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Abstract
In symmetric nano/micro metal slit structures, interference patterns are produced by counter-propagating surface plasmon polaritons (SPPs) in the the center of structures, which can be employed to improve the resolution of microscopy and surface etching and to realize particle trapping. This paper focuses on the shift of the SPP interference patterns in the symmetric arc slit structures. The excitation models with one incident beam and two incident beams are established and analyzed respectively, and methods to shift the SPP interference patterns via adjusting the tilt angle and initial phase of the excitation beams are compared. The FDTD simulation results show that these methods can precisely shift the SPP interference patterns in the symmetrical arc slits. Compared to the linear slits, the SPP waves arising from arc slits are more strongly focused, resulting in a stronger gradient force. The characteristics of stronger focus and dynamic shifting of the focal spot give the symmetric arc slit structure unique advantages in the capture and transfer of the Rayleigh metallic particle.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Surface plasmon polaritons (SPPs) are transverse-magnetic (TM) polarized electromagnetic modes propagating along the interface between a dielectric and a metal [1]. As a special kind of surface wave, they have the characteristics of local enhancement, shorter effective wavelength and abrupt phase change [1,2]. These intriguing characteristics lead them been extensively studied and utilized in the new and exciting applications, such as sub-wavelength optics [3–5], nanolithography [6,7], particle trapping and manipulation [8–12], surface plasmon resonance biosensor [13–14], and plasmonic microscopy [15–21]. The current research interests about SPPs include enhance the excitation efficiency through dispersion engineering and wavefront shaping [22,23], control the propagating direction through changing the geometry or the polarization and wavelength of the excitation beam [24–28], control the interference patterns by adjusting the polarization and phase of the incident light or forming the inhomogeneity boundary by the external electric field [28–32], etc. However, the shift of the SPP interference patterns is also important in many applications. For example, in plasmon structure illumination microscopy, the high frequency information is obtained through shifting the SPP interference fringes to improve the resolution of the microscopic imaging [15–21]. In addition, metal particles are manipulated through shifting the SPP interference focus [8–10]. Therefore, dynamically moving of the SPP interference patterns in symmetric micro/nano structures is significant for the application of SPP waves.
A pair of counter-propagating SPP waves can be generated through exciting two symmetric micro/nano metal slits, generating a SPP interference pattern in the center of the structure. Some more complicated symmetrical structures including squares and hexagons can be decomposed into multiple sets of symmetrical slits. Therefore, two symmetrical slits are the basic structure for study the shift of the SPP interference pattern. The SPP waves generated by arc slits are strong focused, and the corresponding focus can be used as optical tweezers. Particle capture has been achieved in many ways, however the capture and manipulation of Rayleigh metallic particles remains a challenge. Rayleigh metallic particles are the metal particles whose size is much smaller than the wavelength of the laser [33]. Due to its properties of smaller size and stronger absorption and scattering, the gradient force exerted on it is reduced while the radiation pressure is relatively increased. Therefore, to enhance the gradient force is the essential to realize the capture of Rayleigh metal particles. Compared with the linear slit structure, the peak intensity for the focus generated by the four arc slit structure is nearly 4.3 times improved. As a result the gradient force used to trap particles is greatly increased. This paper focus on the shift of the SPP interference pattern based on the symmetric arc slit structures, and its application for trapping and manipulation of Rayleigh metallic particles.
2. Principle analysis
Like the interference of beams in the free space, the SPP interference fringes can be shifted through adjusting the phase difference between two SPP waves [10,15]. However, the phase difference between two SPP waves can only be obtained by the phase difference of excitation beams indirectly. In the following, two models that symmetric arc slits are excited by single and two incident beams will be established and analyzed, respectively.
Figure 1(a) is the three-dimensional diagram of symmetric arc slits. Figures 1(b) and 1(c) are the longitudinal sections along the white dotted line in Fig. 1(a), indicating two slits are illuminated by single and two plane waves. The charge distributions on two sides of the slit are reverse, thus an inherent phase difference π is existed between two counter-propagating SPP waves. Moreover, when the structure is illuminated by single tilt beam as shown in Fig. 1(b), the phases of the excitation beam at two slits are different, thus an extra phase difference is introduces to the SPP waves [34]. Therefore, when two arc slits are illuminated by single beam, the phase difference between SPP waves in the center of the structure is expressed as
Fig. 1. The excitation of symmetric arc slits. The structure is show in (a). Longitudinal sections depict two slits are excited by one incident beam (b) and by two incident beams (c), along the white dotted line in (a).
