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Abstract
The multicore fiber beam shaping devices based on surface plasmon polaritons (SPPs) have been proposed and demonstrated. The gold film is covered on the end face of the optical fiber. An air slit is perforated in the center of each core and the gratings with a fixed period are designed on the gold film on one side of the slit to obtain a deflected beam. Multiple deflected beams based on the multicore fiber interfere and form a periodic field, where the period of the interference field is determined by the deflection angle of the beams and the spacing between the cores. The interference field of the multiple deflected beams can be used to trap the nanosphere. The Maxwell stress tensor method is used to calculate the transverse and longitudinal trapping forces on a nanosphere. The nanosphere can be stably trapped at 45 μm away from the end face of the fiber. Such an all-fiber trapping system is compact and flexible integration, and is promising for long-working-distance and multiple-particle trapping.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Beam shaping techniques have been widely used due to unique light patterns and arbitrary light tailoring. So far, many methods have been proposed to realize optical beam shaping including slit-groove structures [1,2], holographic-based surface plasmon polaritons (SPPs) [3], microlens arrays [4], plasmonic and dielectric metasurfaces [5,6,7,8], as well as periodic structures showing negative refraction [9]. In particular, a considerable amount of efforts have been devoted to beam shaping devices by use of metal slit-grooves structures [10,11], since they can easily excite SPPs wave. The SPP, a kind of electromagnetic surface wave, can be generated by the collective oscillation of free electrons on the metal-dielectric interface, it can break through conventional diffraction limitation and its intensity is evanescent along the direction perpendicular to the interface [12]. Therefore, beam shaping devices based on SPPs have an exceptional potential in optical trapping applications [13,14] and optical micromanipulation [15–17]. In recent years, various structures including periodic circular holes [18], hybrid metal nanorod [19], double nanoholejavascript:; [20] and plasmonic dipole antennas [21] have been demonstrated to trap nanospheres. However, due to inherent losses of metal, the intensity of the diffraction field generated by SPPs rapidly decays along the propagation direction, thus the SPP wave only propagates in a quite short distance, which limits the trapping range of SPP-based optical tweezers [14]. The slab waveguide-based devices have large volumes and need complex optical systems. In addition, the oblique incident light will degrade the performance of the system, which makes the system difficult to integrate, especially for multiple incident beams. The tiny tip of the fiber is a promising platform for micro-nano device integration due to optical fiber's inherent flexibility, low loss, and long-distance transmission. The micro-prism [22,23], miniaturized microphone [24], grating array [25] have been fabricated on the end face of optical fiber to achieve optical tweezer, pressure sensor and beam shaping. Therefore, a compact and flexible all-fiber beam shaping device is more desirable for photonic integration.
In this work, a multicore-fiber beam shaping device based on SPPs has been demonstrated and the trapping ability of the proposed device has been investigated. Multiple deflected beams emitted from the multicore fiber end can interfere and form a periodic field, in which the period of the interference field was determined by the angle between the beams and the spacing between the cores. The interference field consisted of the multiple deflected beams can be used to trap the nanosphere. The transverse and longitudinal trapping forces are calculated. The results demonstrate that the interference field can achieve long-working-distance and multi-particle optical trapping. Compared with the slab-based plasmonic device, the slit-grating structures in the proposed device are easily integrated into the end of the single optical fiber. Therefore, the proposed device has a better stability and can eliminate the influence of oblique incident light on the device performance.
