPPs interference field is,

In the simulation, the design of the phase change of the four incident beams is expected to cause the particles to follow the path of the English letters “NCKU”. Figure 5 shows the simulation results. In the figure, the white object indicates the particle, the blue line shows the trajectory of the manipulated particle, and background shows the SPP interference field at this moment. The particle trajectory in Fig. 5 is not very smooth mainly because the size of the particles used in the simulation is small and thus the D value is large. That is, the particles are subjected to a strong perturbation. If a larger particle is used in a simulation, the trajectory of the particle will be smoother.

 

Fig. 5 Simulated trajectories of a single particle in the SPP interference bright spot array as the incident light field phase changes over time. The white circle indicates the particle and its position, and the blue line in the picture indicates the experienced trajectory of the particle at the time. Figures (A)-(H) plot the particle experienced trajectories at different times and the SPP light field distribution at different time. The time values are marked in the lower right corner of each picture. They are 0.0, 13.0, 26.0, 39, 52.0, 65.0, 78.0, 92.0 ms, respectively.

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The resultant SPP interference bright spot array can simultaneously perform multi-particle manipulation. Figure 6 plots the trajectories of two particles controlled by the SPP interference field. Although the SPP fields surrounded by two particles are the same in time, the trajectories of the two particles are slightly different because the random Brownian motion of the two particles differs with time. The two-particle path difference will decrease as the particle becomes larger and the influence of the Brownian motion becomes smaller. This simulation result confirms the method of laterally shifting the SPP interference field proposed in this paper can be applied to near-field multi-particle capture and manipulation. The precision of manipulating the particles by modulating the phase of the incident light field will depend on the phase modulation ability of the incident light field that excites the SPPs. For example, if a dual Mach-Zehnder interferometer is used to generate the incident light field [30], the phase adjustment ability of the incident light field will depend on the minimum movement step of an electronic motorized stage. If the SLM is used to generate the incident light field [31], the phase adjustment ability of the incident light field will depend on the minimum phase change value that the SLM can modulate.

 

Fig. 6 Simulated trajectories of two particles in the SPP interference bright spot array as the incident light field phase changes over time. The white circle indicates two particles and their positions. The blue and orange line in the picture indicates the experienced trajectories of two particles at the time. Figures (A)-(H) plot the particle experienced trajectories at different time and the SPP light field distribution at different times. The time values are marked in the lower right corner of each picture. They are 0.0, 13.0, 26.0, 39.0, 52.0, 65.0, 78.0, 92.0 ms, respectively.

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The application of the fundamental Gaussian beam to the optical trapping has been well investigated [41]. The comparison of two light fields of a fundamental Gaussian beam and a SPP bright spot, provides an estimate of the ability to manipulate the particle using the SPP bright spot. Figure 7 shows the intensity pattern of a SPP bright spot and a diffraction-limited Gaussian beam of the same wavelength. Here, the beam waist of the Gaussian beam is set as the diffraction limit, Rayleigh criterion $\Delta =1.22\lambda /N_A$[42], and the numerical aperture value $N_A$ is set as 1. It can be clearly seen from Fig. 7 that the SPP bright spot has better spatial locality than the Gaussian beam. Besides, the SPP bright spot has a steeper slope of field intensity with position and thus capable of providing greater optical trapping force (i.e., the gradient force). The slope ratio of the two curves in Fig. 7 at the FWHW (full width at half maximum) of the two light fields is about 2.4. That is, in this case, the gradient force provided by the SPP bright spot will be roughly 2.4 times the gradient force of a diffraction-limited fundamental Gaussian beam. From the above discussion, it can be seen that when two light fields have a same light field intensity, a SPP bright spot produced by this study can provide a greater capture force than what a diffraction-limited Gaussian beam can provide and can more accurately capture particles in space.

 

Fig. 7 The intensity distribution of a fundamental Gaussian beam and a bright spot in the SPP interference field.

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4. Conclusion

This study presents a method for dynamically shifting the SPP interference field by dynamically controlling the phases of the multiple incident beams. That is, a four-slit structure is used, and a plurality of Gaussian beams is used to induce the SPP line source on the metal surface. The correlated phase relationship between the phases of the SPP line sources and the incident light beam phases makes the successful dynamic shifting of the SPP interference pattern with the adjusting of the phases of incident beams. This study theoretically analyzes the physical mechanism of generating and manipulating the 1D SPP interference fringes and the 2D SPP interference bright spots array. Then, the FDTD method was used to demonstrate and verify that the proposed method can dynamically control the aforementioned SPP interference field through the phase adjusting of multiple incident Gaussian beams while maintaining the SPP interference field pattern. The Langevin equation was then used to demonstrate the optical manipulations of single and multi-particles in a static liquid using the dynamic laterally movable SPP interference bright spot array. The method proposed in this paper not only can dynamically control the SPP interference bright spot array, but also the SPP dark spot array or any other transitional SPP interference field between two kinds of SPP interference fields. Therefore, the results of this study also have the potential to be applied to SPP dark capture applications. The dynamic laterally movable SPP interference obtained in this study shows great potential for applications such as biological or nanoparticle capture and manipulation, plasmonic structured illumination microscopy, and other SPP applications.