In the other case, two arc slits are excited by two incident beams respectively, as depicted in Fig. 1(c). It is clear, except for the tilt angle of the excitation beams, the SPP interference pattern is also affected by the initial phase, polarization direction and height of the excitation beams. Because of phase accumulation during the propagation, the phase difference between two excitation beams when reach corresponding slits is expressed as
As showed in Eq. (3), the phase difference between two SPP waves is determined by the tilt angle, initial phase, polarization direction and height of the excitation beams together. In practical, however, it is difficult to control the tilt angle and the height of two excitation beams simultaneously, thus suppose two slits are excited vertically and the polarization directions are outward, i.e. $\textrm{(}{\phi ^{\prime}_1}\textrm{ - }{\phi ^{\prime}_\textrm{2}})\textrm{ = }\pi$, Eq. (3) can be simplified as
So in the case of two excitation beams, it is more flexible and reliable to control the shift of the SPP interference pattern through adjusting the initial phase of the excitation beams, while keep the tilt angle and polarization fixed.
Moreover, as general models, when beams illuminate from the front side at any angle or from back side at an angle close to 90 degrees, the influence of the substrate is negligible. And this paper focuses on the influence of the incident beams on the SPP interference. So in order to simplify the analysis, the influence of the refractive index of the substrate is not considered.
3. Simulation results and discussions
In this section, the finite-difference time-domain (FDTD) simulation is employed to demonstrate the methods proposed above. Two symmetric arc slits with radius of 3 um, width of 100 nm, and central angle 60° are fabricated in 120 nm Ag film. Excitation beam with wavelength of 532 nm and TM polarization are used to illuminate the structure from the back. The intensity distribution of the interference patterns is plotted at 20 nm above the silver surface.
3.1 Shift the SPP interference patterns when two slits are excited by single beam
The exciting scheme that two arc slits are illuminated by single beam is depicted in Fig. 2(a). The polarization direction is indicated by blue arrow, the source area and monitor area are indicated by green rectangle and red frame, respectively. The intensity distributions of electric field are showed in Figs. 2(b)–2(f), in which the red color and blue color indicate the maximum and minimum intensity. It is obviously, when tilt angle is zero, dark fringe appears in the center due to the destructive interference between two SPP waves. If tilt angle is grows at the pace of $\textrm{1}\textrm{.27}^\circ$, the SPP interference fringes move down at the step of 0.25 cycle. When tilt angle is $\textrm{2}\textrm{.54}^\circ$, bright fringe appears in the center because of the constructive interference. It is demonstrated, that SPP interference fringes can be precisely shifted through changing the tilt angle of the excitation beam, when two slits are excited by single beam.
Fig. 2. Two arc slits are excited by single beam (a). The SPP interference pattern is shifted through changing the tilt angle of the incident beam (b-f). White dashed line indicates x = 0.
It is worth noting, when the tilt angle is utilized to control the shifting of the SPP interference fringes, the angle is the smaller the better. Because when the tilt angle is increased, the predicted value will deviate from the actual due to the influence of the SiO2 substrate. In practice, however, this influence can be eliminated or minimized with the use of index matching liquid. Besides, when the tilt angle is increased, the in-plane vector used to excite the SPPs will decrease, resulting in a decrease in excitation efficiency.
3.2 Shift the SPPs interference pattern when two slits are excited by two beams respectively
According to Eq. (3), when supposing h is 2λ, two beams with outward polarization direction are utilized to excite two slits respectively, the SPP interference pattern can be moved through adjusting the tilt angles or initial phases of the incident beams. Figure 3(a) indicates two slits are illuminated by two beams respectively. According to Eq. (4), when the initial phase ${\phi _\textrm{1}}$ is fixed to zero, if ${\phi _\textrm{2}}$ is increased from 0 to 2π, increasing by 0.5π every time, the interference fringes move downward, the corresponding movement speed is 0.25 cycles each time, as shown in Figs. 3(f) to 3(j). Therefore, the SPP interference fringes can be moved through adjusting the initial phase of two excitation beams while keep the vertically illumination.
Fig. 3. Two slits are excited by two incident beams (a). The SPP interference pattern is shifted by changing the tilt angle of the incident beams (b-e). The SPP interference pattern is shifted by changing the initial phase of the incident beams (f-j). White dashed line indicates x = 0.