2. Principle and design of single deflected beam
The proposed all-fiber beam shaping device is shown in Fig. 1(a). Due to the excellent and stable optical properties, the gold nanostructure is selected to support the excited SPPs. A group of grooves with fixed width and depth are placed on one side of an air slit milled into the gold film on the end face of a single mode fiber (SMF). This configuration is developed from the previous works on the electromagnetic wave radiation by slit-groove structures [26–28]. The interface between the gold film and the end face of the fiber is defined as the x-y plane, the radiation field generated from the air slit propagates along the z-axis, in which the coordinate origin is the center of the core. The gold has a refractive index of -40.32 + 2.88i at 980 nm according to the Drude model [29]. Illuminated by perpendicularly incident TM-polarized light from the core of the optical fiber [30], the excited SPPs propagated along the interface between the air and the gold film are recoupled into the free space due to the existence of the grooves and become the transmitted light, as shown in Fig. 1(b). The phase of the radiation wave is dependent on the geometrical structure, particularly the period of the groove grating. When the deflection direction of the transmitted light is the same as the propagation direction of the SPPs wave, the deflection angle is defined as positive, otherwise, the deflection angle is defined as negative. To achieve an expected deflected beam with the deflection angle of θ, according to the geometric relationship and Huygens Fresnel principle, the phase ϕx of the recoupled radiation wave at the coordinate x satisfies the following equation:
Fig. 1. (a) The all-fiber deflected beam shaping device. (b)The slit-groove structure and schematic diagram of the light deflection. (c) The phase difference in the range from 0 to 10 μm. Solid line: analysis result from Eq. (1). Dots: the discretized phases introduced by the microgroove structure.
The wavevector of the SPP wave along the interface between the air and the gold film is ${k_{sp}} = {k_0}\sqrt {{\varepsilon _d}{\varepsilon _g}\textrm{/}({\varepsilon _d} + {\varepsilon _g})}$, where k0 represents the free space wavevector, ksp is the wavevector of the SPP wave on the corrugated surface, ɛd and ɛg are the relative permittivities of the air and the gold film, respectively. The SPP wave is recoupled to the propagating wave via the fixed period grating. Based on the diffraction principle, the relationship between the diffracted angle of the SPP and the grating period p can be expressed by the Eq. (2).
where θ denotes the diffraction angle. Arbitrary diffraction angle can be realized by varying the period. Therefore, the phase distribution introduced by the periodic structure is discrete. The red dots in the Fig. 1(c) represent the phases at the center of the grooves. There is the little difference between the analysis result and calculated value based on groove structure. The dependence of the deflection angle of the beam on the grating period are summarized in Table 1. The required grating period reduces with the increase of the deflection angle. Reducing the grating period leads to the increase of the grating vector (2π/p), thus results in a large beam deflection angle.
Table 1. The dependence of the deflection angle on the grating period
According to previous theoretical and experimental reports [10,22,31], the optimal geometric parameters of the slit-groove structure are determined. The thickness h of the gold film is 200 nm, the width w of the air slit is 130 nm, the depth of the groove is 80 nm, and the duty ratio of the grating is 0.5. In the simulation, 20 grooves are used and the thickness t of the optical fiber substrate is 200 nm, the core diameter of the 980 nm SMF is 8.9 μm and the NA is 0.14, the refractive indices of the core and cladding are calculated through the Sellmeier equation. The fundamental mode of the fiber has a Gaussian profile. In the case of the deflection angles of -20°, -30°, -45°and -60°, the diffraction field intensity distributions calculated by COMSOL Multiphysics based on the finite elements method (FEM) are shown in Figs. 2(a)-(d). In the calculation, the perfectly matched layers (PMLs) were applied on the boundary of the model. As expected, the SPPs are excited by the air slit, and then propagate along the grating surface, the diffracted optical field generated by the grating is confined at a specific angle and the off-axis beam is realized. The several side lobes appear near the main deflected beam, which can be explained as the diffracted light waves from the air slit may radiate into the free space or into another grooves. In this case, the light wave radiated into the grooves diffracts again, therefore some side lobes appear when they propagate in the free space. The absolute value of the deflection angle of the beam increases with the decrease of the grating period, but the transmission distance of the beam decreases. The far-field angle distributions of the deflected beams are shown in Fig. 2(e) for some designed deflection angles from 0 to -60° with the step of -10°. The calculated far-field angles are 1.25°, -8.27°, -18.3°, -27.8°, -37.8°, -46.4°, -55.9°. The difference Δθ between the calculated and designed angles is shown in Fig. 2(f). The Δθ is within 2° when the absolute value of beam deflection angle is less than 20°. However, when the absolute value of the beam deflection angle is greater than 20°, the Δθ increases with increasing the absolute value of the designed direction angle because of the diffraction effect and the depth of the grating. Δθ increases with increasing the grating depth, but the appropriate grating depth is necessary to recouple SPPs and obtain a high efficiency.