On the other hand, the SPP interference fringes can by moved via changing the tilt angles of two excitation beams while keep the same initial phase, which is shown in Figs. 3(b)–3(e). When ${\theta _\textrm{1}}\textrm{ = }{\theta _\textrm{2}}\textrm{ = 0}^\circ$, $\Delta {\phi _{\textrm{spp}}}\textrm{ = 2}\pi$, the maximum intensity appears in the center. When ${\theta _\textrm{1}}\textrm{ = 0}^\circ$ and ${\theta _\textrm{2}}\textrm{ = 29}^\circ$, $\Delta {\phi _{\textrm{spp}}}\textrm{ = 2}\textrm{.5}\pi$, fringes move up 0.25 cycle. When ${\theta _\textrm{1}}\textrm{ = 0}^\circ$ and ${\theta _\textrm{2}}\textrm{ = 41}^\circ$, $\Delta {\phi _{\textrm{spp}}}\textrm{ = 3}\pi$, fringes move up 0.5 cycle and minimum intensity appears in the center. But it is also revealed that the intensity distribution is not symmetric about x = 0. This is because the SPP is excited by the in-plane vector of the incident light. When the tilt angle is increased, the effective in-plane vector and corresponding SPP intensity is decreased. When the amplitudes of the SPP waves from two arc slits are different, the interference fringes are no longer symmetric in the center. Based on these analyses, the tilt angle is not recommended to control the shifting of the SPP interference when two arc slits are illuminated by two incident beams.
The simulation sections 3.1 and 3.2 proved that the precise movement of the SPP interference fringes can be realized no matter single-beam excitation or dual-beams excitation. For comparing, the single-beam excitation is simple, but oblique illumination will affect the excitation efficiency and the tile angle must be controlled accurately; the dual-beams excitation method requires two lasers or two special focal spots generated by phase modulation, however, the interference fringe movement through changing the phase of the incident beams is more flexible and reliable.
3.3 SPP focus and its application to particle trapping
Figure 4(a) is transverse section of SPP intensity distribution arising from four symmetrical arc slit structure, and the transverse profile intensity of the focal spot is extracted and plotted in Fig. 4(b). Clearly, a prominent SPP focus is generated in the center of the structure. The overall intensity distribution of electric field is dominated by the axial component, while the lateral component is almost null. Normally, the optical trapping is based on the competition between the radiation pressure and the gradient force [35]. The gradient is proportional to the intensity gradient of the electromagnetic field and the radiation force is point to the propagating direction of light. However, in the SPP field, because the SPP waves are propagating along the surface pointing to the center and constrained to a nano-scale region in the vertical direction [8], so the SPP focus not only provides a strong gradient force along the vertical direction to attract the particle to the metal surface, but also provide a strong gradient force and a weak radiation force along the horizontal direction, which work together to push the particle to the center of the focus.
Fig. 4. Transverse section of the SPP interference pattern in four arc slit structure (a) and four linear slit structure (c). Transverse profile intensity of the focus in arc slit structure (b). The intensity comparison of focus generated by arc slits and linear slits with different effective length (d). White dotted frames in the center of the structures indicate the focal spots.
Symmetrical linear slits is another structure which can be used to generate surface plasma optical tweezers [10]. Figure 4(c) is corresponding transverse intensity distribution in the four linear slit structure. Obviously, the SPP focus is feeble in the linear slit structure. The transverse profile intensity of the focus generated by both structures is compared in Fig. 4(d). It is show, when the same effective length of slits is 3 μm, the peak intensity produced by arc slits can reach 13.8, while the equivalent from linear slits is only about 3.2. In additionally, when the effective length of arc slits is increased, the intensity of SPP focus increases significantly. But for linear slits, the intensity is hardly changed. Therefore, compared with the linear slit, the arc slit structure can provide stronger focus and stronger gradient force which will enhance the stability and efficiency in particle trapping.
To demonstrate the trapping for Rayleigh metallic particles, we analyze the optical forces acting on the particles. Metal particle has strong scattering and absorption, thus in the SPP field it subjects to gradient force, scattering force and absorption force. For a Rayleigh metallic particle with radius a, the gradient force can be expressed as [33,36–38]
Fig. 5. Optical forces exerted on a gold particle of 35 nm radius in the SPP field. (a) Transverse force along x-axis. (b) Axial force along z-axis.