Fig. 2. (a)-(d) Field intensities |E|2 for different grating periods. The grating periods in (a)-(d) are 724, 648, 570, and 522 nm, respectively. (e) The far-field angles of the diffraction field of the slit-groove structure for various periods. (f) The differences between the designed and simulated angles.
3. Multiple deflected beams based on the multicore fiber
3.1 Symmetrical deflected beams based on twin-core fiber
The multiple deflected beam generators based on the multicore fibers, including twin- and four-core fibers, are further studied. Firstly, twin-core fiber based symmetrical deflected beams are considered. As shown in Fig. 3, the diameter of the core is 8.9 μm and the distance L between two cores is 36 μm. Two air slits are milled in the thin gold film at each core center of the twin-core fiber and the direction of the slits is perpendicular to the line where the two cores are located. Two groups of the gratings with the same period p are symmetrically distributed on the outer side of each slit. The geometric parameters of the slits and the gratings are the same as those in single-core fiber based devices.
Figure 4 shows the intensity distributions of the transmission field for deflection angles of 30°, 40°, and 45°, respectively. TM-polarized light with the same phase is injected into the two cores of the fiber, SPPs are excited in the two air slits. The deflected beams generated from the left and right cores transmit in the symmetry directions, and then intersect. The distance of the intersecting point from the fiber end decreases with increasing the deflection angle. They are 41, 28.3, and 24 μm for deflection angles of -30°, -40° and -45°, respectively. Due to the same polarized direction, two deflected beams interfere in the overlapping area. The period, the contrast of interference fringes, and the length of the overlapping region decrease with increasing the deflection angle. When the deflection angle is -30°, there are five obvious interference fringes with the period of 2 μm and the length of the overlapping region is ∼10 μm. When the deflection angle is -45°, there are nine interference fringes with the period of 700 nm and the length of the overlapping region is 4 μm. The field profile in the interference area is shown in Fig. 5 by the blue line, where the deflection angle is -30°.
Fig. 4. The field distributions of double deflected beams based on twin-core fiber for different deflection angles. (a) -30°, (b) -40° and (c) -45°.
3.2 Asymmetrical deflected beams based on twin-core fiber
Changing the initial phase difference between the two deflected beams, the field profile of the interference region can be adjusted. Within the interference region, the intensity profiles of two deflected beams with opposite directions for different initial phase differences (0, π/2, π, 3π/2, 2π) are shown in Fig. 5. The center fringe in the interference region is bright for the phase differences of 0 and 2π, while is dark for the phase difference of π. For other phase differences, the intensity profile is asymmetric. Within 0∼2π phase difference, the maximum intensity of the interference fringes first appears on the left side of the center line x=0, and then on the right side with increasing the phase difference.