First of all, it can be seen from Fig. 5(a) that the curve of the resultant force ${F_{x - t}}$ completely coincides with the curve of the gradient force ${F_{x - grad}}$ along the x-axis, while the radiation force ${F_{x - radi}}$ is nearly null. This is due to the Gaussian intensity distribution in the transverse plane for the SPP focus. In contrary, the SPP field as a standing wave pattern has almost zero net power flux, leading to negligible transverse radiation force. Thus according to the transverse force distribution, the normalized transverse trapping potential can reach a depth of 10.12 KBT/10 mW/m2. It indicates that the particle can be stably captured to the center. The gradient force distribution in the transverse direction guarantees that the particle can be captured to the center. Secondly, as shown in Fig. 5(b), the axial gradient force ${F_{z - grad}}$ is also much stronger than the axial radiation force ${F_{z - radi}}$. This is because the SPP wave decays exponentially in the direction perpendicular to the interface. The negative forces indicated that particles are attracted towards the interface. Considering the radius of the gold particle and the roughness of the interface, the axial force distribution calculation starts at 0.05 um away from the surface. To sum up, the force calculation clearly demonstrates that three dimensional (3D) stable particle trapping can be realized in the SPP field through exciting two pairs of symmetrical arc slits. This 3D optical trap can capture Rayleigh metallic particles to the center of the focus on the metallic surface. In comparison, the resultant force generated by the linear slit structure is also displayed with pink lines. Clearly the arc slits can provide stronger force to enhance the stability of the particle trapping when both structures have the same effective slit length under the same incident power.
3.4 Shift the SPP focus and its application to particle manipulation
Two pairs of arc slits excited by corresponding beams are showed in Fig. 6(a). Such a multiple-focus excitation can be realized by a vector beam generator [39], which tightly focusing specially designed complex field by high NA objective lens. The SPP interference pattern is a second interference by the horizon and vertical SPP interference fringes. It is clearly, when the structure is illuminated normally, the interference pattern can be modulated by moving one or two sets of SPPs interference fringes through adjusting the phase of the incident beams. As shown in Figs. 6(e)–6(h), when the vertical interference fringes are moved along the horizontal direction through adjusting the phase difference between two horizontal incident beams, the SPP focus in the center of the structure is shifted along the x axis. Similarly, the SPP focus can be shifted along the y axis through moving the horizontal SPP interference fringes along the vertical direction. Moreover, when both the horizontal and vertical interference fringes are moved simultaneously, the SPP focus moves in the directions of 45 degree or 135 degree, as shown in Figs. 6(b)–6(d). It is proved the SPP focus can be preciously shifted by the prediction models in the symmetric multiple arc slit structure, which is crucial to realize the movement and transmission of the particles.
Fig. 6. Four arc slits are excited by corresponding excitation beams (a). The SPP interference pattern is shifted through adjusting the initial phase of the excitation beams (b-h). Black dashed lines indicate x = 0 and y = 0.
The simulation sections 3.3 and 3.4 proved that a stable surface plasma optical tweezers is generated in the symmetrical arc slit structure and the dynamically shifting can be realized through adjusting the initial phase of the exciting beams. This optical tweezers can be used to pull the Rayleigh metallic particle to the center of the focus and manipulate the particle through dynamically moving the focus. Compared to linear slit, the optical tweezers generated by arc slit have higher stability and efficiency.
4. Summary
The dynamical shift of SPP interference patterns is significant in the applications of SPP waves. The methods to shift the SPP interference patterns in the symmetric arc slit structure are analyzed in this paper. In the case of single excitation beam, the interference pattern is affected by the tilt angle of the excitation beam and the radius of the arc slits. The tilt angle can be used to effectively control the shift of SPP interference fringes. In the other case of two excitation beams, the interference pattern is closely related to the polarization direction, tilt angle and initial phase of the excitation beams. Because the tilt illumination will decrease the excitation efficiency, therefore it is more feasible to moving the interference pattern through adjusting the initial phase of the excitation beams. The computed results by numerical software FDTD demonstrated that the SPP interference pattern arising from the symmetric arc slit structure can be precisely and dynamically shifted.
When two pairs of symmetric arc slits are illuminated by corresponding incident beams respectively, the SPP interference pattern can be shifted flexibly by moving one or two sets of SPP interference fringes. In additionally, compared with linear slits, the intensity of SPP focus generated by arc slits is strongly enhanced, resulting in higher gradient force. Strong transverse and axial gradient forces work together to produce a stable 3D optical trap in the vicinity of focus, drawing Rayleigh metal particles to the center of the focus on the metal surface. Moreover, the focus can be moved dynamically. Therefore, the SPP focus generated in the symmetric arc slit structure has stronger gradient force and flexible mobility. These advantageous make it have great potential application in particle trapping and manipulation.
Funding
Shanghai International Science and Technology (19060502500); National Natural Science Foundation of China (61775140, 61805142).
Disclosures
The authors declare no conflicts of interest.
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