3.3 Multiple deflected beams based on four-core fiber
Figure 6 presents the multiple deflected beams generator based on the four-core fiber. The four cores symmetrically distribute in the cladding, and the distance (L/2) of the core from the center of the optical fiber is 18 μm. Similar to the case of the twin-core fiber, four air slits are located in the center of the four cores and the groove gratings are arranged in the radial directions, respectively. The polarization direction of incident light in cores 1 and 3 is perpendicular to that in cores 2 and 4. The input optical powers of cores 1, 2, 3, and 4 are defined as P1, P2, P3 and P4 with the initial phase φ1, φ2, φ3 and φ4, respectively. Injecting the same power light with a wavelength of 980 nm into the four cores (P1=P2=P3=P4, φ1=φ2=φ3=φ4=0), the 3D FDTD simulation results for the deflection angles of -45° and -30° are shown in Fig. 7, where the PML boundary conditions were also used. Due to the symmetry, the intensity distribution of the x-z plane is the same as that in the y-z plane, therefore, only the intensity in the x-z plane is presented. The overlapping position of the four deflected beams coincides well with that for twin-core optical fiber and is ∼23.7 μm (39. 4μm) for the beam with -45° (-30°) deflection angle. The intensity in the x-y plane at z=39.4 μm is shown in Fig. 8 (a) for -30° deflection angle. Because the polarizations in the two directions (x and y) are orthogonal, there is just the superposition of field energy in the overlapping area. The field distribution is similar to the checkerboard grid with alternating bright and dark, and is odd symmetric. The center of the overlapping area is a brightest spot surrounded by eight sub-bright spots. The dark and bright spots are similar to rectangle and square, respectively.
Fig. 6. Schematic illustration of four deflected beam generator based on four-core fiber. The red arrows indicate the polarization direction of the light source.
Fig. 7. The intensities of the beam shaping device based on four-core fiber in x-z plane for the deflection angles of (a) -30°and (b) -45°.
Fig. 8. The electric field intensities of four-deflected beam generator with (a) symmetrical and (b) asymmetrical incident power in the x-y plane at z=39.4 μm. The incident power of cores 1 and 3 is twice that of cores 2 and 4 in (b). (c) Intensity profiles along the line y=-x at z=39.4 μm.
The field distribution in the overlapping region can be adjusted by changing the relative power of incident light in each core. When the incident power in cores 1 and 3 is twice that in cores 2 and 4, P1=P3=2P2=2P4 (φ1=φ2=φ3=φ4=0), the field intensity of four-deflected beam (θ=-30°) is also odd symmetric, as shown in Fig. 8(b). However, the contrast of the grid is changed and the dark spot is no longer square. The intensity profiles of the electric field along the line y=-x at the overlapping region are shown in Fig. 8 (c). Changing the ratio between the incident powers in four cores, the field profile of the overlapping region is unchanged, but the fringe contrast can be adjusted greatly.
The incident light with the same polarization direction can also excite the SPP wave in four air slits simultaneously, as shown in Fig. 9(a), where the angle α between the polarization direction of the incident light (red arrows) and the x axis is 45° (P1=P2=P3=P4, φ1=φ2=φ3=φ4=0). The field intensity in the x-y plane at z=39.4 μm is shown in Fig. 9(b), the pattern of multiple deflected beams in the overlapping region is approximately square. Although the center is a bright spot, it is not the brightest. The number of the brightest spots is 4 and they surround the center. The dark spots still remain in the same position as in Fig. 8(a). Therefore, the same polarization incident lights can excite SPPs to obtain grid pattern field, but the utilization efficiency of incident light is reduced. The contrast of grid-shaped intensity can still be adjusted by the phase difference between the beams. When the phase difference between cores 1, 3 and cores 2, 4 is π/2 (θ=-30°, P1=P2=P3=P4, φ1=φ3=0, φ2=φ4=π/2), the field distribution in the overlapping region is shown in Fig. 9(c). The center brightest spot and eight sub-bright spots can be seen again. Both bright and dark spots are approximately square. The duty cycle of the bright and dark spots is approximately equal. The profiles of the electric field along the line y=-x at z=39.4 μm are shown in Fig. 9(c). For comparison, the case of orthogonal polarization incident is also shown (red line). Through changing the polarization direction and the phase of the incident light, the profile of the intensity is similar to orthogonal polarization incidence, but the relative field intensity and the fringes contrast are significantly reduced.
Fig. 9. (a) The schematic diagram of four-deflected beam generator excited by the incident light with the same polarization angle (α=45°) and incident power (P1=P2=P3=P4). The electric field intensity in the x-y plane at z=39.4 μm for (b) φ1=φ2=φ3=φ4=0 and (c) φ1=φ3=0, φ2=φ4=π/2. (d) The intensity profile along the line y=-x. Blue line: φ1=φ2=φ3=φ4=0. Orange line: φ1=φ3=0, φ2=φ4=π/2. Red line: orthogonal polarization, φ1=φ2=φ3=φ4=0.
4. Trapping ability of the all-fiber multiple deflected interference beams
The optical trapping benefits from the gradient distribution of the light intensity. The gradient force is opposite to the direction of the light field gradient, while the scattering force pushes the particle toward the direction of light propagation and destroys the stability of optical capture. The interference field formed by multiple deflected beams has a gradient characteristic and the period of the interference fringe is the order of wavelength, therefore, the interference field can be used for micro-nano particle trapping. The gradient and scattering forces of the interference field can be calculated by integrating the Maxwell stress tensor (MST) along any closed contour that envelops the particle. For the MST method, the optical force can be written as [32]:
The integration is performed over a closed surface S surrounding the nanoparticle, n is the unit vector outwardly normal to S, T is the Maxwell stress tensor and is given by
Figure 10 depicts the transverse (Fx, z=45 μm) and longitudinal (Fz, x=0 μm) trapping forces of the twin-core symmetrical deflected beam with the deflection angle of -30°on a nanosphere with the radius r of 50 nm (n=1.59). Due to the existence of the interference fringe, there are multiple trapping positions along the transverse orientation. In addition, Fx equals zero at x=0 and is odd symmetrical about the coordinate origin because of the symmetrical distribution of the field. Compared to the other trapping positions, the location x=0 is the most stable one because it has the largest potential well as the largest areas under transverse trapping force curve, the nanosphere is attracted towards the stable equilibrium position. The closer to the equilibrium position, the weaker the attraction force is. With increasing the distance from the fiber end, the longitudinal gradient force varies from positive to negative one. The sphere can be stably trapped at null force positions with a negative slope [33], the trapping distance is nearly 45 μm away from the fiber end and is relatively long. The difference from traditional optical tweezers is that the proposed multiple deflected interference beams can achieve long-distance and multi-position trapping.
Fig. 10. (a) Transverse and (b) longitudinal trapping forces on a nanosphere for twin-core based symmetrical deflected beam device in air.
In order to simulate a more realistic environment, water (n=1.33) is used as the surrounding medium. The results show that twin-core based symmetrical deflected beam system with the grating period of 648 nm cannot trap a particle (n=1.59) with the radius of 50 nm since the scattering force in water is greater than the gradient force. By reducing the period of the grating, the nanoparticle can be trapped. The transverse (Fx, z=20 μm) and longitudinal (Fz, x=0 μm) trapping forces for the grating period of 400 nm are shown in Fig. 11. The longitudinal trapping force is smaller than that in air. The closer the refractive index of the sphere is to that of the environment, the smaller the optical force maximum is.
Fig. 11. (a) Transverse and (b) longitudinal trapping forces on a nanosphere for twin-core based symmetrical deflected beam device in water.
5. Conclusion
In conclusion, we have proposed an all-fiber beam shaping devices based on the slit-grating structures in the gold film on the fiber core end face, showing non-contact long-distance nanosphere capture. The single-, twin- and four-core all fiber-based deflected beams have been numerically demonstrated and complex spatial field distributions are analyzed. By changing the initial phase differences, the relative power and the polarization direction between the input beams, adjustable light field based on multi-core optical fiber can be realized. Moreover, the interference field formed by multiple deflected beams can be used to trap nanosphere at dozens of microns far from the fiber end face with multi-trapping positions. Miniaturized scanning and manipulation optical tweezers can be exploited by the controllable light field based on multi-core optical fiber in the future work.
Funding
National Natural Science Foundation of China (61675054, 91750107, U1931121); Natural Science Foundation of Heilongjiang Province (ZD2018015, ZD2020F002); 111 project to the Harbin Engineering University (B13015); Fundamental Research Funds for the Central Universities (3072021CFT2501).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data availability. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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